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University of Southern California Dissertations and Theses
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Pandemic prediction and control with integrated dynamic modeling of disease transmission and healthcare resource optimization
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Pandemic prediction and control with integrated dynamic modeling of disease transmission and healthcare resource optimization
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Content
Pandemic Prediction and Control
with Integrated Dynamic Modeling of Disease Transmission
and Healthcare Resource Optimization
by
Mingdong Lyu
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the Requirements for the Degree
DOCTOR OF PHILOSOPHY
(INDUSTRIAL AND SYSTEMS ENGINEERING)
May 2023
Copyright 2023 Mingdong Lyu
ii
Table of Contents
Table of Contents .......................................................................................................................................... ii
List of Tables ............................................................................................................................................... iv
List of Figures ............................................................................................................................................... v
Chapter 1. Introduction ................................................................................................................................. 1
1.1 Significance and Importance of Pandemic Research .......................................................................... 1
1.2 Objectives of Proposed Thesis Research ............................................................................................ 2
Chapter 2. Background and Prior Research .................................................................................................. 6
2.1 Transmission Modeling for Infectious Diseases ................................................................................. 6
2.1.1 Models of Community Spread of Disease ................................................................................... 6
2.1.2 Spatial Interaction Modeling in Epidemic Modeling ................................................................. 15
2.2 Vaccine Management and Prioritization during Pandemics ............................................................. 24
2.3 Uncertainties Analysis in Epidemic Modeling.................................................................................. 29
2.4 Summary of Literature ...................................................................................................................... 33
Chapter 3. Research Design and Methods .................................................................................................. 34
3.1 Application of Model to Thesis Aims ............................................................................................... 34
3.2 COVID-19 Data Sources .................................................................................................................. 36
3.2.1 Disease Tracking (Cases, Hospitalization, Fatalities, Recoveries) ............................................ 37
3.2.2 Travel and Mobility ................................................................................................................... 37
3.2.3 Hospital Utilization and Resource Availability ......................................................................... 38
3.3.4 Policy Enactment ....................................................................................................................... 40
Chapter 4 Dynamic Modeling with Time-Varying Transmission and Fatality Rates ................................. 44
4.1 The Proposed Time-Varying Model ................................................................................................. 45
4.2 Parameter Estimation and Model Fitting .......................................................................................... 49
4.3 Model Accuracy ................................................................................................................................ 52
4.4 Effective Reproduction Number Calculation and Trends ................................................................. 55
4.5 Multi-Phase Model ............................................................................................................................ 59
4.6 Sensitivity Analysis for Basic Time-varying Model ......................................................................... 61
4.7 Conclusion and Discussion ............................................................................................................... 70
Chapter 5 Integration of Dynamic Modeling with Spatial Interaction and Effect Analysis ....................... 72
5.1 Introduction ....................................................................................................................................... 72
5.2 Multi-Regional Dynamic Modeling with Spatial Interaction ........................................................... 73
5.2.1 Transportation Data Collection and Its Challenge ..................................................................... 73
5.2.2 Gravity Modeling of the State-Level Transportation ................................................................. 76
5.2.3 Dynamic Modeling with Multi-Regional Spatial Interaction .................................................... 79
iii
5.2.4 Model Accuracy ......................................................................................................................... 83
5.3 Effect Evaluation of Transportation on the Multi-Regional Transmission ....................................... 85
5.3.1 Transmission Export Index ........................................................................................................ 85
5.3.2 Causal Analysis for Increment of Transmission Export Index .................................................. 86
5.3.3 Causal Analysis for Decrement of Transmission Export Index ................................................. 95
5.4 Conclusion and Discussion ............................................................................................................... 99
Chapter 6 Integration of Dynamic Modeling with Vaccination Reallocation .......................................... 101
6.1 Introduction ..................................................................................................................................... 101
6.2 Age-Structured Dynamic Modeling with Vaccination.................................................................... 103
6.2.1 Age-Structured Transmission Data and Vaccine Data ............................................................ 103
6.2.2 Model Structure and Parameter Estimation ............................................................................. 105
6.2.3 Fitting Results of Age-Structured Dynamic Modeling with Vaccination ................................ 107
6.3 Vaccine Allocation Optimization with Dynamic Transmission Pattern ......................................... 110
6.4 Vaccine Allocation Policy under Different Scenarios .................................................................... 114
6.4.1 Healthcare Outcomes without Vaccination.............................................................................. 116
6.4.2 Vaccine Allocation Policy with Original Vaccine Availability ............................................... 117
6.4.3 Vaccine Allocation Policy with 10 Times of Original Vaccine Availability ........................... 122
6.5 Conclusion ...................................................................................................................................... 125
Chapter 7 Conclusion & Discussion ......................................................................................................... 128
7.1 Conclusion and Discussion ............................................................................................................. 128
7.1.1 Dynamic Modeling with Time-Varying Transmission and Fatality Rates .............................. 128
7.1.2 Integration of Dynamic Modeling with Spatial Interaction and Effect Analysis ..................... 129
7.1.3 Integration of Dynamic Modeling with Vaccination Reallocation .......................................... 130
7.1.4 Overall Summary ..................................................................................................................... 132
7.2 Future Study .................................................................................................................................... 133
References ................................................................................................................................................. 136
iv
List of Tables
Table 1: Data sources for COVID-19 disease tracking
Table 2: Public travel and mobility data sources during the COVID-19 pandemic
Table 3: Hospital utilization and resource availability
Table 4: Intervention measures taxonomy
Table 5: Rank of importance of transmission parameter on the cases
Table 6: Rank of importance of transmission parameter on the deaths
Table 7: Major state actions in response to transmission export index increases
Table 8: Major political events in response to transmission export index increases
Table 9: Major festivals/entertainment events in response to transmission export index
increases
Table 10: Major state actions in response to transmission export index decreases
37
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41
63
65
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92
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97
v
List of Figures
Figure 1: Research design for the pandemic prediction and control
Figure 2: Fitting accuracy of the cases and fatality across all states
Figure 3: Fitting results of case number for New York, California, Florida, and Hawaii
Figure 4: Fitting results of death number for New York, California, Florida, and Hawaii
Figure 5: Fitted Rep(t) at the start of the epidemic across all states
Figure 6: Fitted Rep(t) on July 28th
Figure 7: History of effective reproduction number for New York, California, Florida and
Hawaii
Figure 8: History of death rate for New York, California, Florida and Hawaii
Figure 9: Fitting results for the two phases Hawaii
Figure 10: Historical results of the effective reproduction number and death rate
Figure 11: Significance of transmission parameter on the cases
Figure 12: Significance of transmission parameter on the deaths
Figure 13: Simulation results of the 5th to 95th percentile range of cases
Figure 14: Simulation results of the 5th to 95th percentile range of deaths
Figure 15: Ten regions defined by the Federal Emergency Management Agency
Figure 16: Average RRMSE for cases and deaths over 7 months across 50 states
Figure 17:Fitting results of case number for Georgia, New Jersey, Florida, and Maryland
Figure 18:Fitting results of death number for Georgia, New Jersey, Florida, and Maryland
Figure 19: Heatmap of infectious export index for all 50 states in the US from 03/15/2020
to 04/15/2020
36
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vi
Figure 20: Model fitting accuracy across age groups for covid-19 cases in all 50 states
Figure 21: Model fitting accuracy across age groups for covid-19 deaths in all 50 states
Figure 22: Training process of case-prioritized vaccine optimization
Figure 23: Vaccine allocation comparison for case-prioritized vaccine optimization
Figure 24: Training process of death-prioritized vaccine optimization
Figure 25: Vaccine allocation comparison for death-prioritized vaccine optimization
Figure 26: Redistribution of original amount of vaccine among 50 states for case-
prioritized scenario
Figure 27: Redistribution of original amount of vaccine among 50 states for death-
prioritized scenario
Figure 28: Training process of case-prioritized vaccine optimization
Figure 29: Vaccine allocation comparison for case-prioritized vaccine optimization
Figure 30: Training process of death-prioritized vaccine optimization
Figure 31: Vaccine allocation comparison for death-prioritized vaccine optimization
108
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1
Chapter 1. Introduction
1.1 Significance and Importance of Pandemic Research
Pandemics are large-scale outbreaks of infectious diseases that can significantly increase morbidity
and mortality over a wide geographic area, leading to substantial economic, social, and political
disruption. When a new virus emerges, people typically have little or no immunity, allowing
diseases like influenza pandemics to impact a considerable proportion of the global population and
place immense strain on healthcare systems. Regardless of their severity, pandemics affect large
swaths of the population, necessitating a multisectoral response that can last several months or
even years. Besides influenza, novel viral diseases such as COVID-19 have the potential to impact
all segments of the population, potentially resulting in millions of deaths, causing a global
economic recession, and exacerbating political stress. These pandemics are particularly harmful to
vulnerable groups, including people living in poverty, older individuals, and persons with
disabilities.
As global travel and integration have increased, so has the risk of small local epidemics
transforming into global pandemics. In the 21st century, prior to the outbreak of COVID-19, the
world experienced five major epidemics:1) In 2002, Severe Acute Respiratory Syndrome (SARS)
emerged in Guangdong, China, killed 774 people. 2) In 2003, Avian flu broke out in Southeast
Asia, causing more than 400 deaths. 3) In 2009, 18,500 people died of “Swine flu” or H1N1, which
originated in Mexico and the U.S.. 4) In 2012, Middle East Respiratory Syndrome (MERS)
emerged in Saudi Arabia, resulting in nearly 1,000 deaths. 5) In 2013, the Ebola Virus Disease
emerged in West Africa and lasted more than two years, killing more than 11,300 people. A
continuous process of surveillance, reporting, intervention, and resource allocation are critical to
2
effective response to such contagions. Countries should therefore prepare for pandemics
considering their recurring nature.
Generally, individuals will experience five periods for a viral disease, including incubation,
prodromal, illness, decline, and convalescence. Pandemic controls are designed to target these
periods, with the aim of minimizing the impact of the disease on individuals and communities.
Comprehensive surveillance, effective intervention, and optimal healthcare resource allocation can
help control the spread of diseases and mitigate the economic and health consequences to human
life. These measures may include contact tracing, quarantine, public education campaigns, and the
development and distribution of vaccines and treatments. However, in spite of the efforts devoted
to pandemic control from all over the world, there’s still a knowledge gap when a new infectious
disease is transmitted among the public, due to incomplete understanding of transmission methods,
public health inventions, severity and vaccination. Pandemic research could help to monitor a new
outbreak, evaluate the likelihood of transmission, and inform policy intervention and resource
allocation, which could avoid unnecessary casualties and reduce economic hardship.
1.2 Objectives of Proposed Thesis Research
To mitigate the impact of future pandemics on social and economic life, our research will focus
on enhancing surveillance, developing targeted interventions, and optimizing vaccine distribution.
In the initial stages of a novel infectious disease, the lack of biological and medical knowledge
makes it difficult to estimate the extent and rapidity of its spread. A primary job is ascertaining the
case and fatality patterns of the disease based on which we could evaluate the transmissibility and
severity of the diseases.
Moreover, the spread of an epidemic hinges on the probability of infection and the nature of
individual interactions. Mobility networks are instrumental in shaping the temporal and spatial
3
dynamics of disease transmission within populations. The rising global mobility of humans and
increased trade volumes facilitate the introduction of infectious diseases to new regions.
Consequently, our research will employ mathematical modeling that incorporates time-varying
transmission and fatality rates, as well as spatial interactions, to create a comprehensive
representation of transmission dynamics. This modeling approach will generate quantitative
predictions to inform health policies, enabling evaluation of epidemiological outcomes and the
effectiveness of intervention strategies.
Uncertainties during the early stages of infectious diseases, such as insufficient testing and delayed
reporting, can hinder the understanding of a disease's severity and potentially postpone effective
measures to curb large-scale infections. Therefore, another crucial aspect is to evaluate the
likelihood of a disease evolving from a local outbreak to a global pandemic. In addition, it is
essential to account for uncertainties stemming from model parameter estimation and variations in
global events and intervention policies when analyzing disease dynamics. To address these issues,
our research will apply probabilistic uncertainty analysis to a dynamic model of viral disease
transmission, assessing the uncertainty in the healthcare outcomes.
Global pandemics can place immense strain on healthcare systems, necessitating efficient
allocation of limited resources. To enable an effective medical response, healthcare systems must
be well-prepared and organized, considering their capabilities and capacities. Our research will
investigate the optimal distribution of health resources, leveraging reliable historical data from
uncertainty analyses. Ultimately, our research aims to develop a comprehensive surveillance,
evaluation, and guidance platform to better prepare for and respond to future pandemics.
To summarize, the specific aims of the research will be:
4
Aim 1: To comprehensively understand the transmission dynamics of a new infectious disease, we
aim to accurately predict time-varying transmission and fatality rates with a variation of the
Susceptible-Exposed-Infectious-Recovered-Death (SEIRD) model. We will propose a continuous
representation of the transmission rate and case fatality with a small number of parameters. The
model will be validated by checking the goodness of fit to historical data.
Aim 2: To enhance the credibility of the model from Aim 1, we aim to investigate and quantify
the effect of transportation and vaccine allocation. To accomplish this aim, we will incorporate
mobility data and vaccination data into the dynamic model from Aim 1 through introduction of
variables estimating interregional movement and vaccination rate. Based on this comprehensive
model, we will analyze the relationship of transmission dynamics with respect to the mobility
pattern and vaccination progress.
Aim 3: To find the optimal vaccine allocation strategy across different phases of the epidemic, we
aim to dynamically optimize vaccine allocation to maximize the health benefit with equity. To
achieve this goal, we will define a multi-objective function considering both the total infections
and deaths over the whole period of epidemic with the constraints of limited vaccine capacity.
Under reasonable assumptions, the optimal allocation strategy will be solved adaptively.
Aim 4: To inform the choice of intervention policies for decision makers, we aim to evaluate the
healthcare outcomes under different policy scenarios. Based on the model from Aim 3, we will
evaluate the effect of interventions on the spread of disease with uncertainty. According to the
evaluation results, we will assess effectiveness and uncertainty of intervention methods at different
phases of an epidemic.
In summary, by accomplishing the four objectives outlined above, we will establish a
comprehensive modeling framework for analyzing transmission dynamics and optimizing vaccine
5
allocation. This framework, combined with uncertainty analysis, will enable us to evaluate the
scale of an epidemic and the effectiveness of intervention strategies. By accurately describing
historical trends under various scenarios, our approach to vaccine optimization will offer decision-
makers the most effective vaccine allocation plan in the face of uncertainty. Ultimately, this
research will provide a systematic methodology for monitoring, predicting, and responding to
future epidemics, enhancing preparedness, and promoting effective public health responses.
6
Chapter 2. Background and Prior Research
2.1 Transmission Modeling for Infectious Diseases
2.1.1 Models of Community Spread of Disease
Modeling, analyzing, and predicting the impact of infectious diseases is critical to preventing,
controlling, and managing the spread of infectious agents. It is a common perception that infectious
diseases can be transmitted from person to person through a pattern of space and time. In addition
to vaccination and medication, a great amount of effort has been put into the modeling and analysis
of the temporal and spatial dynamics of disease spreading in order to better understand the complex
patterns of infectious diseases and devise effective intervention strategies.
The history of modern epidemiological analysis and modeling can be traced back to the late
nineteenth and early twentieth centuries, when early mathematical models of disease transmission
were developed. One of the earliest examples is John Snow's work in 1849, when he plotted the
cholera epidemic cases on a map and concluded that contaminated water was the predominant
contributor to cholera transmission in London [1]. In a similar vein, Arthur Ransome developed a
discrete-time epidemic model for cholera transmission in 1906 [2]. These early mathematical
models of disease transmission have been helpful in gaining insights into the transmission
dynamics of infectious diseases and the potential role of different intervention policies. Over the
past few decades, various models have been developed and applied to modeling the transmission
dynamics of infectious diseases, including metapopulation models, statistical models, and agent-
based modeling, all of which have contributed to a more comprehensive understanding of the
spread of diseases.
7
Compartmental models are widely used in epidemiology to simulate the spread of infectious
diseases and understand their dynamics. These models divide the population into distinct
compartments based on their health status, such as susceptible, infected, and recovered individuals,
and describe the flow of individuals between these compartments using differential equations. In
this section, we will discuss common compartmental models, their advantages and disadvantages,
and their applications in various epidemiological contexts.
One of the most successful models is Susceptible-Infectious-Recovered (SIR) Model, described
by Kermack and McKendrick in 1927 [3]. Originally, This model was proposed to explain the
rapid rise and fall in the number of infected patients of plague [4]. This type of model is based on
our intuitive understanding of the epidemic transmission and compromises three categories of
individuals: those who are infectious (I) mix among those who are susceptible to disease (S) and
transmit the disease to the susceptible population in a stochastic process, and those who are
infected will recover from the infection and acquire immunity to the disease. People may progress
between compartments, the dynamic of which is illustrated by a series of ordinary differential
equations in Equation (1):
𝑑𝑆
𝑑𝑡
= β
𝑆𝐼
𝑁
𝑑𝐼
𝑑𝑡
= β
𝑆𝐼
𝑁 − γ𝐼
𝑑𝑅
𝑑𝑡
= γ𝐼
(1)
where S, I and R denote the susceptible, infectious and recovered population respectively; N is the
total population; β is the transmission rate, indicating the average effective infection transmitted
by one infectious person; γ is the recovery rate. By adjusting the β and γ, the differential equations
1 could capture the initial rapid increment of infectious population (I) of a given epidemic and
decrement of the number of individuals after the epidemic peak. The compartmental model could
be used with a stochastic framework to better describe the reality of the transmission. The SIR
8
model is particularly useful for studying diseases that confer long-lasting immunity, such as
measles or chickenpox. It has been widely applied to understand the basic reproductive number
(𝑅 0
) of various diseases, which is the average number of secondary infections caused by a single
infectious individual in a fully susceptible population. This parameter is essential for
understanding the potential for disease spread and evaluating the effectiveness of control measures.
The Susceptible-Exposed-Infectious-Recovered (SEIR) Model is an extension of the SIR model
that incorporates an additional compartment for exposed individuals (E) [5]–[8]. Exposed
individuals are those who have been infected but are not yet infectious themselves. The inclusion
of this compartment accounts for the incubation period of the disease, allowing for a more accurate
representation of diseases with a delay between infection and infectiousness, such as COVID-19,
Ebola, or SARS.
The SEIR model's additional compartment enables it to more accurately represent the dynamics of
diseases with latent periods, providing a more detailed understanding of the transmission process.
It has been used in various epidemiological studies to investigate the spread of infectious diseases
and evaluate the impact of control measures, such as quarantine and isolation [9].
The Susceptible-Infectious-Recovered-Death (SIRD) Model expands upon the SIR model by
including a death (D) compartment to account for disease-related fatalities. This model is
particularly useful for studying diseases with significant mortality rates, as it allows for the
estimation of both infection and death rates within a population.
The SIRD model has been applied to various infectious diseases with high mortality rates, such as
H1N1 influenza, to understand the dynamics of disease transmission and death rates [10]. By
incorporating the death compartment, the model can provide insights into the severity of an
9
outbreak and help inform public health decision-making regarding resource allocation, healthcare
capacity, and intervention strategies.
The Susceptible-Exposed-Infectious-Recovered-Death (SEIRD) Model combines the features of
the SEIR and SIRD models, adding both an exposed (E) and a death (D) compartment. This model
provides a more comprehensive representation of disease dynamics, accounting for the incubation
period and disease-related fatalities. The SEIRD model has been used to study various infectious
diseases, including COVID-19, where both latency and mortality play significant roles in disease
dynamics. The model allows researchers to better understand the factors influencing the spread
and severity of a disease, enabling the development of more targeted and effective intervention
strategies.
In addition to the basic compartmental models discussed above, several variations and extensions
have been developed to address specific epidemiological questions or account for additional
factors that influence disease transmission. Some of these variations include:
Age-structured compartmental models divide the population into age groups, reflecting the fact
that disease transmission and severity can vary significantly by age. For example, younger
individuals may have a higher risk of infection due to increased social contacts, while older
individuals may experience more severe outcomes. Age-structured models have been used to study
diseases such as influenza, pertussis, and COVID-19, helping to inform age-specific vaccination
strategies and public health interventions [11]–[13].
Multi-strain compartmental models are used to study diseases caused by multiple strains or
subtypes of a pathogen, such as influenza or dengue fever [14], [15]. These models can help
researchers understand the dynamics of strain competition and the potential for strain replacement
following vaccination or other interventions. Multi-strain models can also provide insights into the
10
evolution of pathogens and the emergence of new strains, informing the development of vaccines
and other control measures.
Network-based compartmental models incorporate social network structures to account for
heterogeneous mixing patterns within a population. These models can more accurately represent
the spread of diseases in populations with complex social structures, such as schools, workplaces,
or communities. Network-based models have been used to study the spread of sexually transmitted
infections, respiratory infections, and other diseases where contact patterns play a critical role in
transmission dynamics [16], [17].
Spatially explicit compartmental models incorporate geographic information to account for the
spatial distribution of individuals and the impact of local population density, environmental factors,
and human mobility on disease transmission. These models can help researchers understand the
role of spatial factors in disease spread and inform targeted interventions, such as travel restrictions,
quarantine measures, or targeted vaccination campaigns. Spatially explicit models have been used
to study a wide range of infectious diseases, including malaria, cholera, and COVID-19 [18]–[20].
Strengths of compartmental models are numerous, with one of the most significant advantages
being their simplicity and flexibility. This allows researchers to easily adapt the models to a wide
range of diseases, populations, and research questions by modifying the number of compartments,
parameters, or assumptions. Additionally, compartmental models are mathematically tractable and
computationally efficient, making them particularly suitable for large-scale simulations, real-time
forecasting, and situations with limited computational resources. Furthermore, these models
provide a solid foundation for more complex models, enabling researchers to systematically build
upon and refine the basic compartmental structure to better represent specific aspects of a disease
or population of interest. Importantly, the parameters and variables in compartmental models often
11
have clear biological or epidemiological interpretations, making the models easy to understand
and communicate to a broad audience, including policymakers, public health officials, and other
stakeholders who may not have an extensive background in mathematical modeling [21].
In spite of the strengths of the compartmental model, the limitations of the compartmental model
have gained growing attention. One primary drawback is the homogeneous mixing assumption,
which can be unrealistic in real-world populations with complex contact patterns due to factors
such as age, social networks, or spatial distribution. This assumption can lead to inaccurate
predictions of disease spread, particularly for diseases where contact patterns play a critical role in
transmission dynamics. Another limitation is the lack of spatial consideration in most basic
compartmental models, which can restrict their ability to accurately represent the spread of
diseases with strong spatial components, such as vector-borne diseases or those influenced by
environmental factors. While spatially explicit compartmental models have been developed to
address this issue, they can be more complex and computationally demanding. Compartmental
models can also oversimplify disease dynamics by neglecting important factors such as individual
heterogeneity, which can result in an inaccurate understanding of the true dynamics of disease
transmission. Lastly, parameter uncertainty is a significant concern in compartmental models,
particularly for emerging diseases with limited data available. This uncertainty can limit the
reliability of model predictions and may impact their usefulness in informing public health
decision-making. Researchers must carefully assess parameter uncertainty and its potential impact
on model results and should consider using techniques such as sensitivity analysis or Bayesian
approaches to account for this uncertainty.
Besides the compartmental modeling, various statistical analysis methods have also been applied
to the modeling of infectious diseases. Usually, statistical modeling formalizes relationships
12
among variables that may influence the spread of disease, describes how the variables are related
to each other, and test the hypothesis or statements about the disease transmission. Statistical
models could involve the statistical analysis and modeling of the disease observation with a
broader variety of variables including the special factors, economy factors, and environmental
factors.
Various complicated statistical models have been constructed. For example, time series analysis is
a statistical method that focuses on analyzing data points collected over time to predict future
trends. This approach is useful for understanding temporal patterns in infectious diseases, such as
seasonality or long-term trends [22]. However, the main drawback of time series analysis
compared to compartmental modeling is that it does not consider the underlying disease
transmission mechanisms, making it challenging to identify specific factors driving the spread of
the disease or evaluate the impact of control measures.
Regression models, including linear regression and logistic regression, are used to establish
relationships between disease spread and various factors, such as demographic characteristics or
environmental factors. They can help identify risk factors and guide public health interventions.
For example, generalized linear mixed model (GLMM), an extension of linear mixed models,
could incorporate both individual level and integrated level data by allowing the response variables
from different distributions, such as binary responses [23].However, compared to compartmental
modeling, regression models are limited by their reliance on a fixed functional form, which may
not accurately capture the complex dynamics of disease transmission. Additionally, they may not
account for spatial dependencies, limiting their applicability in spatially heterogeneous settings.
Bayesian hierarchical models offer a flexible framework for incorporating various sources of
information and uncertainty in infectious disease modeling [24]. These models can account for
13
both spatial and temporal dependencies and integrate multiple data sources. However, the main
drawback of Bayesian hierarchical models compared to compartmental modeling is their
computational complexity. As these models often involve a high number of parameters and require
Markov chain Monte Carlo (MCMC) methods for estimation, they can be computationally
intensive and time-consuming, especially for large-scale applications [25].
Generally, statistical models are flexible regarding the format of the input data and the parameters
selection, which makes them more powerful for analyzing disease on a detailed level. However,
statistical model is a data-based method, requiring a large variety and quantity of high-quality data.
Missing data, underreporting, and uncertainty are common for the disease data, especially at the
early stage of the transmission. In practice, it is challenging to build a complicated statistical model
incorporating different aspects of transmission without the right data. Meanwhile, some of the
advanced statistical methods, like Markov Chain Monte Carlo, is usually very computationally
expensive and time consuming while the model considers many factors.
With the development of computational technology and accessibility to big data, computational
simulation approaches have been used increasingly in the epidemic analysis and forecasting. One
of the popular beliefs of the disease transmission is that population is heterogeneous and each
individual exhibit unique spread pattern. Agent-based modeling (ABM) is a powerful
computational tool for simulating complex systems, including infectious diseases. This approach
allows for the representation of individual agents, such as people or animals, and their interactions
within a population [26]. Different from the metapopulation model, agent-based models assume:
(a) individuals are characterized uniquely by their age, race, income, and so on;(b) the number of
interactions varies by person; (c) individuals are spatially distributed and mobile.[27] Based on the
14
assumptions mentioned above, agent-based models represent each individual uniquely by their
characteristics, behaviors and interaction with others.
Object-oriented modeling is one of the most popular and widely used agent-based modeling
approaches.[28] This approach constructs the agents using a collection of features and organize
the agents in a hierarchical way. Bian used the object-oriented GIS framework to model the
individual fish movement and growth in a heterogeneous aquatic environment. As shown in object-
oriented modeling, agent-based modeling could provide a clear framework to incorporated
individual characteristics, individual interaction patterns, which ultimately in total determine the
transmission results [28]. Agent-based model, combined with social contact networks, have been
widely used to simulate influenza-type diseases like West Nile virus in the United States and
Canada with the agent representing mosquitoes, avian hosts, mammalian host, and human.[29]
These models simulate the interaction patterns between the agents by considering habitat location
and weather conditions.
One of the primary strengths of agent-based modeling is its ability to represent heterogeneity
within a population, including variations in demographics, behavior, and disease susceptibility.
This flexibility can lead to more accurate predictions and better understanding of the dynamics of
infectious diseases. However, a drawback of this approach compared to compartmental modeling
is that it can be computationally expensive, particularly for large populations or when simulating
detailed individual-level interactions.
Meanwhile, agent-based modeling can explicitly model individual behaviors and their impact on
disease transmission, such as social distancing or vaccination uptake. This capability allows
researchers to study the effects of behavior change interventions on disease spread. However,
compared to compartmental modeling, a limitation of agent-based modeling is that it often relies
15
on assumptions about individual behaviors and their interactions, which may not always be
accurate or generalizable to other populations.
Agent-based models are well-suited for capturing spatial dynamics in infectious disease
transmission, as they can represent individuals' locations and the impact of spatial structure on
disease spread. This ability can lead to more accurate predictions of disease spread in spatially
heterogeneous environments. However, compared to compartmental modeling, a drawback of
agent-based modeling is that they may require detailed spatial data, which can be difficult to obtain
or may be subject to privacy concerns.
Agent-based methods in infectious disease modeling offer several advantages over compartmental
modeling, such as capturing heterogeneity, modeling individual behaviors, and representing
spatial dynamics. However, these methods also have their drawbacks, such as computational
expense, reliance on assumptions about individual behaviors, and the need for detailed data. In the
cases when new diseases first appear, compartmental modeling, which simplifies the disease
transmission process by grouping individuals into compartments, may offer a more
computationally efficient and easily interpretable approach.
2.1.2 Spatial Interaction Modeling in Epidemic Modeling
Commuting patterns and travel behavior underlies the spread of infectious diseases among
locations. Spatial interaction models of human mobility have been used in the study of the
epidemic transmission dynamics. While the temporal dynamics of epidemics have always been a
primary focus of most models, over the years, efforts have been made to model spatial dynamic in
epidemic processes. The efforts could be divided into two parts: (1) modeling of spatial interaction
between different groups of research units, like livestock or human. (2) incorporating the spatial
interaction into the epidemic models. In this part of review, we will first review the most common
16
interaction models used in epidemic modeling and then review the main categories of spatial
oriented epidemic model.
2.1.2.1 Spatial Interaction models
For the consideration of the spatial dynamic of the epidemic, one of the main focuses is the
modeling of the spatial interaction among modeling units. Many spatial interaction models have
been proposed to uncover the relationships between spatiotemporal infectious disease patterns and
environmental characteristics. These models rely mostly on two frameworks, the gravity model
and radiation model. In the absence of easily available data on travel behavior, these models
intuitively describe the mobility in the population.
The gravity model is the most common spatial interaction model, based on the principle of gravity
as described in Issac Newton’s law of gravity. The model has two basic assumptions that the
interaction between two places is proportional to the product of population totals of the places and
inversely proportional to the square of distance. The gravity model is generalized as follows in
Equation (2) [30]:
𝐼 𝑖𝑗
= 𝑘 𝑝 𝑖 . 𝑝 𝑗 𝐷 𝑖𝑗
𝑏
(2)
where 𝑝 𝑖 is the population size in region 𝑖 , 𝐼 𝑖𝑗
is the estimate of the volume of spatial interaction
between place of origin 𝑖 and place of destination 𝑗 , 𝑘 is a constant scaling factor, 𝐷 𝑖𝑗
is the
distance between 𝑖 and 𝑗 , 𝑏 is the distance effect acting as an impedance on the interactions. Many
empirical studies have been conducted on spatial interaction models. For example, different
measures of distance (straight-line distance, Euclidean distance, city-block metric and travel time)
have been suggested. Meanwhile, different functions for distance effects and extra components
have been proposed to specify the attractiveness of the destination considering the benefits of
economies scale.
17
The advantages of using gravity approach for modeling transition process during the epidemic are
the ability to explain interregional trade patterns under the conditions of comparatively sparse data
and the validity of theoretical background of the model to human interactions in a large scale.
In spite of the advantages of the gravity model, the spatial interaction model is typically based on
aggregate, zonal data under the assumption of homogeneity. The model might fail to accordingly
represent description for regions with high heterogeneity and uncertainty[31]. Meanwhile, the
gravity model requires tedious tunning process based on the real observations. However, in reality,
mobility data are difficult to complete.
To overcome limitations of gravity models, Simini [32] proposed the radiation model, which
originally appeared in physics to study the process of energetic particles or wave travel through
vacuums. The radiation model describes the mobility patterns without any parameter estimation.
The traffic from site 𝑖 to 𝑗 , with populations 𝑃 𝑖 and 𝑃 𝑗 , is given by Equation (3).
𝑇 𝑖𝑗
= 𝑇 𝑖 𝑃 𝑖 𝑃 𝑗 (𝑃 𝑖 + 𝑠 𝑖𝑗
)(𝑃 𝑖 + 𝑠 𝑖𝑗
+ 𝑃 𝑗 )
(3)
Where 𝑇 𝑖 = ∑ 𝑇 𝑖𝑗 𝑗 ≠𝑖 is the outgoing traffic from the site 𝑖 . Compared to the gravity model, the
radiation model has broader flexibility without the need of parameter estimation, and better
prediction for long distance travel. However, as formulated in equation (3), the radiation model
only considers the large-scale parameters and does not incorporate the parameters to calibrate for
smaller scales. Thus, the radiation model has relatively poor predictability on short-distance travel
[33]. Moreover, the radiation model requires additional information on 𝑇 𝑖 , in contrast to the gravity
model.
The radiation and gravity models have been compared to each other in terms of the predictability
of mobility patterns observed in various empirical data sets.[33]
18
In terms of the application of the spatial interaction model in the epidemic research. Li et al. [34]
validated the performance of the gravity model in predicting the global spread of H1N1 by
formulating the global transmission between major cities and Mexico. The variables, such as
population sizes, per capita gross domestic production (GDP), and the distance between the
countries and Mexico were incorporated in the gravity model, which is validated by the estimation
of global transmission trend. Kraemer et al. [35] described a flexible transmission model to test
the utility of generalized human movement models in estimating Ebola virus disease (EVD) cases
and spatial spread over the course of the outbreak. Gravity model, radiation model and adjacent
network model were applied to show the generalized movement models have the ability to improve
the modeling of spread of EVD epidemic.
2.1.2.2 Spatial oriented models
Besides modeling of spatial interaction, another concern is the design of the spatially oriented
epidemic models incorporating different kinds of spatial interactions. The spatially oriented
models could be categorized according to the scale of the modeling unit and the mobility of the
modeling units. According to the scale of the modeling unit, the models could be divided into three
types: (1) Population-based models, (2) Sub-population models, and (3) Individual-based models.
In each category, models could be further divided according to the level of mobility.
1) Population-based models
Population-based models divide the whole population into different segments with various
assumptions of the characteristics of each segment. The simplest version of the family of
population-based models is the SIR model, which consists of three exclusive segments:
Susceptible(S), Infectious(I), and Recovered(R). However, the classic population-based models
do not consider spatial dynamics at all. In the 1980s, geographers proposed a spatial framework
19
for epidemiological models that explicitly considers the spatial dispersion of infectious
diseases[36]. A simple form of these spatial models is a three-ring wave model, where the first
infection case occurs at the center of a space and spread towards all directions like a wave. The
infection segment stays at the center of the wave, the susceptible segment surrounds the infection
segment, and the recovery segment is at the outermost ring surrounding the infection segment[36],
[37]. The location of the three-ring changes dynamically as the infectious wave spread through the
space. The three-ring wave model differs from the classic model by how it projects the three
populations into space, which also marks the beginning of spatially explicit epidemiological
models. However, like the other population-based models, the inherent homogeneity assumptions
impede the models’ ability to describe the heterogeneity in disease transmission[38]–[40]. One
application for population-based wave models is studying pandemic waves in a large space level,
such as the 1918-1920 Spanish flu that spread globally [41].
2) Sub-population models
Instead of assuming the whole population as an identical model unit, sub-population models divide
the population into a substantial number of subpopulations. The increased number of model units
(i.e., the subpopulations) could increase the heterogeneity considered in the model in order to
produce more realistic results than those produced by the classic population-based models. One
branch of the sub-population models divides the population according to the spatial distribution of
each group, called spatially structured models[37], [42], [43]. Spatially structured models usually
divide the space into regular grid cells. Each cell, containing one subpopulation, inherits the same
identical and homogeneously mixed assumption as the classic population-based model. With the
increased number of spatially distributed units, the interactions between the subpopulations are
also added to the model enhancing the model ability to account for the heterogeneity in certain
20
levels. Due to the homogeneity assumption within each cell, the sub-population models suit more
for the modeling of diseases between high-density, immobile communities, such as the diseases
transmitted in livestock[44], [45]. For example, Doran and Laffan implemented the SIR model in
the spatially structured sub-population framework to evaluate the season impact on the
transmission of foot and mouth disease between different groups of livestock [44].
To address the limitations of the classic models, a variety of revisions have been used to improve
the sub-population models. One revision is adding heterogeneity within a subpopulation according
to the characteristics of subpopulations (such as population size, age structure, and education level).
Different from the individual-based models, these characteristics are only represented in a
statistical way (e.g., statistical distribution) across each subpopulation. Another revision is the finer
representation of the transmission between subpopulations by statistical representation of the
interaction between subpopulations or explicitly tracking of the exchanging individuals [46].
Subpopulations could then become mobile and move to different locations. One or multiple
infected subpopulations could split and then merge with other subpopulations, where the location
of units are explicitly recorded. In this way, diseases are transmitted across space through time.
Specifically, the homogeneity assumption within each subpopulation still holds for the revised
models. Hence, these models are more effective when implemented in mobile and high-density
population, such as military or refugee camps, where the subpopulations will merge and split
dynamically in the transmission process. [46], [47].
3) Individual-based models
Individual-based models could be considered as a further extension of the sub-population models
by dividing the whole population of the individual level [41]. In addition to treating individuals as
the modeling unit, the interactions between the units are explicitly represented. Individual-based
21
models break the homogeneous assumption held by the population-based models or sub-
population models. Each individual is unique, interacts with a limited number of other individuals,
and collectively contributes to the dynamic pattern of transmission in a population level. For
example, when the individuals are immobile and only interact with the adjacent individuals,
Martins et al. proposed a cellular automata model to simulate the progress of the citrus variegated
chlorosis disease by adjusting the parameters controlling vectors motility, plant stress, and
initialization[48]. When the disease transmission depends heavily on the structure of social
networks and the spatial movement of the units is not explicit represented, Eames and Keeling
developed pair-wise network equations utilizing the essential characteristics of the mixing network
to estimate the effectiveness of various control strategies towards sexually transmitted diseases[49].
However, individual-based models require a great amount of detailed information about
individuals, which is hard to obtain at the early stage of the epidemic. To compensate for this
scarcity of data, efforts have been made to investigate individualized contact behavior from mobile
device data[50]. The development of the individual-based models relies on the build-up of a
secured, united, real-time information tracking system.
In the section, spatial oriented models are categorized according to the scale of the modeling units
and illustrated by their major implementation on the analysis of spatial dynamics in the epidemic
process. The model selection should depend on the characteristics of research target, availability
of the data and the scope of the analysis.
2.1.2.3 Applications of Spatial Interaction Models in Epidemiology
Spatial interaction models have demonstrated significant potential in various aspects of infectious
disease epidemiology, from disease spread modeling to healthcare resource allocation and
surveillance. This part will dive deeper into the applications of spatial interaction models in
22
epidemiology, examining their role in enhancing epidemic management and highlighting
opportunities for future research.
1) Disease Spread Modeling
Spatial interaction models simulate the spread of infectious diseases over space and time by
incorporating the movement of people, goods, and services [51]. These models are built on the
foundation of gravity and entropy maximization principles [52], which assume that the interaction
between two locations is proportional to the product of their masses (e.g., population size) and
inversely proportional to the distance between them. Recent developments in spatial interaction
modeling have incorporated additional variables, such as socio-economic factors, transportation
infrastructure, and regional characteristics [53].
Spatial interaction models were used to study the global spread of the 2009 H1N1 influenza
pandemic [54]. By incorporating air travel data and population distribution, the models
successfully predicted the initial wave of the pandemic and provided valuable insights into the
potential effectiveness of various intervention strategies, such as travel restrictions and vaccination
campaigns. During the Ebola outbreak in West Africa (2014-2016), spatial interaction models
were employed to analyze the spread of the virus and inform containment measures [55]. The
models incorporated data on population movement, healthcare infrastructure, and social factors,
enabling a better understanding of the epidemic's dynamics and informing the deployment of
resources to affected regions. The COVID-19 pandemic has underscored the importance of spatial
interaction models in predicting and managing the spread of infectious diseases [56]. These models
have been used to assess the role of international travel in the early stages of the pandemic and to
evaluate the potential impact of different containment measures, such as social distancing and
lockdowns, on disease transmission.
23
2) Healthcare Resource Allocation
Spatial interaction models can help identify areas with a high demand for healthcare services,
enabling more efficient allocation of resources during an outbreak [57]. By accounting for factors
such as population density, disease prevalence, and transportation networks, these models can
predict the spatial distribution of healthcare needs, informing decisions on resource allocation and
facility planning.
In malaria-endemic regions, spatial interaction models have been employed to optimize the
allocation of resources for malaria control interventions, such as insecticide-treated bed nets and
indoor residual spraying [58]. By incorporating data on population distribution, environmental
factors, and transportation networks, these models can help identify areas with the highest potential
impact of interventions, thereby maximizing their cost-effectiveness. During the COVID-19
pandemic, spatial interaction models were used to identify areas with high potential for hospital
overload, allowing policymakers to prioritize resources and implement containment measures [59].
These models incorporated data on population distribution, healthcare infrastructure, and mobility
patterns, enabling a more targeted response to the crisis.
3) Surveillance and Early Detection
Spatial interaction models can be used to optimize surveillance networks, ensuring that resources
are allocated efficiently to detect and monitor disease outbreaks [60]. By incorporating data on
population movement, transportation networks, and disease prevalence, these models can identify
high-risk areas and vulnerable populations, enabling the design of more effective surveillance
systems.
Spatial interaction models can help detect the early stages of an outbreak by modeling the
movement of people and the flow of information [61]. Early detection is critical for implementing
24
timely interventions and preventing the spread of infectious diseases. These models can also be
used to identify potential hotspots for targeted interventions, such as vaccination campaigns and
public health messaging.
2.1.2.4 Challenges for spatial interaction modeling in epidemiology
Spatial interaction models offer significant potential for epidemic forecasting and management.
However, addressing several challenges is essential to fully harness their potential. Firstly,
obtaining high-quality, real-time data on population mobility and transportation networks is
critical for accurate spatial interaction modeling [62]. As data availability continues to expand
through mobile devices and social media, future research should prioritize improving data quality
and incorporating these novel data sources into models.
Secondly, ensuring the accuracy of predictions and the reliability of policy recommendations in
epidemiological models requires thorough validation and calibration using historical data [63].
Future research should focus on devising robust validation and calibration techniques, as well as
incorporating uncertainty quantification into spatial interaction models.
In conclusion, the promotion of interdisciplinary cooperation among epidemiology, transportation,
and urban planning domains has the potential to significantly augment the advancement of
thorough spatial interaction models. Facilitating such partnerships may ultimately result in more
efficient tactics for managing epidemics [64].
2.2 Vaccine Management and Prioritization during Pandemics
In recent years, large outbreaks of new emerging epidemics, such as the SARS-CoV-2 virus in
2019, the Ebola virus in 2014, and the HINI virus in 2009, bring about deaths, health losses and
economic damage. To combat the potential crisis caused by emerging epidemics, public health
interventions have been limited to the non-pharmaceutical interventions at the early stage,
25
including travel restriction, contact tracing and lockdown [65], [66]. While critical to slowing
down the viral spread, non-pharmaceutical interventions have brought about occupation
misfortune and financial difficulties [67], [68]. Given the considerable political and financial
expenses related with non-pharmaceutical interventions, long-term solutions such as vaccines that
protect from viral infection are needed. Vaccines provide direct protective benefits to the public,
including protection from infection, reduced symptom development, and lower mortality rates.
Meanwhile, vaccination can also bring about indirect benefits by decreasing the risk of exposure,
even for the unvaccinated susceptible individuals.
The manufacturing of the vaccine, such as influenza vaccine, follows a tight schedule to produce
sufficient doses in the initial phases[69], [70]. Developing a healthy and reliable vaccine involves
many steps including initial experiments, three phases of clinical trials, US Food and Drug
Administration (FDA) authorization, manufacturing and distribution. And it usually takes between
6 and 36 months to get a healthy vaccine available to the public. However, once the vaccine is
authorized by the FDA, the first batch of vaccine is limited. The large number of susceptible
individuals result in overwhelming demand over supply. Therefore, it is critical to find out the best
decision-making method on vaccine distribution.
Two popular ways to prioritize vaccine distribution: (i) directly vaccinate the high-risk individuals
for severe outcomes, such as people over 65 years old, and (ii) indirectly protect them by
vaccinating individuals with the highest numbers of potentially disease-causing contacts. Most of
the model-based vaccination allocation research investigates trade-offs between these two
strategies. Considering the high contact rate related to healthy school children and their expected
remaining life, epidemiologists have suggested vaccinating school age children first to moderate
the spread of disease and thereby indirectly decrease mortality [71]–[73]. Monto et al. vaccinated
26
85% of younger students in Tecumseh, Michigan with inactivated flu vaccine in 1968 and
discovered two-third lower rate of illness than the neighbor town of Adrian during a rush of flu A
[72]. Reichart et al. argue that routine influenza vaccination of school-aged children in Japan from
1962 to 1994 has prevented 10,000–12,000 deaths annually from pneumonia and influenza [74].
Longini et al. utilized stochastic epidemic simulations to investigate the effectiveness of target
antiviral prophylaxis to contain influenza. They found that vaccinating 80% of all school-age
children is almost as effective as vaccinating 80% of the entire population [71]. In addition,
modeling studies have estimated that annual influenza epidemics could be contained if 50%–70%
of children were vaccinated [75], [76]. Moreover, similar modeling analysis has found that the
optimal balance between direct and indirect protection depends on the reproduction numbers of
the disease and vaccination efficiency. Bansal et al. have shown that direct protection strategies
outperform the other if the transmission level (reproduction number) is moderate, while the reverse
is true for highly transmissible diseases [77].
Considering the limited availability of vaccine stockpiles, several studies focus on the optimal
allocation policies to contain an outbreak in its early stages using the smallest amount of vaccine.
One general mathematical formulation adopted by researchers is formulated in Equation (4) [78]–
[81].
𝑚𝑖𝑛𝑚𝑖𝑧𝑒 ∑ 𝑛 𝑖 𝑖 𝑓 𝑖
𝑠 . 𝑡 . 𝑅 𝑓 ≤ 1
0 ≤ 𝑓 𝑖 ≤ 1
(4)
where 𝑛 𝑖 refers to the number of people in each subgroup, 𝑓 𝑖 denotes the vaccine coverage in
subgroup i. 𝑅 𝑓 denotes the effective reproduction number, the expected number of secondary
infections transmitted by an infectious individual. The epidemic will prevail if 𝑅 𝑓 > 1 [79]. Hence,
the first constraint in (4) forces 𝑅 𝑓 < 1, which is non-convex or non-concave Hill and Longini
27
characterized the threshold surface of critical vaccine allocations in a population with interacting
subgroups [78]. They proposed a solution approach based on Lagrangian multipliers applied with
Maple to minimize the amount of distributed vaccine. They pointed out that 𝑅 𝑓 is equal to the
spectral radius of the product matrix of vaccination fraction matrix and next generation matrix,
which has been widely adopted in later research. Duijzer et al. utilized the SIR compartmental
model considering the geographically separated subgroups and interactions to analyze the relation
between the vaccination fraction and herd effect [80]. They conclude that to successfully alleviate
a continuous episode with a restricted vaccine reserve, policymakers should focus on subgroups
where the disease has not made much progress yet. However, they solved the optimization model
with a nonlinear programming solver that does not guarantee an global optimal. Based on similar
assumptions, Enayati and Ozaltin further improved the solution quality by discretization and
reformulation of the constraints. They iteratively solve the upper and lower bound of the mixed-
integer program to find the approximated global optimum.
Besides the problem of containing the epidemic with the minimal vaccine, another branch of
research is seeking to find the optimal allocation policy to get the best healthcare outcomes given
limited stockpile of the vaccine. The general formulation for this type of question is formulated as
below in Equation (5):
𝑚𝑖𝑛𝑚𝑖𝑧𝑒 ∑ σ
𝑖 𝑖 𝑛 𝑖 𝑤 𝑖
𝑠 . 𝑡 . ∑ 𝑣 𝑖 𝑖 𝑛 𝑖 ≤ 𝑉
0 ≤ 𝑣 𝑖 ≤ 1
(5)
where σ
𝑖 measures the proportion of people get infected among subgroup 𝑖 , 𝑛 𝑖 refers to the
population of subgroup 𝑖 , 𝑤 𝑖 denotes the weights assigned to each subgroup, 𝑣 𝑖 refers to the
vaccination proportion in each subgroup and 𝑉 is the total stockpile of the vaccination. Under this
optimization framework, Patel et al. utilized genetic algorithms and random mutation hill climbing
28
to find optimal distributions with the stochastic epidemic simulations to minimize the number of
infections and deaths. However, due to the non-linearity and stochasticity of the epidemic models,
both of the algorithms could only approximately find the local optimum. Medlock and Galvani
developed a compartmental model that incorporates both age groups and vaccination status for the
outbreak of swine-origin influenza in 2009 [73]. They enumeratedly compared different vaccine
allocation policies by the amount of vaccine needed to reduce the effective reproduction number
less than one. Dalgic et al. takes advantage of the mesh-adaptive direct search (MADS) algorithm
to numerically compare the effectiveness of vaccine policies derived from agent-base models and
compartmental models for influenza pandemic. Starting from the initialization, the MADS
algorithm iteratively enhances the current best solution by generating test points on the variable
space around current best solution. The algorithm will decrease the step size if it fails to find a
better solution until the limits of the mesh size has been reached. The MADS algorithm used in
this study is derivative free, which makes it more flexible when the gradient of the epidemic
models are not available.
As pointed out in previous research, vaccine allocation problems are usually non-convex (or non-
concave) due to the nonlinearity, complexity, and stochasticity of the epidemic. Therefore, the
derivative of the model may not be calculated analytically, which makes the gradient-based
method inapplicable. Most algorithms will converge to a local optimum.
For the modeling of COVID-19 vaccination, Matrajt et al. used an age-stratified mathematical
model paired with optimization algorithms to study the relationship between the effectiveness of
the vaccination and health outcomes [82]. They pointed out that direct vaccination of elders brings
out the best health outcomes when the vaccination efficiency is low or the stockpile of the vaccine
is limited. Gallagher et al. emphasized that the indirect benefits of the SARS-CoV-2 vaccines has
29
the potential to quell the pandemic even the vaccine has a weaker direct protection but stronger
indirect effects [83]. Under the constraints of limited availability, uncertain vaccination efficiency,
and ethical equity problems, a fair and credible decision-making method with feasible optimal is
still well-motivated.
Unlike what has been done before for influenza, the vaccine of COVID-19 could only be available
after the epidemic is widely spread. Hence, vaccine allocation should consider the severity of the
infection and the regional characteristics (such as age structure) in each region. Meanwhile,
different from previous research, the manufacturing and distribution of the COVID-19 vaccine
should be represented as a continuous process. The allocation of the vaccine is then a dynamic
process. The optimal dynamic allocation policy for the vaccine to minimize the overall burden of
disease under the constraints of limited vaccine is a serious and urgent problem to be addressed.
2.3 Uncertainties Analysis in Epidemic Modeling
Epidemiological models serve as vital tools in understanding and forecasting the dynamics of
infectious diseases, which in turn enables policy makers to effectively allocate resources and
implement intervention strategies. However, the credibility of these models is contingent upon the
quality of data used for parameter estimation, which is influenced by factors such as monitoring,
timeliness, privacy restrictions, and reporting accuracy. Thus, addressing uncertainty in model
predictions is essential. By conducting parameter sensitivity and uncertainty analyses for model
outcomes, confidence in result interpretation and decision-making can be bolstered. To achieve
best practice in uncertainty analysis, general modeling standards recommend employing
probabilistic sensitivity analyses that address both global parameter uncertainty and output
uncertainty [84].
30
Previous uncertainty analysis applications in dynamic models can be classified according to
transmission models. The deterministic SIR model [85]–[87] serves as the most prevalent approach
for evaluating interventions in dynamic systems. Although deterministic SIR models provide
useful intervention effect estimates, the optimal parameter estimates are often imprecise and non-
unique. To address this issue, sensitivity analyses explore the relationship between model
parameters and corresponding outcomes [88]. These analyses typically involve perturbing one or
more parameters and investigating the resulting outcomes. The perturbation can either involve
assessing the impacts of arbitrarily small changes in parameter values [89] or evaluating the effects
across a range of realistic probability density functions [90]. For example, Zhang et al. applied
Sobol’s method to a compartmental COVID-19 model to identify key model parameters and
controlling parameters, which can help policy makers explore various intervention options [91].
Similarly, Sarabaz et al. utilized the SimBiology Toolbox in MATLAB with three different
techniques to analyze parameter sensitivity in a system of differential equations built for the same
purpose of identifying key spreading parameters [92]. However, these methods fail to account for
interaction effects in non-linear dynamic models and do not evaluate global uncertainty in
parameters or outcomes since other parameter values remain constant at their best point
estimations.
In contrast to deterministic dynamic model sensitivity analyses that only address variation in each
parameter, global probabilistic sensitivity analyses examine each parameter's contribution to
model outcomes while accounting for the uncertainty of other model parameters [93], [94]. This
allows modelers to convey the robustness of their predictions to policy makers. Uncertainty in
parameter values can be addressed by randomly sampling from empirical data or fitting probability
density functions to empirical data using methods such as bootstrapping, Monte Carlo sampling,
31
and Latin hypercube sampling [88], [93]. To assess each parameter's contribution to the variance
in output values, model output from parameter samples can be examined using linear, monotonic,
and non-monotonic statistical tests [88], [93]–[95]. By incorporating data-driven parameter
uncertainty into model outputs, probabilistic uncertainty analyses produce probabilistic
distributions rather than single-value estimates of potential outcomes, which enables modelers to
communicate the robustness of their predictions to policy makers. Amaku et al., for example,
calculated the force of infection for six distinct viruses in a Brazilian community using
seroprevalence study data and estimated parameter confidence intervals using Monte Carlo
simulations. Similarly, Ciufolini and Paolozzi used an empirical Gauss error function model for
COVID-19 cases in Italy, with 150 Monte Carlo simulations providing a more robust peak day
prediction [96].
During the COVID-19 pandemic, numerous non-pharmaceutical interventions (NPIs) have been
implemented worldwide to mitigate viral spread, and uncertainty analysis plays a crucial role in
quantifying the effects of these NPIs. Chinazzi et al. applied the global epidemic and mobility
model (GLEAM) to understand the impact of travel restrictions on COVID-19 transmission in
Wuhan and assessed the robustness of their results through extensive sensitivity analysis [97]. In
another study, Quilty et al. employed a stochastic branching process with a negative binomial
offspring distribution to estimate the intervention effects of cordon sanitaire and holiday travel,
simulating outbreaks generated by arrivals in four representative major cities [98]. Sensitivity
analyses for the overdispersion factor and serial interval were performed to provide confidence
intervals for the effective reproduction number. Their findings suggest that the cordon sanitaire
alone did not significantly impact epidemic progression in major cities.
32
Children often serve as key transmitters in viral epidemics like influenza due to their high levels
of direct contact with other children and parents [99]. Consequently, all 50 U.S. states implemented
statewide school closures in March 2020. Evaluating the effect of school closures on viral spread
control is therefore essential. Auger et al. conducted a population-based time series analysis of all
50 states from March 9 to May 7, 2020, which included a lag period to account for policy-related
changes [100]. They examined the sensitivity of the lag period for incidence and mortality by
varying its length, finding that school closures led to the largest reduction in incidence and
mortality when implemented as early as possible. Burns and Gutfraind introduced a novel SEIR
model incorporating student location and grade to compare the effect of two intervention policies
in schools [101]. To ensure the robustness of their results, they sampled parameters such as start
day, contact rate from normal distributions, and simulated the epidemic 500 times. Their findings
suggest that shortening the school week could serve as an important tool for controlling COVID-
19 in schools.
In summary, addressing uncertainties in epidemiological models is crucial for ensuring robust
predictions and informing policy decisions. Sensitivity and uncertainty analyses allow researchers
to identify key parameters and assess the impact of various interventions while accounting for
uncertainties in the data. Probabilistic sensitivity analyses offer a more comprehensive approach
by considering the uncertainty of multiple parameters simultaneously. Ultimately, these analyses
contribute to more effective and targeted intervention strategies in the face of infectious disease
outbreaks, such as the COVID-19 pandemic.
33
2.4 Summary of Literature
In summary, great effort has been placed on research and policies about the modeling, forecasting,
and containing infectious diseases. In the literature review above, we summarized the research
about transmission modeling of infectious diseases and the optimization of vaccine allocation.
For the transmission modeling of infectious diseases, compartmental model, statistical model, and
agent-based model are most widely adopted. At the early stage of a new epidemic, the relevant
data usually suffer from a lack of monitoring and inconsistency between datasets from different
surveillance systems. Considering the sufficiency and correctness of the epidemic data, it is
challenging to build a complicated statistical model incorporating various aspects of transmission.
Meanwhile, it is hard to use an agent-based model due to the tedious parameter tuning in practice.
The compartmental model is still a good choice to capture the transmission dynamics at a high
level, though it is limited by the homogeneity assumption. However, most of the deterministic
compartmental models fail to capture the time dependency of the transmission dynamics. A
concise and general form of the compartmental model with the consideration of time dependency
and uncertainty plays a critical role in the modeling of future epidemics.
Regarding research on vaccine allocation, most of the previous studies have only focused on the
epidemic that has not been widely spread, such as influenza. For an ongoing epidemic, such as
COVID-19, prioritizing the vaccine allocation with the consideration of current and future
infection has not been addressed too much. Meanwhile, the vaccine manufacturing and allocation
of an ongoing epidemic is a dynamic process, which is different from allocating a fixed amount of
vaccine in the previous studies. Dynamic optimization of vaccine allocation is still an urgent need
for decision-makers.
34
Chapter 3. Research Design and Methods
3.1 Application of Model to Thesis Aims
In this section, we introduce our comprehensive modeling framework for transmission dynamics
and vaccine allocation, which will be utilized in the thesis to accomplish the four research aims
stated in Chapter 1. The methodology is illustrated in Figure 1.
Utilizing disease tracking data, such as case numbers, hospitalizations, fatalities, and recoveries,
we first create a dynamic model incorporating time-varying transmission and fatality rates based
on the widely recognized Susceptible-Exposed-Infectious-Recovered-Death (SEIRD) framework.
By introducing these time-varying rates, we capture the evolving nature of the disease and the
effects of various interventions, enabling a more comprehensive understanding of regional
transmission dynamics and facilitating more accurate predictions of the epidemic's progression.
We refine our model using advanced statistical and machine learning techniques for parameter
estimation and incorporate real-time data updates to ensure responsiveness to changes in the
disease landscape. This dynamic approach offers valuable insights into the effectiveness of
interventions and their potential to alter the course of the epidemic within a specific region, as well
as allowing for the investigation of potential correlations between different variables, such as the
relationship between transmission rates and socioeconomic factors or the impact of pre-existing
health conditions on fatality rates. Through this detailed examination of regional transmission
dynamics, we can identify trends and patterns that inform public health policies and targeted
interventions to curb the spread of the disease more effectively.
After developing a dynamic model for one region, we broaden our analysis to consider interactions
between different regions and the impact of vaccinations. To accomplish this, we introduce a
spatial interaction model and a vaccinated compartment to the existing SEIRD framework. The
35
spatial interaction model captures population movements between regions, which can contribute
to disease spread. The vaccinated compartment represents the portion of the population that has
been vaccinated, making them less susceptible to contracting and spreading the virus. By
leveraging various data sources, such as mobility data and vaccination records, we can
parameterize the spatial interaction model and the vaccinated compartment. This integrated
approach enables us to analyze the effects of vaccination and regional interactions on transmission
dynamics and generate more precise analysis of epidemic progression.
To assess the impact of intervention policies on transmission dynamics, we will conduct a multi-
phase analysis for selected regions, based on local policy implementation timelines. This analysis
will allow us to evaluate the effects of measures such as lockdowns, social distancing, and
vaccination campaigns. Furthermore, we will examine the uncertainty of historical transmission
under conditions of limited data and varying public responses to interventions. By employing a
combination of sensitivity analysis and Monte Carlo simulations, we can quantify this uncertainty
and determine its implications for our predictions and policy recommendations.
With an accurate representation of historical trends, we will optimize vaccine allocation, taking
into account constraints on vaccine availability. We will devise mathematical models to identify
optimal allocation strategies that maximize health outcomes, such as minimizing infections,
hospitalizations, and fatalities. Subsequently, we will evaluate health outcomes under different
policy scenarios, including varying levels of vaccine coverage, alternative prioritization strategies,
and the effects of non-pharmaceutical interventions. This assessment will offer valuable insights
for decision-makers in designing effective policies to control the epidemic.
To demonstrate the practical application of our methodology for pandemic prediction and control,
we will apply our comprehensive modeling framework to the COVID-19 pandemic. By using real-
36
world data to calibrate our models and validate our predictions, we will showcase the relevance of
our approach and its potential to contribute to improved pandemic preparedness and response in
the future.
Figure 1: Research design for the pandemic prediction and control
3.2 COVID-19 Data Sources
In this section, we will introduce the datasets we have identified related to COVID-19 and
systematically summarize the main categories of the data. We have identified four types of public
data sets to support the research: (1) Disease tracking, including the daily infectious cases,
hospitalization, fatalities and recoveries; (2) Travel and mobility data; (3) Hospitalization and
utilization data; (4) Policy enactment data. All the data resources could be indexed by our website:
www.covid19datasource.usc.edu. As we explained in the end of this section, not all data sources
are in quality and we will focus on the disease tracking data, mobility data, and policy enactment
data.
37
3.2.1 Disease Tracking (Cases, Hospitalization, Fatalities, Recoveries)
The basic data requirement for the COVID-19 related analysis is the disease tracking of the time-
series data with respect to the daily case/fatality monitoring, PCR and antibody testing,
hospitalization, vaccination and recovery. We could leverage the disease tracking data to analyze
the transmission pattern, evaluate the effect of intervention, allocate medical resources, and control
the immunization process. Table 1 below shows the most credible and wildly cited data sources
for disease tracking.
Table 1: Data sources for COVID-19 disease tracking
Data source Data information Brief Description
The COVID Tracking
Project [102]
Cases, testing, hospitalization,
and patient outcomes,
demographic, and long-term care
facilities information
Current daily time-series data on
cases, fatalities, tests, and
hospitalizations.
Johns Hopkins University:
COVID-19 Data
Repository [103]
Cases, fatality, incidence rate,
testing rate
Interactive dashboard with
downloadable real-time data.
Our World in Data [104]
Cases, deaths, hospitalizations,
testing, and vaccinations
Comprehensive COVID-19 pandemic
data on cases, deaths, hospitalizations,
tests, and vaccinations.
Facebook Data for
Good[105]
Population density, movement
range, forecasting, social
connectedness index
Tools and initiatives for
organizational COVID-19 response.
Google COVID-19 public
dataset program [106]
Demographics, economy,
epidemiology, geography,
health, hospitalizations,
mobility, government response,
and weather
Global COVID-19 daily time-series
datasets for US and EU countries.
3.2.2 Travel and Mobility
Recent research points out the significant and positive association between the frequency of air,
automobile, rail and transit travel with the daily increase of infections [107], [108]. To better
understand the mutual impact of the COVID-19 transmission and transportation related behavior,
we identified several general public datasets. In Table 2 below, we summarize data sources from
different aspects of transportation, including migration, airline, daily travel and change of mobility.
38
Table 2: Public travel and mobility data sources during the COVID-19 pandemic
Data source Data type Brief description
Airline On-Time
Statistics[109]
Airline data
Departure/arrival statistics (including delays) by airport and
airline, airborne time, cancellations, and diversions
Daily Travel
during the
COVID-19 Public
Health Emergency
Airline, driving,
rail and transit
data
Daily percentage and count of people staying home/not
staying home, trips taken across 10 distance groupings
FHV Trip
Records[110]
Taxi Trip records of For-Hire Vehicles (FHV) in New York City
Facebook
movement
map[111]
Mobility
changes
Movement change comparisons, using a pre-social distancing
baseline
Google
Community
Mobility
Reports[112]
Mobility
changes
Visits to places (e.g. grocery stores, parks) in each region,
showing changes over time
Baidu China
Migration
Data[113]
Migration
Changes in population movement between Chinese cities,
measured by Baidu Migration index
3.2.3 Hospital Utilization and Resource Availability
With the outbreak and resurgence of COVID-19, the medical system has been stressed by multiple
waves of patients. Preparing, monitoring and allocating medical resources, such as ICU beds,
ventilators, expendable supplies, and vaccines, affect the capability to response to the surge of
patients. Table 3 below summarizes hospital resources, monitoring data and possible strategies for
increasing relevant capacity.
Table 3: Hospital utilization and resource availability
Key Covid-19 Healthcare Resources
University of Southern California COVID-
19 Data Source
Resource
Category
Resource Types Data Source Strategies for Increasing Capacity
Personnel
Doctors -- certain
types
STATISTA[114]
Implement overtime, postpone non-urgent
care, extend staff availability, recruit
temporary medical professionals
Emergency Medical
Technicians
NREMT
Implement overtime, recruit additional
EMTs, enhance training programs
Skilled Nursing
BUREAU OF LABOR
STATISTICS
Hire traveling nurses, implement overtime,
extend staff availability, postpone non-
39
urgent care, recruit temporary nursing
professionals
Medical Devices
Antibody Test
CV19 LAB TESTING
DASHBOARD
Increase production, streamline
distribution, prioritize high-risk
populations
Infusion Pumps
Optimize use, manage inventory
efficiently, invest in additional equipment
Oxygen Tanks
Optimize use, improve supply chain,
prioritize critical cases
PCR Tests HEATHDATA.GOV
Enhance manufacturing and supply chain,
prioritize testing for symptomatic or high-
risk individuals, increase testing capacity
Rapid Testing
Increase production, streamline
distribution, expand testing sites
Ventilators
DEFINITIVE HEALTH
CARE[115]
Use alternative devices (CPAP,
Noninvasive Ventilation), prioritize
ventilator allocation, invest in additional
equipment
Pharmaceuticals
Anticoagulation
Streamline distribution, prioritize high-risk
patients, optimize treatment protocols
Convalescent
Plasma
Incentivize COVID-recovered patients to
donate plasma, raise awareness, improve
plasma collection infrastructure
Dexamethasone VIZIENTIC
Reserve for patients in critical need,
establish stricter criteria for patient
selection, increase production
Monoclonal
Antibody
HHS PROTECT PUBLIC
DATA HUB[116]
Coordinate primary care for high-risk
patients with underlying conditions,
streamline distribution, enhance production
Remedesivir
Diversify production sources, streamline
distribution, prioritize high-risk patients
Expendable Supplies
Bulk Oxygen GETUSPPE[117]
Optimize supply chain, prioritize
allocation, invest in additional
infrastructure
Disinfecting wipes GETUSPPE
Conservation, Substitution, Re-use, Limit
HCP face-to-face interaction,
Telemedicine, Proper use training,
Velarization rooms, Physical barriers,
Selective airborne infection room use,
Cohosting patients
Face Shields GETUSPPE
Conservation, Substitution, Re-use, Limit
HCP face-to-face interaction,
Telemedicine, Proper use training,
Velarization rooms, Physical barriers,
Selective airborne infection room use,
Cohosting patients
Gloves GETUSPPE
Conservation, Substitution, Re-use, Limit
HCP face-to-face interaction,
Telemedicine, Proper use training,
Velarization rooms, Physical barriers,
40
Selective airborne infection room use,
Cohosting patients
Gowns GETUSPPE
Hand Sanitizer GETUSPPE
N95 Masks STATISTA
Surgical Masks GETUSPPE
Vaccination
Moderna CDC M
Standardize vaccine distribution
approaches, expand vaccination sites,
increase production
Pfizer-BioNTech CDC P
Standardize vaccine distribution
approaches, expand vaccination sites,
increase production
Hospital Beds
ICU Beds
UNIVERSITY OF
MINNESOTA[118]
Convert standard beds/PACU to ICU,
deploy modular hospitals, reduce patient
length of stay, focus on preventing severe
disease
Licensed Beds
COVID-19 CARE
MAP[119]
Repurpose non-COVID areas (hotels,
alternate spaces), utilize virtual care,
deploy modular hospitals, reduce patient
length of stay, focus on preventing severe
disease
Skilled Nursing,
After Discharge
HEALTHDATA
Reduce post-discharge recovery stay
length, prepare skilled nursing facilities for
COVID patients, promote home-based
recovery
Emergency
Response
Ambulance
HOMELAND
SECURITY
Reduce ER wait times, expand
telemedicine use, prevent severe
hospitalization by early intervention
3.3.4 Policy Enactment
In response to the COVID-19 pandemic, governments have implemented a wide range of
intervention policies. The intervention policy, especially non-pharmaceutical interventions, plays
a key role in flattening the infection number and reducing the stress on the healthcare system when
the effective vaccines and medications are still under development.
During the COVID-19, government intervention policy has shown a large divergence in the
response to the rapidly changing and unprecedented circumstances of COVID-19. The UK
government and Swedish governments initially adopted a herd immunity approach, which implies
doing little or nothing to stop the epidemic. [120]. The predictive model and the transmission
results show such a strategy fails to flatten the curve and prevent overwhelming the healthcare
41
system. [121] Some countries applied stringent interventions, sometimes limiting residents’ liberty.
In China, for example, mandatory quarantine, social distancing, contact tracing and testing, have
successfully mitigated the COVID-19 transmission.
In the United States, non-pharmaceutical intervention measures fall into five categories:
Movement restrictions, Public health measures, Social and economic measures, Social distancing
and Lockdown [122]. The subtypes of each category have been summarized below in Table 4.
Table 4: Intervention measures taxonomy
CATEGORY
MEASURES BRIEF DESCRIPTION
Movement restrictions
Health/doc
requirements
Health declarations or doctor's certifications required upon arrival to
assess traveler's health and minimize the risk of spreading the virus.
Border inspections
Travel and ID document checks at land and sea entry points to control
the movement of people and curb the spread of the virus.
Partial border closure
Borders closed to non-nationals/residents to minimize the influx of
potentially infected individuals.
Full border closure
Borders closed to everyone, including nationals, to prevent any cross-
border spread of the virus.
Internal checkpoints
Checkpoints within a country for health checks and controlling
internal movement to limit the spread of the virus between regions.
Flight suspensions
Government suspension of international/domestic flights to restrict
travel and reduce the risk of importing new cases.
Domestic movement
limits
Limited movement within the country to reduce the virus's spread
within communities.
Visa limitations
Entry limitations for specific nationalities or new visa restrictions to
control the flow of people from high-risk areas.
Curfew implementation
Regional or country-wide curfews to limit the movement of people
and reduce potential exposure to the virus.
Movement monitoring
Electronic surveillance for case tracing or movement monitoring to
ensure compliance with restrictions and track potential virus spread.
Public Health Measures
Public awareness
campaigns
Media campaigns promoting hygiene, social distancing, and other
preventive measures to raise public awareness and reduce the virus's
spread.
Quarantine/isolation
policies
Self-quarantine/isolation for arrivals, symptomatic individuals, or
contact cases to prevent the spread of the virus among the general
population.
Hygiene
recommendations
Government-issued hygiene guidelines and precautions to educate the
public on effective ways to protect themselves and others.
Health screenings
Temperature controls and health screenings at airports/border
crossings to identify potentially infected individuals before they enter
the country.
42
Non-COVID medical
tests
Forced health checks unrelated to COVID-19 (e.g., HIV) to assess the
overall health of the population and address other health concerns.
Psychosocial support
Support for patients, families, and quarantined/locked down
individuals to address mental health needs and alleviate stress related
to the pandemic.
Large-scale testing
Country or regional population screening to identify infected
individuals, track the virus's spread, and inform public health
measures.
Public health system
reinforcement
Hiring more medical personnel, expanding facilities, and increasing
resources to better handle the pandemic and provide adequate
healthcare.
Infection testing
Tests to identify infected individuals, enabling targeted isolation and
quarantine measures.
Public protective gear
Mandatory masks/gloves, etc., to reduce the risk of transmitting the
virus in public spaces.
Additional health
measures
Transport sanitation, additional health regulations to improve public
health and reduce the potential spread of the virus.
Burial regulation
changes
Changes to burial regulations/attendance limits to minimize large
gatherings and reduce the risk of virus transmission.
Social economic measures
Economic interventions
Measures to mitigate the economic and societal impact of the
pandemic, such as financial aid, tax breaks, and subsidies.
Emergency structures
Emergency Response committees for coordination, decision-making,
and monitoring the implementation of pandemic response measures.
Import/export
restrictions
Restrictions on food/health item imports/exports to ensure adequate
supply and distribution within the country.
Emergency declaration
Declaration allowing the implementation of extraordinary measures
to address the pandemic that may not be allowed under normal
circumstances.
Military assistance
Military deployment to support medical operations, enforce
restrictions, and ensure compliance with public health measures.
Social distancing
Public gathering limits
Canceling public events and limiting gathering sizes to reduce close
contact between individuals and minimize virus transmission.
Business/service
closures
Closure of businesses and public services, with online alternatives
when possible, to reduce human interaction and curb virus spread.
Prison policy changes
Changes to prison policies, such as early release, suspension of day-
release programs, and visitation restrictions, to mitigate disease
spread within prisons.
School shutdowns
Authorities closing schools to limit close contact between students
and staff, reducing potential virus transmission.
43
Lockdown
Partial lockdown
Limited reasons for leaving home (e.g., essential shopping, medical
appointments); non-essential stores closed to minimize public
movement and virus spread.
Complete lockdown
Limited reasons for leaving home; non-essential services/production
halted to enforce strict social distancing and minimize virus
transmission.
Camp/minority
lockdown
Movement limitations for populations in camps or camp-like
conditions, such as refugees or internally displaced persons, to reduce
virus spread within vulnerable communities.
In our research, we will utilize the COVID Tracking Project [102] and Our World in Data [104]
for the daily/cumulative confirmed cases, deaths, and demographic. For spatial interaction data
between states, we will use the Daily Travel during the COVID-19 Public Health Emergency from
BTS and daily percentage of out-of-state trips from Maryland Transportation Institute. For the
vaccination data, we will use COVID-19 Vaccine Distribution Allocations by Jurisdiction from
CDC, which provides the total distributed and administered vaccine of Janssen, Moderna, and
Pfizer by Jurisdiction. For the analysis of policy effects, we will utilize the time series policy data
from COVID-19 Government Response Tracker of Oxford University.
44
Chapter 4 Dynamic Modeling with Time-Varying
Transmission and Fatality Rates
Dynamic modeling with time-varying transmission rates is a powerful tool in the field of
epidemiology, enabling insights into complex disease outbreak dynamics. By accounting for
fluctuations in transmission rates and fatality rates over time, these models offer a robust
framework for anticipating the future trajectory of epidemics and informing evidence-based public
health decision-making. Furthermore, dynamic modeling allows for the evaluation and comparison
of various intervention strategies, such as vaccination campaigns, social distancing measures, and
testing, tracing, and isolation protocols, enabling the optimization of resource allocation and the
development of targeted, effective public health policies. This is particularly pertinent in the
context of emerging infectious diseases, where initial information is often scarce, and the potential
for rapid spread and severe consequences necessitates swift, informed action. By leveraging
dynamic modeling, researchers and policymakers can better understand the interplay between
time-varying transmission and fatality rates, assess the potential impacts of different interventions,
and ultimately, help to mitigate the devastating effects of disease outbreaks on global health and
wellbeing.
In this chapter, we present an enhanced SEIRD (Susceptible-Exposed-Infectious-Recovered-
Death) model that incorporates time-varying case fatality and transmission rates for confirmed
cases and deaths, aiming to provide a more comprehensive understanding of infectious disease
dynamics. Our analysis demonstrates that, by representing case fatalities and transmission rates as
simple Sigmoid functions, historical cases and fatalities can be accurately fit with a root-mean-
squared-error accuracy on the order of 2% for the majority of American states during the period
45
from the initial cases up to July 20, 2020. For states experiencing multiple waves of infection, we
propose an alternative multi-phase model, which allows for a nuanced understanding of the
varying dynamics in these regions and the potential effects of intervention strategies throughout
the epidemic [123]. The enhanced SEIRD model offers a valuable method for explaining historical
reported cases and deaths using a compact set of parameters, thereby enabling the analysis of
uncertainty and variations in disease progression across different regions. This approach provides
crucial insights for public health officials and policymakers, supporting the development of
targeted and effective strategies to control the spread of infectious diseases.
4.1 The Proposed Time-Varying Model
We draw from the SEIRD compartmental model, which divides the population into five groups:
susceptible(S), exposed (E), infected (I), recovered (R) and dead (D). The SEIRD model is selected
due to its simplicity and flexibility. The model can be easily adapted to capture the unique
characteristics of different diseases and populations without requiring detailed individual level
data, which is usually not available at beginning of an epidemic.
SEIRD utilizes differential equations to model the evolution of the number of people in these states
over time. Initially, individuals are classified as susceptible, meaning they have not been infected
with the disease and are at risk of becoming infected. When an individual comes into contact with
an infected person, they may become exposed, meaning that the disease is in its incubation period
and the individual is not yet infectious. After the incubation period, the individual becomes
infectious and is classified as infected. Finally, if the individual recovers from the disease, they are
classified as recovered and become immune to the disease.
The transmission of the disease is governed by the transmission rate 𝛽 (𝑡 ), which represents the
probability of an infected individual transmitting the disease to a susceptible individual. The
46
transmission rate is typically assumed to be constant in the basic SEIR model, but it is modeled as
a time-varying parameter to capture changes in behavior or the impact of interventions on disease
transmission. Death rate α(t) is also treated as a time varying function, representing the proportion
of infectious individuals who eventually die from the disease, by date. Those who eventually die
transfer from the infected to the died state at a rate of ρ, representing the inverse of the time from
becoming infectious until time of death. In our model, ρ is assumed to be constant over time.
Those who eventually recover do so at the γ, representing the inverse of the time from becoming
infectious until recovery. We will also later derive the effective reproduction number 𝑅𝑒𝑝 (𝑡 ),
representing the average number of persons who are exposed to the disease by each infectious
person, as a function of time.
Taking these factors into account, the system of equations of the proposed SEIRD model is given
by Equation (6):
∂S(t)
∂t
= −β(𝑡 ) ∙ 𝐼 (𝑡 ) ∙
𝑆 (𝑡 )
𝑁
𝜕𝐸 (𝑡 )
𝜕𝑡
= β(𝑡 ) ∙ 𝐼 (𝑡 ) ∙
𝑆 (𝑡 )
𝑁 − σ ∙ 𝐸 (𝑡 )
𝜕𝐼 (𝑡 )
𝜕𝑡
= σ ∙ 𝐸 (𝑡 ) − (1 − α(𝑡 )) ∙ γ𝐼 (𝑡 ) − α(𝑡 ) ∙ ρ ∙ 𝐼 (𝑡 )
𝜕𝑅 (𝑡 )
𝜕𝑡
= (1 − α(𝑡 )) ∙ γ ∙ 𝐼 (𝑡 )
𝜕𝐷 (𝑡 )
𝜕𝑡
= α(𝑡 ) ∙ ρ ∙ 𝐼 (𝑡 )
(6)
where:
𝑆 (𝑡 ) = number of people in susceptible state at time t
𝐸 (𝑡 ) = number of people in exposed, but uninfected at time t
𝐼 (𝑡 ) = number of people in infectious state at time t
𝐷 (𝑡 ) =number of people who have died at time t
𝑅 (𝑡 ) = number of people who have recovered at time t
47
N = total number of people
𝛽 (𝑡 ) = transmission rate at time t
𝜎 = transformation rate from exposed to infectious, which is the reciprocal of the
incubation period
𝛼 (𝑡 ) = likelihood of eventual death of a person who is infected at time t
𝛾 = transformation rate from infectious to recovered, which is the reciprocal
of the recovery time
𝜌 = transformation rate from infectious to death
Transmission rates in epidemics are influenced by several factors, including pathogen
characteristics, population density, mobility, and social behaviors. The virulence and
transmissibility of a pathogen directly impact its spread, with more infectious pathogens resulting
in higher transmission rates. High population densities, particularly in urban centers, and increased
movement of people due to travel and globalization facilitate transmission by enabling more
frequent contact between individuals. Social behaviors, such as handshaking or attending large
gatherings, can increase the likelihood of transmission, while cultural practices, like funeral rituals,
can also contribute to disease spread if they involve close contact with infected individuals or their
bodily fluids. Public awareness and adherence to hygiene practices, such as handwashing and
sanitizing, can further influence transmission rates.
Death rates during epidemics can be affected by factors such as age distribution, healthcare
infrastructure, and the prevalence of comorbidities. Older individuals and those with underlying
health conditions, like diabetes or heart disease, are more susceptible to severe outcomes, leading
to higher death rates in populations with a greater proportion of vulnerable individuals. The
capacity and quality of healthcare systems also play a crucial role in determining death rates.
48
Access to timely diagnosis, adequate hospital capacity, and effective treatments can mitigate the
severity of the disease, reducing the likelihood of fatalities. Furthermore, socioeconomic factors,
such as access to healthcare, nutrition, and living conditions, can influence the overall health and
resilience of a population, affecting death rates during an epidemic.
Intervention policies, such as non-pharmaceutical interventions (NPIs), vaccination programs, and
testing, tracing, and isolation strategies, can have a significant impact on transmission and death
rates during epidemics. NPIs like social distancing, mask-wearing, and school or workplace
closures help reduce transmission by limiting contact between individuals, slowing the spread of
the disease and preventing healthcare systems from becoming overwhelmed. Vaccination
programs are vital for controlling infectious diseases, as they lower the number of susceptible
individuals in a population, indirectly reducing death rates by protecting vulnerable individuals
from infection. Effective testing and tracing programs can identify and isolate infected individuals,
further limiting the spread of the disease, providing critical information for public health decision-
making, and potentially saving lives. Public health communication and community engagement
are also essential for the successful implementation of these intervention policies, as they help to
build trust and ensure adherence to guidelines.
Recognizing the importance of these intervention policies, it is crucial to understand how changes
in such policies, global events, and medical care affect transmission dynamics, represented by 𝛼 (𝑡 )
and 𝛽 (𝑡 ). While these functions could potentially change erratically due to discrete events such as
the introduction of new public health measures, we hypothesize that such events do not cause
abrupt alterations in either function. Therefore, we explore whether a simple continuous model,
with a minimal set of parameters, can accurately represent historical data. For instance, when a
new intervention policy is enacted, the public may not react immediately, and the transmission
49
parameters do not shift instantaneously. Over time, the public adapts to the policy, and the effective
reproduction number eventually stabilizes. Furthermore, the public's response is influenced not
only by government policies but also by effective communication about the disease.
Communication comes from many, sometimes conflicting, sources. How the public at large
absorbs and responds to such often confusing messages may be gradual. The public will get used
to the policy after a period of adaptation, and eventually the effective reproduction number will
stabilize.
A natural function to describe this pattern of change is the Sigmoid function. Equation (7) is the
general form of the Sigmoid function, where 𝑘 determines the slope of the function and 𝑎
determines the x value at the middle point (i.e., point of time when y=.5).
𝑆 (𝑥 ) =
1
1 + 𝑒 𝑘 (𝑥 −𝑎 )
(7)
Thus, we define the function for transmission rate and death rate as Equation (8) and (9).
β(𝑡 ) = 𝛽 𝑒𝑛𝑑 +
𝛽 𝑠𝑡𝑎𝑟𝑡 − 𝛽 𝑒𝑛𝑑 1 + 𝑒 𝑚 ∙(𝑥 −𝑎 )
(8)
𝛼 (𝑡 ) = 𝛼 𝑒𝑛𝑑 +
𝛼 𝑠𝑡𝑎𝑟𝑡 − 𝛼 𝑒𝑛𝑑 1 + 𝑒 𝑛 ∙(𝑡 −𝑏 )
(9)
where,
𝛽 𝑠𝑡𝑎𝑟𝑡 is the starting reproduction number
𝛽 𝑒𝑛𝑑 is the ending reproduction number
𝛼 𝑠𝑡𝑎𝑟𝑡 is the starting death rate, ranging from 0 to 1
𝛼 𝑒𝑛𝑑 is the ending death rate, ranging from 0 to 1
𝑚 , 𝑛 , 𝑎 , 𝑏 are the shape parameters
4.2 Parameter Estimation and Model Fitting
Parameters in Eqs. 1 will be estimated with the objective of minimizing the weighted summation
of squared error between cumulative predicted and measured confirmed cases and the summation
50
of squared error between cumulative predicted and cumulative confirmed deaths. Our analysis
encompasses the period from the day of the first reported case in each state until July 28, 2020,
covering all 50 American states. For each state, we selected a start date four days prior to the date
of the first confirmed case, in accordance with a report by the Centers for Disease Control and
Prevention (CDC), which indicates that the median incubation period is 4 days, with a range of
2~7 days.
To estimate the shape parameters 𝑚 , 𝑛 , 𝑎 , 𝑏 and the starting/ending parameters
𝛽 𝑠𝑡𝑎𝑟𝑡 , 𝛽 𝑒𝑛𝑑 , 𝛼 𝑠𝑡 𝑎 𝑟𝑡
, 𝛼 𝑒𝑛𝑑 , we fit Eqs. 1 to the cumulative confirmed case numbers and the
cumulative confirmed death numbers with the nonlinear least square method. Other parameters
were derived from prior research.
Among 305 hospitalized patients and 10,647 recorded deaths, the median time of hospitalization
was 8.5 days and the median interval from illness onset to death was 10 days (IQR = 6 - 15 days).
We assume the median hospitalization time is the median time for infectious people to stop being
contagious. Hence, we set these parameters as the inverse of these time values: 𝜎 = ¼, 𝛾 =
1/8.5, 𝜌 = 1/10.
The remaining parameters are derived for each American state by optimizing the fit of the model
to historical case and death data, where the objective is to minimize a weighted sum of daily
squared error over the analysis period. We utilized a search algorithm that required initialization
and a constrained search space, as explained below.
We define the model function M(t; [𝛽 𝑠𝑡𝑎𝑟𝑡 , 𝛽 𝑒𝑛𝑑 , 𝑚 , 𝑎 , 𝛼 𝑠 𝑡𝑎𝑟𝑡 , 𝛼 𝑒𝑛𝑑 , 𝑛 , 𝑏 ]): 𝑡 → 𝑅 2
, where M(t;
[𝛽 𝑠𝑡𝑎𝑟𝑡 , 𝛽 𝑒𝑛𝑑 , 𝑚 , 𝑎 , 𝛼 𝑠𝑡𝑎𝑟𝑡 , 𝛼 𝑒𝑛𝑑 , 𝑛 , 𝑏 ]) = [𝐼 ̂
(𝑡 ) + 𝑅 ̂
(𝑡 ) + 𝐷 ̂
(𝑡 ), 𝐷 ̂
(𝑡 )] and the reported case number
and death number at time t is [Cases(t), Deaths(t)].
51
Because it is unlikely for transmission and death rates to change drastically in a single day, we set
upper bounds for 𝑚 𝑎𝑛𝑑 𝑛 at 0.33 (meaning that rates do not suddenly change in less than three
days) and initialize the search at 0.25. We permit the turning point of the sigmoid function to
occur on any day in the timeline; we set 𝑎 , 𝑏 ∈ [0,125], where 125 is the length of the period from
March 1
st
to July 28
th
, in days (as of March 1 few states had reported cases). Prior research suggests
that the initial effective reproduction number is around 3 [124], equivalent to a transmission rate
of 0.75, which we use for initialization. Because transmission rates vary significantly among
locations due to local conditions (such as crowding), we bound β
𝑠𝑡𝑎𝑟𝑡 ∈ [0.5,7.5] and β
𝑒𝑛𝑑 ∈
[0,2.5], thus permitting a wide range of results.
To summarize, the parameters set 𝑃 = [β
𝑠𝑡𝑎𝑟𝑡 , β
𝑒𝑛𝑑 , 𝑚 , 𝑎 , 𝛼 𝑠𝑡𝑎𝑟𝑡 , 𝛼 𝑒𝑛𝑑 , 𝑛 , 𝑏 ] is initialized as
[0.75,0.5,0.25,10,0.4,0.1,0.25,10]. Then the parameter optimization problem is formulated in
Equation (10).
𝑚𝑖 𝑛 𝑃 ‖𝑀 (t; 𝑃 ) − [𝐶𝑎𝑠𝑒𝑠 (𝑡 ), 𝐷𝑒𝑎𝑡 ℎ𝑠 (𝑡 )]‖
2
2
𝑠 . 𝑡 . 0.5 ≤ β
𝑠𝑡𝑎𝑟𝑡 ≤ 7.5
0.1 ≤ β
𝑒𝑛𝑑 ≤ 2.5
0 ≤ α
start
≤ 1
0 ≤ α
𝑒𝑛𝑑 ≤ 1
0.01 ≤ 𝑚 ≤ 0.33
0.01 ≤ 𝑛 ≤ 0.33
0 ≤ 𝑎 ≤ 125
0 ≤ 𝑏 ≤ 125
(10)
The number of reported deaths is smaller than the number of reported cases in all locations. Thus,
treating errors in death estimation and case estimation the same will lead to underfitting of the
death data, in preference to minimizing the errors in case data. Therefore, considering the accuracy
of the reported death data and the fitting accuracy, we optimized a weighted sum of squared death
and case data, multiplying w by deaths during the fitting process. The adjusted objective function
is formulated in Equation (11):
mi𝑛 𝑝 ‖(𝐼 ̂
(𝑡 ) + 𝑅 ̂
(𝑡 ) + 𝐷 ̂
(𝑡 ) − 𝐶𝑎𝑠𝑒𝑠 (𝑡 ))
2
+ 𝑤 ∗ (𝐷 ̂
(𝑡 ) − 𝐷𝑒𝑎𝑡 ℎ𝑠 (𝑡 ))
2
‖
2
(11)
52
The parameters are estimated by solving the nonlinear constrained least-squares problem in
Equation (10), utilizing the Levenberg–Marquardt algorithm (LMA). The Levenberg-Marquardt
Algorithm (LMA) is a numerical optimization algorithm used to solve non-linear least squares
problems by combining the steepest descent method and the Gauss-Newton method. It starts by
using the steepest descent method to make large corrections in the model parameters, then switches
to the Gauss-Newton method to make more accurate adjustments as the parameters get closer to
the optimal solution. The LMA also uses a damping factor to balance the step size between the
steepest descent and Gauss-Newton methods, ensuring convergence to the minimum of the
objective function. Its hybrid approach and damping factor make it robust and efficient, and it has
many practical applications in science and engineering[125]. The LMA will be implemented to
our model fitting by the lmfit package in Python.
4.3 Model Accuracy
The first case of COVID-19 in the United States was reported on January 20, 2020[126]. As of
July 31, 2020, a total of 4,665,469 cases and 155,863 deaths had been reported across the states
and territories of America[102]. We fit the model with the dataset of 7-day moving average cases
and deaths for the 50 states, provided by the COVID-19 tracking project lead by The Atlantic
(derived from the Center for Disease Control), for the period from the date of the first reported
cases to July 31
st
. The fitting accuracy across all states is presented in Figure 2, measured by the
relative root mean square error (RRMSE) in Equation (12).
𝑅𝑅𝑀𝑆𝐸 =
[∑ (𝑦 𝑖 ̂ − 𝑦 𝑖 )
2 𝑁 𝑖 =1
/𝑁 ]
1/2
𝑦 𝑁 − 𝑦 1
(12)
where 𝑦 𝑁 is the case/death on the 𝑁 𝑡 ℎ
day.
The fitting accuracy of the reported cases ranges from 0.54% to 7.34% and of the reported deaths
ranges from 0.29% to 7.28%. The average and median RRMSEs for deaths are 1.61% and 1.33%;
53
for cases, the average and median values are 2.30% and 1.88%. RRMSE fell below 5% by both
measures for all states except Hawaii, Idaho, Louisiana, Montana and Wyoming.
Figure 2 shows that the proposed SEIRD model with time-dependent transmission rate and death
rate captured the pattern of the transmission dynamics well across most states with only 8 fitted
parameters.
Figure 2: Fitting accuracy of the cases and fatality across all states
54
Figure 3 and Figure 4 show the specific fitting results for cases and deaths by day for the two states
with the largest number of cases (New York and California) as well as two other states for which
the fit is less accurate (Florida and Hawaii). For New York and California, the fitting results almost
coincide with the CDC data. Examining Florida and Hawaii, the CDC data follows a pattern of
two phases, which is not as well captured by our model. Especially for Hawaii, the curve flattened
for a period and then rose. As discussed later, our model characterizes the transmission dynamic
for a period with one phase, i.e. the curve should become flat at most once.
Figure 3: Fitting results of case number for New York, California, Florida, and Hawaii
55
Figure 4: Fitting results of death number for New York, California, Florida, and Hawaii
4.4 Effective Reproduction Number Calculation and Trends
The effective reproduction number, which we define as 𝑅𝑒𝑝 (𝑡 ), is a key metric used in
epidemiology to describe the transmission potential of an infectious disease. It represents the
number of secondary infections that can be caused by a single infected individual in a population
that is partially susceptible to the disease. In other words, 𝑅𝑒𝑝 (𝑡 ) is a measure of how many people
an infected person will go on to infect, on average [127]. When 𝑅𝑒𝑝 (𝑡 ) > 1, the rate of new cases
will increase over time, until the population loses susceptibility to the disease. When 𝑅𝑒𝑝 (𝑡 ) < 1,
the rate of new cases will decline over time.
There are several factors that can influence 𝑅𝑒𝑝 (𝑡 ), including the infectiousness of the disease, the
duration of infectiousness, the population density, and the effectiveness of control measures such
as social distancing, mask-wearing, and vaccination. Public health officials use 𝑅𝑒𝑝 (𝑡 ) as a tool
for monitoring the progress of an outbreak and for making decisions about when and how to
56
implement control measures. Hence, during this section we will calculate the trend of effective
reproduction number and analyze the possible factors that influence the evolution of the
transmission dynamics.
𝑅𝑒𝑝 (𝑡 ) c an be estimated with the Next Generation Matrix(NGM) method[128]. The Next
Generation Matrix method is a powerful tool for estimating 𝑅𝑒𝑝 (𝑡 ) from compartmental models.
The NGM method is based on the idea that the distribution of secondary infections can be
described by a matrix, with each element representing the probability of an infected individual
transmitting the infection to another individual in a specific population subgroup. The NGM matrix
can be calculated from the parameters of the compartmental model, such as the transmission rate
and the distribution of individuals in different compartments. The resulting matrix is then used to
calculate the spectral radius, which is a measure of the effective reproduction number 𝑅𝑒𝑝 (𝑡 ).
We define 𝑋 as the vector of infected class (i.e. E, I) and Y as the vector of uninfected class (i.e.
S, R, D). Let
𝑑𝑋
𝑑𝑡
= ℱ(𝑋 , 𝑌 ) − 𝒱 (𝑋 , 𝑌 ), where ℱ(𝑋 , 𝑌 ) is the vector of new infection rates (flows
from Y to X) and 𝒱 (𝑋 , 𝑌 ) is the vector of all other rates. Then for our model, the next generation
matrix is expressed in Equation (13).
𝑀 = (
𝜕 ℱ
𝜕𝑋
)
(𝑁 ,0,0,0,0)
(
𝜕𝑉
𝜕𝑋
)
(𝑁 ,0,0,0,0)
−1
= [
0 β(𝑡 )
0 0
] [
𝜎 0
𝜎 (1 − α(𝑡 )) ∙ γ + α(𝑡 ) ∙ 𝜌 ]
−1
(13)
Then the effective reproduction number is the spectral radius of M, which is
β(𝑡 )
(1−α(𝑡 ))∙γ+α(𝑡 )∙ρ
At the beginning of the epidemic, 𝑅𝑒𝑝 (𝑡 ) reflects the natural transmissibility of COVID-19, i.e.
the basic reproduction number 𝑅 0
in the absence of intervention. With the evolution of the
epidemic, 𝑅𝑒𝑝 (𝑡 ) changes dynamically, as do the transmission rate β(𝑡 ) and death rate α(𝑡 ),
57
which are influenced by both the intervention policy and population immunity. Figure 5 and Figure
6 show the fitted 𝑅𝑒𝑝 (𝑡 ) at the start of the epidemic across all states and fitted 𝑅𝑒𝑝 (𝑡 ) on July 31
st
.
We see that 𝑅𝑒𝑝 (𝑡 ) ranges from 1.27 to 16.49, with a median value of 2.87. It should be kept in
mind that this optimal fit is a reflection of the reported data on cases. Increasingly aggressive
testing may make it appear that 𝑅 𝑒𝑝 (𝑡 ) grows faster than the actual (unknown) number of cases.
Figure 5: Fitted Rep(t) at the start of the epidemic across all states
Figure 6: Fitted Rep(t) on July 28th
For illustration, Figure 7 shows our estimated history of for New York, California, Florida and
Hawaii. Time 0 in these graphs is the day of the first reported case, which varies from state to state.
In these cases, the effective reproduction number both stabilized and became smaller than 1 with
time, with the change occurring over a period of 10 to 30 days.
58
Figure 7: History of effective reproduction number for New York, California, Florida and Hawaii
As noted, in the early stages of an epidemic, the reproduction number may seem particularly large
not only because the disease spreads rapidly but also because the rate of testing is increasing. In
this sense, the estimated reproduction number is a reflection of both changes in the data collection
process and the actual spread of disease.
Death rate is another measure that shows the change in virus outcomes over time, reflecting the
health system’s ability to deal with the flood of infected people. Figure 8 provides examples. From
the historical plot, we see the hardest-hit states, like New York and Florida, experienced a much
higher death rate in the early stage than the average 3% death rate in the United States. The
relatively high death rate could be caused by the lack of effective medical treatment and hospital
overload. It could also reflect limited testing of patients, whereby only the sickest patients were
recorded as cases. With improvement of medical treatment, and increased testing, the death rate
per confirmed case for most states decreased to a much smaller value.
59
Figure 8: History of death rate for New York, California, Florida and Hawaii
4.5 Multi-Phase Model
Our model, as initially presented, demonstrates a strong fit for reported data on cases and deaths,
with an error margin of less than 2% in the majority of states. Nevertheless, the model's
foundational assumption—that transmission rates do not initially decrease before eventually
increasing—necessitates modification when applied to states that have experienced multiple waves
of the disease. Data from Hawaii, where our model exhibits the least accurate fit, exemplifies this
pattern.
In response to these limitations, we introduce a multi-phase model specifically designed for
locations exhibiting multiple waves of the disease. The first positive COVID-19 case in Hawaii
was announced on Oahu on March 6th, prompting the Hawaii Department of Health to implement
a stay-at-home order on March 25th. As a result, the case curve flattened between April 19th and
May 7th. Upon announcing the commencement of the first phase of reopening on May 7th, data
began to reflect a second wave of the virus.
60
To account for this pattern, we divided the Hawaii timeline into two distinct periods: the first from
March 6th to May 7th, and the second from May 7th to July 28th. For the first phase, we fit the
model with the assumption of only one exposed individual at the beginning. To initialize the
second phase, we used the predicted numbers of exposed, infectious, and recovered individuals
from the first phase, incorporating the reported deaths as of May 7th. This modification resulted
in a decrease in the RRMSE for cases to below 2.5% and the RRMSE for deaths to below 2.7%.
As illustrated in Figure 9, our two-phase model more accurately captures the transmission pattern
in Hawaii than the single-phase model. This approach allows us to better account for fluctuations
in transmission rates, reflecting the complexities of disease spread in locations with multiple waves
of infection. Further research and refinements to our multi-phase model may enable even more
precise predictions and facilitate better-informed policy decisions for managing the ongoing
pandemic.
Figure 9: Fitting results for the two phases Hawaii
61
The history of effective reproduction number and death rate are shown in Figure 10. The first
phase showed a decline in the reproduction number after the initial announcement of the stay-at-
home order. However, with the reopening, the reproduction number increased, explaining
increases in case rates. Death rates, by contrast, exhibit a peculiar behavior, increasing over time
in each phase, with a discontinuity when transitioning from the first phase to the second. Beyond
exhibiting two phases, Hawaii has a small number of deaths, with no deaths occurring in the
transition period between phases. We surmise that the function, while representing the data well,
is peculiar because of the unusual pattern in deaths within Hawaii.
Figure 10: Historical results of the effective reproduction number and death rate
4.6 Sensitivity Analysis for Basic Time-varying Model
Sensitivity analysis is a critical step in transmission modeling of pandemics. Through sensitivity
analysis, researchers and policymakers can determine the impact of changes in input parameters
or assumptions on the model output. By varying input parameters and assessing the resulting
output, sensitivity analysis can identify which parameters or assumptions have the most significant
62
impact on the model's results, such as the number of cases or deaths predicted. This process helps
researchers and policymakers understand how changes in key parameters can affect the spread of
the virus and the effectiveness of various interventions.
One common method for conducting sensitivity analysis is one-way sensitivity analysis. This
method involves varying one input parameter while keeping all other parameters constant to
evaluate its effect on the model output. For example, one-way sensitivity analysis can be used to
assess how changes in the rate of transmission affect the number of cases or deaths predicted by
the model. In our model, we have eight parameters with four parameters (i.e. β
𝑠𝑡𝑎𝑟𝑡 , β
𝑒𝑛𝑑 , 𝑚 , 𝑎 )
related to case and four parameters (i.e. 𝛼 𝑠𝑡𝑎𝑟𝑡 , 𝛼 𝑒𝑛𝑑 , 𝑛 , 𝑏 ) related to death. In order to check the
significance of each parameter to the case and death, we vary each parameter by ±5% while keep
other parameters constant as the best-fitted values. We sample each parameter 500 times within
this ± 5% range, and compute the maximum discrepancy in case/death numbers on the 30th day
since the onset. This discrepancy is represented as ∆
𝑝𝑎𝑟𝑎 = 𝑂 𝑚𝑎𝑥
− 𝑂 𝑚𝑖𝑛 , where 𝑂 𝑚𝑎𝑥
and 𝑂 𝑚𝑖𝑛
denote the highest and lowest case/death numbers among the 500 samples on the 30
th
day,
respectively. For a comparative analysis of significance across different parameters, we normalize
each parameter's maximum discrepancy (∆
𝑝𝑎𝑟𝑎 ) among the eight parameters. This normalized
significance is denoted as 𝛼 𝑝𝑎𝑟𝑎 =
∆
𝑝𝑎𝑟𝑎 Σ∆
𝑝𝑎𝑟𝑎 . Figure 11 and Figure 12 illustrate the significance of
the transmission parameter on case and death numbers. Here, the height of each color bar signifies
the level of significance associated with the corresponding parameter.
63
Figure 11: Significance of transmission parameter on the cases
Figure 12: Significance of transmission parameter on the deaths
In Table 5, we observe that the four most significant parameters influencing the number of cases
are β
𝑠𝑡𝑎𝑟𝑡 , β
𝑒𝑛𝑑 , 𝑚 , 𝑎 , which are all related to case transmission. Conversely, the parameters
associated with death (𝛼 𝑠𝑡𝑎𝑟𝑡 , 𝛼 𝑒𝑛𝑑 , 𝑛 , 𝑏 ) exhibit minimal impact on the number of cases. Although
the ranking of the first two parameters (β
𝑠𝑡𝑎𝑟𝑡 , β
𝑒𝑛𝑑 ) may vary across states, their significance
levels remain comparable.
Table 5: Rank of importance of transmission parameter on the cases
State Rank1 Rank2 Rank3 Rank4 Rank5 Rank6 Rank7 Rank8
Alabama 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Alaska 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
64
Arizona 𝑎 𝛽 𝑒𝑛𝑑 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Arkansas 𝛽 𝑒𝑛𝑑 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
California 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Colorado 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼
Connecticut 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑘 𝛼
Delaware 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑎 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
District of
Columbia
𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑎 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
Florida 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Georgia 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼
Hawaii 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑎 𝛽 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡
Idaho 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Illinois 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑎
Indiana 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
Iowa 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Kansas 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼 𝑎
Kentucky 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼 𝑎
Louisiana 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼
Maine 𝑎 𝛽 𝑒𝑛𝑑 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Maryland 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑎 𝑘 𝛼
Massachusetts 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼
Michigan 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝑚 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑘 𝛼
Minnesota 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼 𝑎
Mississippi 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼 𝑎
Missouri 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑘 𝛼
Montana 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Nebraska 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑚 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Nevada 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
New
Hampshire
𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
New Jersey 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
New Mexico 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
New York 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
North Carolina 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
North Dakota 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Ohio 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑎 𝑘 𝛼
Oklahoma 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
Oregon 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Pennsylvania 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Rhode Island 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝑎 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑘 𝛼
South Carolina 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
South Dakota 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Tennessee 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
65
Texas 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Utah 𝑎 𝛽 𝑒𝑛𝑑 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡
Vermont 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼 𝑎
Virginia 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑘 𝛼 𝑎 𝛼 𝑒𝑛𝑑
Washington 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑘 𝛼
West Virginia 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑘 𝛼
Wisconsin 𝛽 𝑒𝑛𝑑 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Wyoming 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝑥 𝛼 𝑘 𝛼 𝛼 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡
Regarding the influence of the eight parameters on the number of deaths in Table 6, it is evident
that the four case-related parameters (β
𝑠𝑡𝑎𝑟𝑡 , β
𝑒𝑛𝑑 , 𝑚 , 𝑎 ) continue to play a critical role. This is
primarily because the number of cases determines the baseline number of deaths. However, when
comparing the local sensitivity analysis for cases and deaths, it becomes apparent that the death-
related parameters (𝛼 𝑠𝑡𝑎𝑟𝑡 , 𝛼 𝑒𝑛𝑑 , 𝑛 , 𝑏 ) have a more pronounced effect on the number of deaths.
Table 6: Rank of importance of transmission parameter on the deaths
State Rank1 Rank2 Rank3 Rank4 Rank5 Rank6 Rank7 Rank8
Alabama 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑚 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Alaska 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Arizona 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Arkansas 𝛽 𝑒𝑛𝑑 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑎 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑘 𝛼
California 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Colorado 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑎 𝑘 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼
Connecticut 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
Delaware 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑎 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
District of
Columbia
𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑎 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
Florida 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Georgia 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼
Hawaii 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑎 𝛼 𝑒𝑛𝑑 𝛽 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡
Idaho 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Illinois 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑘 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑎
Indiana 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
Iowa 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑎 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
Kansas 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑥 𝛼 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑘 𝛼 𝑎
Kentucky 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼 𝑎
Louisiana 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑘 𝛼
Maine 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Maryland 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑎 𝑘 𝛼
Massachusetts 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑎 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Michigan 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝑚 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
66
Minnesota 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼 𝑎
Mississippi 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝑘 𝛼 𝑎
Missouri 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Montana 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
Nebraska 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑚
Nevada 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
New
Hampshire
𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
New Jersey 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
New Mexico 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
New York 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
North Carolina 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
North Dakota 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑚 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Ohio 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑎 𝑘 𝛼
Oklahoma 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
Oregon 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Pennsylvania 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Rhode Island 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
South Carolina 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
South Dakota 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑥 𝛼 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Tennessee 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼
Texas 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Utah 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑚 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡
Vermont 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑎
Virginia 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝑘 𝛼 𝑎 𝛼 𝑒𝑛𝑑
Washington 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑎 𝑚 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼
West Virginia 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝑚 𝛽 𝑒𝑛𝑑 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼
Wisconsin 𝑚 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑎 𝑥 𝛼 𝛼 𝑠𝑡𝑎𝑟𝑡 𝛼 𝑒𝑛𝑑 𝑘 𝛼
Wyoming 𝑎 𝛽 𝑠𝑡𝑎𝑟𝑡 𝛽 𝑒𝑛𝑑 𝑥 𝛼 𝑘 𝛼 𝛼 𝑒𝑛𝑑 𝑚 𝛼 𝑠𝑡𝑎𝑟𝑡
Another method is multi-way sensitivity analysis, which examines the interactions between
multiple input parameters and their effect on the model output. Multi-way sensitivity analysis can
help identify the combined effects of multiple input parameters, which can provide more
comprehensive insights into the factors driving the spread of the virus. Monte Carlo simulation
methods are increasingly popular in the field of epidemiology and data science for conducting
sensitivity analysis. Monte Carlo methods involve generating multiple sets of input parameters by
randomly sampling from probability distributions. This approach allows for the assessment of the
uncertainty associated with input parameters and the propagation of that uncertainty through the
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model to obtain a distribution of model outputs. Monte Carlo sensitivity analysis can provide a
more comprehensive understanding of the uncertainty associated with model predictions, which is
critical for informing decision-making in pandemic response planning.
One of the advantages of Monte Carlo methods over other sensitivity analysis methods is that they
can handle complex models with many input parameters and non-linear relationships between
those parameters. Monte Carlo methods can also account for correlation between input parameters
and identify the most influential parameters on the model output. This approach enables the
identification of the most critical factors that drive the spread of the virus, such as the reproduction
number (R0), the rate of transmission, the incubation period, and the severity of the disease.
In pandemic modeling, Monte Carlo sensitivity analysis can help policymakers develop more
effective strategies for controlling the spread of the virus. By understanding the impact of changes
in key parameters on the model output, policymakers can develop targeted interventions to
mitigate the spread of the virus. For example, policymakers can use Monte Carlo sensitivity
analysis to assess the impact of different social distancing measures, vaccination campaigns, or
targeted testing on the spread of the virus. Overall, sensitivity analysis is an essential tool for
modeling the spread of pandemics, and Monte Carlo methods can provide a more comprehensive
understanding of the uncertainty associated with model predictions, enabling informed decision-
making in pandemic response planning.
In our analysis, we utilize the Monte Carlo simulation method to assess the combinational effect
of parameter uncertainties on the two primary outputs of each state: Cases and Deaths. Each
parameter is assumed to follow a uniform distribution within a range of ± 5% of their optimal fitted
values. For every simulation iteration, we randomly draw samples of these parameters from their
respective distributions and incorporate them into our time-varying transmission model. By
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conducting 3,000 Monte Carlo simulations, we are able to effectively evaluate the collective
impact of parameter uncertainties and provide a robust analysis of the model's outcomes. We
highlight the results from four representative states (California, Florida, New York, Michigan),
showcasing the 5th to 95th percentile range of both cases and deaths in Figure 13 and Figure 14.
Figure 13: Simulation results of the 5th to 95th percentile range of cases
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Figure 14: Simulation results of the 5th to 95th percentile range of deaths
In our Monte Carlo Analysis, we discovered that even minor fluctuations (±5%) in the parameters
can result in significant changes in healthcare outcomes. This finding highlights the inherent
challenges associated with forecasting future transmission dynamics of infectious diseases, as
transmission parameters can be easily altered by a multitude of factors. For example, the
emergence of new virus strains, local events, government policies, and environmental conditions
can all lead to sudden shifts in transmission rates.
Likewise, death rates can be influenced by various factors, such as the availability of healthcare
resources, advancements in treatment protocols, or the emergence of more virulent strains. The
dynamic nature of these factors further complicates the prediction of disease transmission, as each
element can have a considerable impact on the overall trajectory of the outbreak. Additionally,
changes in individual and collective behaviors, such as adherence to social distancing guidelines,
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mask-wearing, and hygiene practices, can have substantial effects on transmission rates, further
contributing to the complexity of predicting future disease dynamics.
Considering the multitude of factors that can influence transmission parameters, it is not
implausible that these parameters may change by more than 5% within a day or two. Such sudden
shifts can significantly increase the uncertainty associated with forecasting future disease trends.
4.7 Conclusion and Discussion
In this chapter, we presented an extension of the SEIRD model, which incorporates dynamic
changes in death and transmission rates over time using a continuous Sigmoid function. We
hypothesized that these rates change continuously, rather than abruptly, in response to public
health policies or treatment implementation. Our model demonstrated a strong fit to historical data
for the early months of the pandemic in the United States, with a median RRMSE of 1.33% for
deaths and 1.88% for cases across the 50 states. The median effective reproduction rate at the onset
of the pandemic was 2.87, which we estimated had dropped below 1 for all states by July 28, 2020.
We observed that states with poorer model fits typically experienced multiple waves of the disease.
To account for these discrepancies, we proposed a multi-phase extension of the model, in which
transitions between phases are marked by changes in public health policy. Applying this two-phase
model to Hawaii as a case study, we observed a significant improvement in the model's accuracy,
with RRMSE values decreasing to 2.5% for cases and 2.7% for deaths.
A key advantage of our model is the minimal number of parameters required to represent dynamic
changes in transmission and death rates, enabling efficient quantification of regional and temporal
differences in disease spread and outcomes. By examining historical trends, our model offers
valuable insights into how variations in simple parameters can influence the number of cases and
deaths, informing future policy and intervention strategies.
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Additionally, our sensitivity analysis underscores the importance of certain parameters in shaping
the predicted number of cases and deaths. This understanding can aid researchers and
policymakers in anticipating potential shifts in disease transmission and devising effective
interventions. Moreover, our Monte Carlo simulations reveal the substantial impact of
uncertainties in parameter values on model outcomes, highlighting the difficulty in forecasting of
the future trend.
In conclusion, our extended SEIRD model, which accounts for continuous changes in transmission
and death rates, provides a powerful tool for understanding the complex dynamics of infectious
diseases and guiding evidence-based public health decision-making.
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Chapter 5 Integration of Dynamic Modeling with Spatial
Interaction and Effect Analysis
5.1 Introduction
Transportation plays a crucial role in the transmission of pandemics such as COVID-19. The virus
can be spread through respiratory droplets that are released when an infected person talks, coughs,
or sneezes. These droplets can then be inhaled by other individuals in close proximity to the
infected person. Transportation modes such as buses, trains, and airplanes are high-risk areas for
the transmission of the virus, as they often involve large numbers of people in enclosed spaces for
extended periods of time. In addition, long-distance travel can significantly impact the
transmission of diseases to new areas. When people travel long distances, they can bring infectious
agents with them, such as bacteria, viruses, and parasites. Modes of transportation, such as planes,
trains, and ships, can increase the risk of disease transmission due to the close proximity of
travelers and the confined spaces they share. Additionally, travelers may come into contact with
infectious agents through contaminated surfaces or by interacting with infected individuals. These
agents can then spread to new populations and new area, potentially causing outbreaks of disease.
Therefore, studying the impact of long-distance travel on the transmission of diseases is crucial
for understanding how infectious diseases can spread across different regions and populations.
This knowledge can inform public health policies and strategies aimed at preventing and
controlling the spread of infectious diseases. Studying the impact of long-distance travel on disease
transmission can also inform broader discussions about global health and the interconnectedness
of populations around the world. As travel becomes more common and widespread, it is
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increasingly important to understand how diseases can be transmitted across borders and how to
prevent the spread of infectious agents.
This chapter aims to investigate the transmission of diseases through spatial interaction, with a
specific focus on state-level travel. We propose transmission export index related to transportation
to assess the impact of long-distance travel on disease transmission. By gaining a better
understanding of the effects of travel on disease transmission, we can analyze historical disease
outbreaks and develop effective strategies for preventing and controlling future outbreaks.
5.2 Multi-Regional Dynamic Modeling with Spatial Interaction
5.2.1 Transportation Data Collection and Its Challenge
In addition to fundamental data pertaining to disease transmission, including case and mortality
statistics and basic information about the causative pathogen, transportation data is an essential
variable for examining the effect of spatial interactions on the spread of infectious diseases. The
COVID-19 pandemic has presented significant challenges for collecting transportation data and
analyzing the impact of travel on disease transmission. The pandemic has caused a sharp decrease
in travel activity, with many countries implementing travel restrictions and lockdown measures to
slow the spread of the virus. This has made it difficult to collect reliable and comprehensive
transportation data to analyze the effects of travel on disease transmission.
First, many traditional data sources, such as surveys and manual counts, may not be feasible due
to social distancing measures and restrictions on non-essential activities. This makes it difficult to
obtain accurate information on travel patterns and transportation demand.
Moreover, the pandemic has also changed the way people travel. With more people working from
home, there has been a shift from public transit to private vehicles, which may not be captured in
traditional transportation data. Additionally, the pandemic has also led to changes in the timing
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and frequency of travel, which may further complicate the collection and analysis of transportation
data.
Another challenge is the issue of privacy and data protection. Collecting transportation data,
particularly data related to individuals' movements, raises concerns about privacy and data
protection. This can make it difficult to obtain the necessary data to conduct meaningful analyses
of the impact of travel on disease transmission, particularly in countries with strict data protection
laws.
To address these challenges, transportation researchers and practitioners have turned to alternative
data sources, such as mobile phone location data, to track changes in transportation patterns during
the pandemic. With the help of innovative data sources and methods, it is possible to gain insights
into the impact of COVID-19 on transportation systems. This knowledge is crucial for developing
effective policies and strategies to address the ongoing pandemic and future public health crises.
After a thorough search, we found two main data sources for the spatial interactions: i) Trips by
Distance Data, ii) The COVID-19 Impact Analysis Platform.
i) Trips by Distance Data
Bureau of Transportation Statistics collected and curated data on the number of trips taken in the
United States by distance, mode of transportation, and purpose of trip, provided on the website of
https://data.bts.gov/Research-and-Statistics/Trips-by-Distance/w96p-f2qv. The data is available
for the years 2019 to 2022, and the daily travel estimates are based on a merged mobile device
data panel that addresses issues with geographic and temporal variation.
Trips are defined as movements that include a stay of longer than 10 minutes at an anonymized
location away from home, and the data captures travel by all modes of transportation. It is
considered multiple trips when a movement involves multiple stops of more than 10 minutes
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before returning home. The data is analyzed at the national, state, and county levels, and a
weighting procedure is used to ensure the sample of mobile devices is representative of the entire
population in a given area. To protect confidentiality and support data quality, data for a county is
not reported if there are fewer than 50 devices in the sample on any given day.
It is important to note that the data is experimental and may not meet all quality standards. However,
these experimental data products are created to provide valuable insights to data users in the
absence of other relevant products. We will combine the dataset with the following dataset
provided by the COVID-19 Impact Analysis Platform to generate the spatial interaction flow.
ii) The COVID-19 Impact Analysis Platform
The COVID-19 Impact Analysis Platform is developed by Maryland Transportation Institute (MTI)
and Center for Advanced Transportation Technology Laboratory (CATT Lab) cooperatively,
which is a comprehensive data analysis tool designed to provide insights into the impact of the
COVID-19 pandemic on communities across the United States. The platform provides a range of
data and analytical tools, including interactive maps, visualizations, and dashboards, to help users
better understand the spread of the virus and its impact on various social and economic indicators.
One of the key features of the platform is its ability to integrate multiple data sources, including
public health data, mobility data, and socioeconomic data, to provide a more comprehensive
picture of the pandemic's impact. Specifically, the mobility data tracks daily visits to different
types of locations, such as retail and recreation areas, transit stations, workplaces, and grocery
stores, and compares them to pre-pandemic levels. The mobility data is derived from anonymized
and aggregated data from mobile devices, such as smartphones and tablets, that have opted into
location tracking services. The data is aggregated at the county level in the United States, and at
the national level for other countries. For the analysis of spatial interactions, the platform
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specifically provides the state/county level percentage of out-of-state/out-of-county trips per day
from Jan 1
st
, 2020 to April 30
th
, 2021.
We can combine the mobility data provided by the COVID-19 Impact Analysis Platform and the
daily trips from BTS to calculate the daily out-of-state trips for each state, which can be useful for
understanding how people's movements have changed during the pandemic, and for identifying
areas where social distancing measures may be effective or where there may be increased risk of
COVID-19 transmission. However, it's important to note that both the mobility datasets are based
on a sample of mobile devices and may not be representative of the entire population, and that the
datasets are anonymized and aggregated to protect user privacy.
5.2.2 Gravity Modeling of the State-Level Transportation
Based on the best datasets we mention above, we can only get the daily out-of-state trips for each
state. However, for the analysis of spatial interaction, we should also calculate the daily trips that
go into each state. One common model that is widely applied in various transportation contexts to
estimate the flow of passenger or goods between cities, which is based on Newton's law of
universal gravitation. The model is relatively straightforward and involves estimating the
transportation flow between two locations based on their mass and distance.
The basic principle behind the gravity model is that the flow of transportation between two
locations is proportional to the product of their masses and inversely proportional to the distance
between them.
The gravity model is widely used in transportation modeling due to its simplicity and ability to
provide accurate predictions of transportation flows. One of the key advantages of the gravity
model is that it can be applied to a wide range of transportation contexts, including freight
transportation, passenger transportation, and tourism. The model can also be applied to various
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transportation modes, including air, sea, and land transportation. Additionally, the gravity model
can be adapted to include other variables that may influence transportation flows, such as
population density, income, or trade barriers. The model's simplicity and ability to provide accurate
predictions of transportation flows make it a valuable tool for transportation planners and
policymakers seeking to understand and optimize transportation networks.
Despite its usefulness, the gravity model has some limitations that should be considered. One of
the main limitations of the model is that it assumes that transportation flows are solely dependent
on the mass and distance between two locations, and it does not consider other discrete factors that
may affect transportation demand, such as differences in regional economies, cultural preferences,
or political factors. In addition, another limitation of the gravity model is that it may be challenging
to estimate accurate values for the model parameters. For instance, estimating the exponents of the
model can be challenging since they are typically determined through statistical regression analysis,
and the estimates may vary depending on the data used for calibration. Furthermore, the
parameters' values may also vary depending on the transportation mode or the nature of the goods
being transported.
Despite these limitations, the gravity model remains a valuable tool for transportation modeling
and has been extensively used in research and policy analysis. Various extensions of the model
have been proposed to address some of the limitations mentioned earlier. For example, some
researchers have proposed incorporating the network topology of transportation systems or
accounting for heterogeneity in transportation preferences across different population groups.
In conclusion, the gravity model is a powerful and versatile tool for transportation modeling that
has been widely used in various transportation contexts. The model's simplicity and ability to
provide accurate predictions of transportation flows make it a valuable tool for transportation
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planners and policymakers seeking to optimize transportation networks. However, it is essential
to consider the model's limitations and potential extensions when applying it to real-world
transportation planning and policy analysis.
The gravity model has several common features, including the Gross Domestic Product (GDP)
and distance. The GDP is a measure of the economic activity within a particular country. The GDP
is used in the gravity model to represent the mass of a particular location. A higher GDP is often
associated with a higher demand for transportation goods and services. This is because a higher
GDP typically implies higher economic activity, which may increase the demand for transportation
of goods and services. In the gravity model, the GDP of each location is used to estimate the mass
of the location, which is then used to estimate the flow of transportation between the two locations.
Distance is another essential feature in the gravity model. The model posits that transportation
flows between two locations are inversely proportional to the distance between them. This means
that as the distance between two locations increases, the flow of transportation between them
decreases. Distance is often considered a critical factor in transportation demand and can
significantly impact transportation infrastructure planning and investment decisions. In the gravity
model, distance is incorporated into the model through the denominator of the formula, where
transportation flows decrease as the distance between two locations increases.
The formula for the gravity model of trip distribution is expressed in Equation (14):
𝑀 𝑖𝑗
= 𝑀 𝑖 ∗
𝐺 𝑗 𝛼 𝐷 𝑖𝑗
𝛾 ∑
𝐺 𝑘 𝛼 𝐷 𝑖𝑘
𝛾 𝑘
(14)
Where 𝑀 𝑖𝑗
represents the flow of trips from region 𝑖 to region 𝑗 , 𝐺 𝑘 is the GDP of the location 𝑘 ,
𝐷 𝑖𝑘
represents the distance between the region 𝑖 and region 𝑘 , and α and γ are exponents that
determine the relative influence of the variables.
𝐺 𝑗 𝛼 𝐷 𝑖𝑗
𝛾 indicates the attraction index of region 𝑗
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calculated by gravity model. The ratio
𝐺 𝑗 𝛼 𝐷 𝑖𝑗
𝛾 ∑
𝐺 𝑘 𝛼 𝐷 𝑖𝑘
𝛾 𝑘 shows the proportion of the total trips goes from
region 𝑖 to region 𝑗 . Due to the limitation of real trip flow data from region 𝑖 to region 𝑗 , we cannot
fit the best-fitted values of α and γ for the real case. However, a reasonable assumption that follows
the correlation is both the α and γ equal to 2, which is same as the Newton’s gravity model.
5.2.3 Dynamic Modeling with Multi-Regional Spatial Interaction
Based on the dynamic modeling with time-varying transmission and fatality rates, we will further
increase the applicability of model (1) to incorporate the effect of multi-regional spatial interaction.
In general, state-level transportation can influence the movement of people between regions,
increasing the likelihood of contact between individuals from different regions and leading to the
spread of the virus.
For the susceptible population, who have not been infected with the virus and can become infected
if exposed, state-level transportation can increase the size of the susceptible population by bringing
individuals from different regions into contact with each other, increasing the likelihood of
exposure to the virus.
For the exposed population, who have been infected with the virus but have not yet developed
symptoms, state-level transportation can increase the size of the exposed population by facilitating
the movement of infected individuals across regions, increasing the likelihood of exposure to
susceptible individuals in other regions. For example, individuals who are infected with the virus
and travel through airports or highways may spread the virus to other regions, leading to an
increase in the number of individuals in the exposed population.
Meanwhile, some states that are geographically close to each other tend to have more spatial
interactions compared to far-connected states. For example, New York states have higher volume
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of transportation with New Jersey due to geographic proximity, strong transportation infrastructure,
and cultural and social connections. Hence, for the simplicity of the modeling, it is reasonable to
assume the effect of state-level transportation on the transmission is consistent within the same
region.
We use the Standard Federal Regions to aggregate the 50 united states into 10 parts. The regions
were defined based on geographic, economic, and cultural factors. They were designed to promote
efficient and effective delivery of federal programs and services by bringing together federal
agencies, state and local governments, and private organizations to work collaboratively and
address regional issues and concerns. The states within the same region have proven to be able to
share resources, expertise, and best practices across state lines and jurisdictions. The 10 Standard
Federal Regions (shown in Figure 15) are as follows:
Region 1: Connecticut, Maine, Massachusetts, New Hampshire, Rhode Island, and Vermont.
Region 2: New Jersey, New York, Puerto Rico, and the Virgin Islands.
Region 3: Delaware, Maryland, Pennsylvania, Virginia, West Virginia, and the District of
Columbia.
Region 4: Alabama, Florida, Georgia, Kentucky, Mississippi, North Carolina, South Carolina, and
Tennessee.
Region 5: Illinois, Indiana, Michigan, Minnesota, Ohio, and Wisconsin.
Region 6: Arkansas, Louisiana, New Mexico, Oklahoma, Texas.
Region 7: Iowa, Kansas, Missouri, and Nebraska.
Region 8: Colorado, Montana, North Dakota, South Dakota, Utah, and Wyoming.
Region 9: Arizona, California, Hawaii, Nevada, the Pacific Trust Territories.
Region 10: Alaska, Idaho, Oregon, and Washington.
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Figure 15: Ten regions defined by the Federal Emergency Management Agency
in the continental United States and territories
According to the above assumptions, the modified model is shown in Equation (15).
∂S
i
(t)
∂t
= −β(𝑡 ) ∙ 𝐼 𝑖 (𝑡 ) ∙
𝑆 𝑖 (𝑡 )
𝑁 𝑖 + ∑ μ
𝑘𝑗
∑
𝑀 𝑖𝑗
𝑆 𝑗 𝑁 𝑗 − 𝐼 𝑗 𝑗 ≠𝑖 𝑗 ≠𝑖 − 𝜇 𝑘𝑖
∑
𝑀 𝑗𝑖
𝑆 𝑖 𝑁 𝑖 − 𝐼 𝑖 𝑗 ≠𝑖
𝜕 𝐸 𝑖 (𝑡 )
𝜕𝑡
= β(𝑡 ) ∙ 𝐼 𝑖 (𝑡 ) ∙
𝑆 𝑖 (𝑡 )
𝑁 𝑖 − σ ∙ 𝐸 𝑖 (𝑡 ) + ∑ μ
kj
∑
𝑀 𝑖𝑗
𝐸 𝑗 𝑁 𝑗 − 𝐼 𝑗 𝑗 ≠𝑖 𝑗 ≠𝑖 − 𝜇 𝑘 i
∑
𝑀 𝑗 𝑖 𝐸 𝑖 𝑁 𝑖 − 𝐼 𝑖 𝑗 ≠𝑖
𝜕 𝐼 𝑖 (𝑡 )
𝜕𝑡
= σ ∙ 𝐸 𝑖 (𝑡 ) − (1 − 𝜏 ∙ α(𝑡 )) ∙ γ𝐼 𝑖 (𝑡 ) − τ ∙ α(𝑡 ) ∙ ρ ∙ 𝐼 𝑖 (𝑡 )
𝜕 𝑅 𝑖 (𝑡 )
𝜕𝑡
= τ ∙ (1 − α(𝑡 )) ∙ γ ∙ 𝐼 𝑖 (𝑡 )
𝜕 𝐷 𝑖 (𝑡 )
𝜕𝑡
= τ ∙ α(𝑡 ) ∙ ρ ∙ 𝐼 𝑖 (𝑡 )
(15)
where 𝑆 𝑖 (𝑡 ), 𝐸 𝑖 (𝑡 ), 𝐼 𝑖 (𝑡 ), 𝑅 𝑖 (𝑡 ), 𝐷 𝑖 (𝑡 ) 𝑎𝑛𝑑 𝑁 𝑖 are the susceptible, exposed, infected, recovered, dead
and total population in region 𝑖 at time t. θ is defined as the protection rate of the vaccine,
indicating the percentage of vaccinated people who are truly immune. τ is a scalar factor that
shows the reduction effect of the vaccination on the death rate. Spatial interaction between cities
is represented by the daily number of people traveling from region 𝑗 to region 𝑖 and an adjustable
82
factor 𝜇 𝑘𝑗
. It is still possible that infectious people, who have symptoms and is conscious about
their status infection, still travel locally and sometimes between different regions. However, for
simplicity, we assume the infectious people will not commit a state-level travel in general. The
rare case of the traveling infected people will be represented by the adjustable factor 𝜇 𝑘𝑗
.
Meanwhile, given that the total number of individuals moving in and out of region 𝑖 is substantially
smaller than the region's overall population, we've simplified our model by disregarding changes
in total population due to multi-regional transportation.
Comparing to the parameter estimation for a single state, finding the best-fitted values for the
transmission parameters simultaneously for all 50 states is more complicated and time-consuming.
For a single state, we only have 8 parameters to be fitted. However, due to the travel of exposed
and susceptible population, multi-regional spatial interactions make the compartments (i.e.
susceptible, exposed, infectious, recovered) for one state depend on the compartments of other
states. In order to get the best-fitted values for the transmission parameters considering the spatial
interaction, we need to simulate the 50 states together and find out all 410 parameters at the same
time. Parameter estimation procedure from Chapter 4 basically consists of two parts: i) calculate
the value for the cases and death by solving the transmission differential equations with the scipy
odeint solver, ii) iteratively optimize the transmission parameters with Levenberg–Marquardt
algorithm. However, solving the 200 differential equations using the fourth-order Runge-Kutta
method with the odeint solver is time-consuming. Additionally, the fourth-order Runge-Kutta
method necessitates interpolation of data points between two days, which significantly prolongs
the simulation time and introduces the issue of gradient vanishing due to the implicit function
involved in the interpolation process. Therefore, we opted to replace the third-party ode solver
with our own custom function that employs the first order Runge-Kutta method to solve the
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ordinary differential equations. This modification reduces the simulation time from 10 hours to 2
hours while yielding improved fitting results.
5.2.4 Model Accuracy
We fit the model with the dataset of 7-day moving average cases and deaths for the 50 states,
provided by the COVID-19 tracking project lead by The Atlantic (derived from the Center for
Disease Control), for each 30 days from 03/15/2020 to 10/15/2020. The fitting accuracy across
all states is presented in Figure 16, measured by the relative root mean square error (RRMSE)
defined in Equation (12).
The average fitting accuracy of the reported cases over 7 months ranges from 0.54% to 3.78% and
of the reported deaths ranges over 7 months from 0.24% to 2.49%. The average and median
RRMSEs for cases are 1.54% and 1.48%; for deaths, the average and median values are 1.20%
and 1.14%. Figure 16 shows the average RRMSE for cases and deaths over 7 months across 50
states of the dynamic model with multi-regional spatial interaction.
Figure 16: Average RRMSE for cases and deaths over 7 months across 50 states
Figure 17 and Figure 18 display the fitted results for COVID-19 cases and deaths in example states
(Georgia, New Jersey, Florida, and Maryland), during the period from October 15, 2020, to
November 15, 2020.
84
Figure 17:Fitting results of case number for Georgia, New Jersey, Florida, and Maryland
Figure 18:Fitting results of death number for Georgia, New Jersey, Florida, and Maryland
85
In summary, the dynamic modeling with multiregional spatial interaction, demonstrates a high
degree of accuracy in capturing the historical transmission dynamics of infectious diseases. This
method effectively accounts for the complexities and interactions between various regions, leading
to a more comprehensive understanding of the factors influencing disease spread and the
effectiveness of control measures.
5.3 Effect Evaluation of Transportation on the Multi-Regional
Transmission
5.3.1 Transmission Export Index
According to the model (15), the parameter 𝜇 𝑖 represents the average number of individuals from
other regions who could potentially contract the infection upon contact with a single infectious
traveler originating from region 𝑖 . The parameter 𝛽 𝑖 , on the other hand, denotes the average
number of people who could become infected through contact with a single infectious individual
within region 𝑖 , thus reflecting the local transmissibility of the disease.
To evaluate the risk of disease transmission from one region to another due to multi-regional
spatial interaction, we introduce a transmission export index for region 𝑖 , defined as 𝛽 𝑖 ∗ 𝜇 𝑖 . This
index incorporates both 𝜇 𝑖 and 𝛽 𝑖 , with 𝜇 𝑖 accounting for the potential spread of the infection to
new regions by an infectious traveler and 𝛽 𝑖 indicating the local increase in the number of infected
individuals.
The transmission export index provides valuable insights into the extent to which a region poses a
transmission risk to other regions through travel. If a region is experiencing a surge in transmission
(i.e., high 𝛽 𝑖 ) and travelers from that region exhibit a higher propensity to spread the disease to
other areas (i.e. high 𝜇 𝑖 ), it is considered to pose a greater risk to other regions via travel.
86
Policymakers in low-risk regions must be proactive in implementing strict travel regulations or
quarantine measures for travelers originating from such high-risk areas.
In summary, the transmission export index is a vital tool in assessing the potential risk of disease
transmission from one region to another through multi-regional spatial interaction. By considering
both the local transmissibility (𝛽 𝑖 ) and the ability of infectious travelers to spread the disease to
new regions (𝜇 𝑖 ), this index serves as a valuable guide for healthcare experts and policymakers
alike. By identifying high-risk regions, appropriate interventions such as stringent travel
restrictions or quarantine measures can be put in place, ultimately mitigating the spread of
infectious diseases and safeguarding public health. Figure 19 shows the heatmap of infectious
export index for all 50 states in the US from 03/15/2020 to 04/15/2020.
Figure 19: Heatmap of infectious export index for all 50 states in the US from 03/15/2020 to 04/15/2020
5.3.2 Causal Analysis for Increment of Transmission Export Index
In the following discussion, we will delve into the possible reasons for the increase in the
transmission export index, focusing on three primary aspects: Coronavirus State Actions, Political
Events, and Festivals/Entertainment events. Our objective is to shed light on the complex interplay
87
of these factors and their potential impact on the transmission of infectious diseases, with the goal
of informing future decision-making processes and guiding public health policy.
To begin, we will examine the various state-level actions and policies enacted in response to the
coronavirus pandemic. These actions encompass a wide range of measures, including the
imposition and relaxation of social distancing protocols, the closure and reopening of public spaces
and businesses, and the adjustment of transportation capacities. By assessing the effectiveness and
potential consequences of these measures, we aim to identify the ways in which they may have
contributed to the observed fluctuations in the transmission export index.
Next, we will explore the role of political events in influencing the transmission of infectious
diseases. During the pandemic, numerous political gatherings, rallies, and protests have taken
place, often attracting large crowds and creating environments conducive to disease transmission.
By examining the specific contexts and circumstances surrounding these events, we hope to better
understand the extent to which they may have affected the transmission export index and the
broader public health landscape.
Finally, we will investigate the impact of festivals and entertainment events on disease
transmission dynamics. These gatherings, which frequently involve large numbers of attendees in
close proximity, have the potential to serve as major drivers of infectious disease spread. We will
consider various factors, such as event size, location, and duration, as well as the implementation
of preventive measures, in order to assess the potential influence of festivals and entertainment
events on the transmission export index.
Through a comprehensive analysis of these three aspects, we aim to provide a nuanced
understanding of the factors contributing to the increase in the transmission export index during
the coronavirus pandemic. This in-depth exploration will not only shed light on the complex
88
dynamics at play but also serve as a valuable resource for policymakers and public health officials
as they navigate the ongoing challenges posed by infectious diseases.
5.3.2.1 State Actions in Response to Transmission Export Index Increases
Table 7 summaries the top 2 states with the most-increased transmission export index for each
month and their related state-level actions and policies that could potentially lead to an increase in
the transmission export index of infectious diseases. These actions can be grouped into several
categories: relaxation of social distancing measures, reopening of public spaces and businesses,
expansion of transportation capacity, and easing of regulations for certain industries.
Firstly, some actions, such as the closure of state parks and forests (Executive Order 118) and New
York City playgrounds, initially aimed to strengthen social distancing measures. However, the
subsequent reopening of county beach parks in the County of Hawaii and the approval of
businesses and operations on O'ahu represent a relaxation of these measures, which could
potentially contribute to an increased transmission export index.
Similarly, the expansion of allowable outdoor recreational activities (Executive Order 20-38) and
the resumption of contact practices and competitions in outdoor settings for organized sports
(Executive Order No. 168) may also lead to increased transmission rates. These activities may
involve close contact between individuals and a higher likelihood of disease transmission.
Another factor that could contribute to the increase of the transmission export index is the
reopening of public spaces, such as allowing food trucks to operate at highway rest stops in
Minnesota (Executive Order 20-49). This may encourage people to gather in these locations,
increasing the possibility of disease transmission.
In the realm of transportation, the lifting of 50% capacity limits on NJ TRANSIT and private-
carrier buses, trains, and light rail vehicles (Executive Order No. 165) could lead to increased
89
transmission risks. As more individuals use public transportation, the likelihood of close contact
between passengers and subsequent disease transmission may rise.
Moreover, easing regulations for certain industries, such as providing emergency relief from
regulations for motor carriers and drivers operating in Minnesota (Executive Order 20-80), could
potentially contribute to the increase of the transmission export index. This might result from
increased movement and interactions between individuals in these industries.
On the other hand, some state-level actions may help mitigate transmission risks. For example, the
distribution of more than 4 million masks to businesses, their customers, and those who are unable
to afford or easily obtain one (July 29, 2020) promotes the use of face coverings, which can reduce
disease transmission.
In conclusion, while some state-level actions and policies have been implemented to control the
spread of infectious diseases, others may inadvertently contribute to an increase in the transmission
export index. These actions include the relaxation of social distancing measures, reopening of
public spaces and businesses, expansion of transportation capacity, and easing of regulations for
certain industries. To effectively manage and mitigate disease transmission risks, it is crucial for
state authorities to balance the need for economic recovery with public health considerations.
Table 7: Major state actions in response to transmission export index increases
Date State
Change of
Transmission
Export index
(beta*mu)
Coronavirus State Actions
0320-
0415
New Jersey 1.931885
April 7, 2020 - State & County Parks, Statewide -
Executive Order 118 announced, closing all parks and
forests to enforce social distancing measures.
0320-
0415
New York 1.561676
0415-
0515
Minnesota 1.610151
April 1, 2020 - New York City, New York - Governor
announced the closure of all playgrounds due to
insufficient social distancing compliance.
April 3, 2020 - New York State - Governor introduced a
90
website with daily updates on the state's comprehensive
coronavirus testing data.
0415-
0515
Wisconsin 1.239867
April 20, 2020 - Wisconsin - Health Services Secretary-
designee issued an Emergency Order for the Badger
Bounce Back reopening plan.
April 27, 2020 - Wisconsin - Governor signed Emergency
Order #34, expanding operations for essential businesses
and permitting curbside drop-off for nonessential
businesses.
0515-
0615
Hawaii 0.2480408
May 19, 2020 - County of Hawaii - Governor approved
reopening of county beach parks island-wide with social
distancing restrictions.
May 27, 2020 - Oahu, Hawaii - Governor approved
Mayor's proposal to reopen more businesses and
operations under safety guidelines.
0515-
0615
Nevada 0.1858456
May 26, 2020 – Nevada -The Governor announced that
Nevada is ready to move into Phase 2 of the state’s
Nevada United: Roadmap to Recovery reopening plan on
Friday, May 29.
0615-
0715
New Jersey 2.86914
June 19, 2020 - Statewide - Administration announced
outdoor visits for long-term care facility residents starting
June 21.
July 13, 2020 - New Jersey - Executive Order No. 165
signed, lifting 50% capacity limits on public transit and
requiring carriers to limit vehicles to maximum seated
capacity.
0615-
0715
Minnesota 1.196351
July 13, 2020 - Statewide - Executive Order 20-78 signed,
extending the COVID-19 peacetime emergency.
July 14, 2020 - Statewide - Governor announced $100
million housing assistance program, funded by CARES
Act, to prevent evictions, homelessness, and maintain
housing stability.
0715-
0815
New Jersey 0.48593
July 20, 2020 - New Jersey - Executive Order No. 168
signed, permitting resumption of outdoor contact
practices and competitions for high-risk sports.
0715-
0815
Minnesota 0.4159602
July 17, 2020 - Minnesota - Executive Order 20-80
signed, extending provisions in Executive Order 20-76
for emergency relief for motor carriers and drivers
transporting livestock.
July 29, 2020 - Statewide - Governor highlighted
distribution of over 4 million masks to businesses,
customers, and those unable to afford or obtain a mask.
5.3.2.2 Major Political Events in Response to Transmission Export Index Increases
During the study period from March 15, 2020 to Oct, several major events occurred in the top 2
states with the most-increased transmission export index, summarized in Table 8, which may have
influenced the transmission of the virus.
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1. May 26 George Floyd Protests:
Major protests began in the Minneapolis–Saint Paul area following the murder of George Floyd.
These protests quickly spread to other cities across the United States and around the world. Large
gatherings of protesters, often in close proximity and sometimes without masks or face coverings,
created an environment conducive to the transmission of COVID-19. The risk of transmission
increased due to the difficulty of maintaining physical distancing and proper hygiene practices
during the protests. Additionally, law enforcement's use of tear gas and other crowd control
measures could have exacerbated respiratory issues, further contributing to the spread of the virus.
2. July 7 Primary Elections in New Jersey:
The primary elections in New Jersey, rescheduled from June 2, also posed a potential risk for
COVID-19 transmission. The act of voting typically involves people gathering in polling stations,
standing in lines, and touching shared surfaces such as voting machines and pens. Although
election officials implemented various safety measures, such as social distancing, providing hand
sanitizer, and encouraging the use of face coverings, the risk of transmission could not be entirely
eliminated. Moreover, some voters might have been discouraged from participating due to fear of
infection, impacting overall voter turnout.
3. June 12 Minneapolis City Council Vote:
The Minneapolis City Council's vote to disband the police department and replace it with a
"community" safety department was a significant political event. Although the vote itself likely
had minimal direct impact on the transmission of COVID-19, the associated public meetings,
discussions, and debates could have contributed to the spread. Such gatherings often involve
people in close proximity, speaking passionately and potentially projecting respiratory droplets.
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The risk of transmission would be higher if these gatherings took place indoors or if proper safety
measures, such as wearing masks and maintaining physical distance, were not followed.
In summary, all three of these events had the potential to influence the transmission of COVID-
19. The George Floyd protests, in particular, posed a significant risk due to the large gatherings
and close contact between participants. The primary elections and the Minneapolis City Council
vote also presented potential risks, although safety measures were likely implemented to mitigate
the spread. These events highlight the challenges of balancing essential social and political
activities with public health concerns during a pandemic.
Table 8: Major political events in response to transmission export index increases
Date State
Change of
Transmission
attack index
(beta*mu)
Major Political Events
0320-
0415
New Jersey 1.931885
0320-
0415
New York 0.561676
0415-
0515
Minnesota 1.610151 May 26 - Minneapolis-Saint Paul, Minnesota - Major
protests begin in response to George Floyd's murder.
0415-
0515
Wisconsin 1.239867
0515-
0615
Hawaii 0.2480408
0515-
0615
Nevada 0.1858456
0615-
0715
New Jersey 2.86914
July 7 - New Jersey - Primary elections held, rescheduled
from June 2.
0615-
0715
Minnesota 1.196351
June 12 - Minneapolis, Minnesota - City Council votes to
disband Police Department and replace with a
community safety department, but is prevented by city
charter.
0715-
0815
New Jersey 0.48593
0715-
0815
Minnesota 0.4159602
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5.3.2.3 Major Festivals/Entertainment Events in Response to Transmission Export Index
Increases
The potential for infectious disease transmission at large-scale events and festivals is a significant
concern, given the unique challenges these gatherings present. Several factors contribute to the
risk of disease spread, including the close contact between attendees, indoor venues, shared
surfaces and equipment, food and beverage consumption, travel and accommodation arrangements,
and insufficient hygiene practices. In the subsequent analysis, we will delve into these factors
comprehensively and offer suggestions to reduce the risks linked to festivals and entertainment
events in the top two states with the highest monthly increase in transmission export indices, as
presented in
Table 9.
Large-scale events and festivals, such as CineKink NYC, Queens World Film Festival, Tribeca
Film Festival, Roots and Bluegrass Music Festival, Lakes Jam, Stone Arch Bridge Festival, and
Uptown Art Fair, are characterized by the congregation of vast crowds, leading to increased close
contact between attendees. This proximity can facilitate the transmission of infectious diseases,
particularly those that are airborne or spread through respiratory droplets. Notably, events held in
indoor venues, like the Roots and Bluegrass Music Festival and CineKink NYC, pose an elevated
risk of disease transmission due to limited air circulation and confined spaces.
Another factor that contributes to disease transmission at these events is the presence of shared
surfaces and equipment. For example, events like Ecofest, where visitors interact with exhibits, or
the Jersey Surf Film Festival, which offers surf lessons using shared equipment, may expose
attendees to contaminated surfaces. Food and beverage consumption at events featuring tastings,
such as Restaurant Week La Crosse or the Roots and Bluegrass Music Festival, can also increase
94
the risk of transmission due to shared utensils, plates, or cups, as well as close contact during food
preparation and serving.
Furthermore, the travel and accommodation arrangements associated with these events can
contribute to the spread of infectious diseases. Attendees often travel from different regions or
countries, potentially introducing new disease threats to the event location. Shared
accommodations, such as hotels or hostels, can further facilitate disease transmission.
Compounding these factors, insufficient hygiene practices at large events can exacerbate the risk
of disease spread. Maintaining proper hygiene can be challenging, especially in areas like
restrooms, food service stations, or communal spaces where inadequate handwashing or sanitizing
facilities may be present.
To mitigate the risk of disease transmission during such events, both organizers and attendees
should consider implementing preventive measures. These measures may include promoting
proper hand hygiene, providing sanitizing stations, enforcing physical distancing, requiring masks
or face coverings, implementing health screenings, and ensuring adequate ventilation in indoor
venues. Additionally, event organizers should stay informed about emerging disease threats and
collaborate with local health authorities to make informed decisions regarding event planning and
execution. By adopting these strategies, the risk of disease transmission at large-scale events and
festivals can be significantly reduced.
Table 9: Major festivals/entertainment events in response to transmission export index increases
Date State
Change of
Transmission
Export index
(beta*mu)
Festivals
/Entertainment events
0320-
0415
New Jersey 1.931885
95
0320-
0415
New York 0.561676
March 18-22, 2020 - New York City, NY - CineKink
NYC, a four-day film event celebrating diverse sexuality
with movies, panel discussions, and parties.
March 19-29, 2020 - Queens, NY - Queens World Film
Festival, showcasing innovative films by maverick
filmmakers from around the world.
April 4-5, 2020 - New York City, NY - Ecofest, a free
event featuring alternative energy exhibits, green
vehicles, food, and entertainment in Times Square.
April 15-26, 2020 - Lower Manhattan, NY - Tribeca Film
Festival, offering movie screenings, celebrity talks, and
exclusive content.
0415-
0515
Minnesota 1.610151
0415-
0515
Wisconsin 1.239867
April 17-19, 2020 - Wisconsin - Roots and Bluegrass
Music Festival, a free three-day indoor event with
regional and local bands, workshops, and tastings.
April 20-26, 2020 - La Crosse, WI - Restaurant Week La
Crosse, a week-long food festival celebrating local
restaurants and eateries.
0515-
0615
Hawaii 0.2480408
0515-
0615
Nevada 0.1858456
0615-
0715
New Jersey 2.86914
June 19-20 - Mont Grantez - Sunset Concerts, summer
music event with pop, blues, and jazz in a picturesque
setting.
July - New Jersey - Jersey Surf Film Festival, celebrating
surfing with outdoor film screenings, workshops, surf
lessons, and talks (2020 festival unconfirmed).
0615-
0715
Minnesota 1.196351
June 25-27 - Minnesota - Lakes Jam, featuring two days
of country music and a day of rock.
June 19-21 - Minneapolis, MN - Stone Arch Bridge
Festival, a three-day event celebrating art, food, and live
music over Father's Day weekend.
0815-
0915
New Jersey 0.48593
July 18 - New Jersey - Wonky Town, a post-apocalyptic
themed one-day music festival with immersive
experiences.
0915-
1015
Minnesota 0.4159602
August 7-9 - Uptown, Minnesota - Uptown Art Fair, a
juried arts festival celebrating the Uptown community
with over 380,000 visitors.
5.3.3 Causal Analysis for Decrement of Transmission Export Index
According to Table 10 about the top 2 states with most decrement in transmission export index
every month, a variety of state-level actions and policies have been implemented to curb the
transmission of infectious diseases and reduce the transmission export index. Some common
96
strategies include promoting public health guidance, enacting travel restrictions, and implementing
social distancing measures. The following paragraphs provide a summary and discussion of these
policies.
One common policy implemented by state governments is urging residents to follow guidance
from health authorities such as the Centers for Disease Control and Prevention (CDC) and state
health departments. By promoting adherence to these guidelines, states aim to minimize the spread
of infectious diseases within their jurisdictions. Examples of such policies include encouraging
residents to stay at home as much as possible and extending stay-at-home orders for specified
durations.
Another approach taken by states is the implementation of social distancing measures to limit the
spread of infections. Some of these measures include suspending in-person voting for elections,
issuing shelter-in-place orders for specific counties with increased cases, closing schools for the
remainder of the academic year, and limiting social, community, recreational, leisure, and sporting
gatherings. In some cases, states have permitted the reopening of certain businesses and
establishments, such as salons, barbershops, massage and tattoo parlors, restaurants, and fitness
centers, but only with strict public health measures in place.
States have also introduced phased plans for reopening their economies in a gradual and safe
manner. These plans typically involve assessing the reopening of businesses and activities based
on the level of disease transmission and essential classification. For example, some states have
adopted multi-stage approaches to reopening, with each stage permitting a specific set of
businesses or activities to resume operations under certain conditions. Additionally, some states
have authorized businesses to deny entry to individuals who do not wear masks or face coverings,
further emphasizing the importance of personal protective measures.
97
Travel restrictions have been another key policy employed by states to reduce the transmission
export index. These restrictions often involve mandatory quarantines for travelers entering the
state or requiring travelers to present proof of a negative COVID-19 test prior to their arrival. In
some cases, states have delayed the implementation of pre-travel testing programs or extended the
duration of mandatory quarantines for incoming travelers.
In conclusion, state governments have adopted a range of policies to mitigate the transmission of
infectious diseases and lower the transmission export index. These strategies include promoting
public health guidance, implementing social distancing measures, introducing phased plans for
economic reopening, and imposing travel restrictions. By adopting these measures, states aim to
protect their residents and limit the spread of infections both within their borders and across the
nation.
Table 10: Major state actions in response to transmission export index decreases
Date State
Change of
Transmission
attack index
(beta*mu)
Coronavirus State Actions
0320-
0415
Wisconsin -4.1314
March 20, 2020 - Wisconsin: The Governor urged residents
to follow CDC and state health department guidance to stay
home as much as possible.
April 6, 2020 - Wisconsin: The Governor signed an executive
order postponing in-person voting for the April 7 election
until June 9 and called for a special legislative session to
address the election date.
0320-
0415
Mississippi -3.91885
March 31, 2020 - Mississippi: The Governor issued a shelter
in place order for a county due to increased cases in the
region.
April 14, 2020 - Mississippi: The Governor announced that
schools will remain closed for the rest of the school year.
0415-
0515
Kansas -0.855282
April 15, 2020 - Kansas: The Governor extended the stay-at-
home order until May 1st.
April 30, 2020 - Kansas: The Governor presented a detailed
framework for gradually reopening the economy starting May
4, 2020, with Executive Order 20-29, lifting the statewide
stay-home order in Executive Order 20-16.
May 14, 2020 - Kansas: The Governor signed Executive
Order 20-31, establishing a new "1.5" Phase effective May
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18, 2020, continuing reopening efforts with some restrictions
to prevent community transmission of COVID-19.
0415-
0515
Iowa -0.849734
April 16, 2020 - Iowa: The Governor signed a proclamation
continuing the State Public Health Emergency Declaration,
requiring additional protective measures in Region 6
(Northeastern Iowa), including limiting social and
recreational gatherings.
April 19, 2020 - Iowa: The Governor announced that all
schools will be closed for the remainder of the school year.
May 13, 2020 - Iowa: The Governor signed a proclamation
continuing the Public Health Disaster Emergency, allowing
certain businesses to reopen with restrictions and extending
the prohibition on gatherings of more than 10 people until
11:59 p.m. on May 27, 2020.
0515-
0615
New
Jersey
-4.353733
May 18, 2020 - New Jersey: The Governor unveiled a multi-
stage approach for a responsible and strategic economic
reopening and signed Executive Order No. 147, allowing
certain outdoor activities at recreational businesses and
community gardens with social distancing measures.
June 1, 2020 - New Jersey: The Governor announced that the
state will enter Stage Two on June 15, including outdoor
dining for restaurants and indoor, non-essential retail.
0515-
0615
New York -2.264473
May 20, 2020 - New York: The Governor announced that
religious gatherings of no more than 10 people will be
allowed starting May 21.
May 21, 2020 - New York The Governor announced that
summer school will be conducted through distance learning
and that meal programs and childcare services for essential
employees will continue.
May 28, 2020 - New York: The Governor issued an executive
order allowing businesses to deny entry to individuals not
wearing masks or face coverings.
0615-
0715
Hawaii -0.1641658
June 24, 2020 - Hawai‘i: The Governor announced a pre-
travel testing program for out-of-state travelers starting Aug.
1 and approved the proposal to allow singing and playing of
wind instruments at indoor and outdoor restaurants/bars with
restrictions.
July 13, 2020 - Hawai‘i: The Governor delayed the launch of
the pre-travel testing program to Sept. 1, extending the
mandatory 14-day quarantine for travelers entering the state
until then.
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0615-
0715
Arizona -0.1007183
June 29, 2020 - Arizona: The Governor issued an executive
order prohibiting large gatherings, ceasing new special event
licenses, and pausing operations of bars, gyms, movie
theaters, waterparks, and tubing rentals, and delayed in-
person learning until August 17, 2020.
July 9, 2020 - Arizona: The Governor issued an executive
order requiring restaurants with indoor seating to operate at
less than 50% percent capacity.
0715-
0815
Hawaii -3.147501
July 17, 2020 - Hawai‘i: The Governor signed the 10th
Emergency Proclamation, keeping the mandatory 14-day
quarantine in effect for travelers entering the state, and
travelers will continue to undergo mandatory screening at
airports.
July 20, 2020 - Hawai‘i: The Governor confirmed the state's
plans to move ahead with school reopening for students on
August 4.
July 29, 2020 - Hawai‘i: The Governor announced plans to
reinstate some of the measures relaxed in recent weeks to
combat COVID-19 in Hawaii.
0715-
0815
Arizona -1.167903
July 23, 2020 - Arizona: The Governor extended an executive
order pausing operations on gyms, bars, nightclubs, movie
theaters, water parks, and tubing and announced a statewide
campaign promoting mask use and other precautions.
July 30, 2020 - Arizona: The Governor extended a statewide
mask order until August 31, mandating masks in schools and
colleges for employees and students in second grade and
above.
5.4 Conclusion and Discussion
The investigation of disease transmission through spatial interaction, particularly state-level travel,
has provided valuable insights into the complex dynamics that govern the spread of infectious
diseases. By developing a multi-regional dynamic model with spatial interaction, we have been
able to accurately capture historical transmission patterns and evaluate the impact of long-distance
travel on disease transmission. This understanding is crucial for developing effective prevention
and control strategies for future outbreaks.
The introduction of the transmission export index, which combines local transmissibility and the
potential for infectious travelers to spread diseases to new regions, has proven to be an important
tool for assessing the risk of disease transmission between regions. By identifying high-risk areas,
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appropriate interventions, such as travel restrictions or quarantine measures, can be put in place to
mitigate disease spread and protect public health.
Furthermore, our causal analysis of factors influencing the transmission export index has
highlighted the importance of considering a wide range of influences, including state-level actions,
political events, and festivals/entertainment events. By examining the interplay between these
factors and their potential impact on disease transmission, we can better inform future decision-
making processes and guide public health policy.
Additionally, our analysis of state-level policies aimed at reducing the transmission export index
has demonstrated the effectiveness of various measures, such as promoting public health guidance,
implementing social distancing measures, introducing phased plans for economic reopening, and
imposing travel restrictions. These strategies play a vital role in protecting residents and limiting
the spread of infections both within individual states and across the nation.
In conclusion, this chapter has shed light on the critical role of spatial interaction in disease
transmission and the importance of understanding these dynamics for effective prevention and
control efforts. The methods and findings presented here can serve as a foundation for future
research, policy development, and public health interventions aimed at mitigating the impact of
infectious diseases on a regional, national, and global scale.
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Chapter 6 Integration of Dynamic Modeling with Vaccination
Reallocation
6.1 Introduction
Vaccination is one of the most effective public health interventions for controlling the spread of
infectious diseases. Vaccines work by inducing immunity to a pathogen, thereby reducing the
likelihood of transmission. Vaccination can have a significant impact on disease transmission.
When a large proportion of the population is vaccinated, the likelihood of transmission decreases,
as the pathogen has fewer hosts in which to replicate and spread. This phenomenon is known as
herd immunity. The level of herd immunity required to control the spread of a disease varies
depending on the disease and the vaccine efficacy. In general, a higher proportion of the population
needs to be vaccinated for diseases with higher transmissibility.
The introduction of vaccines has led to a renewed interest in transmission modeling in
epidemiology. Modeling the impact of vaccination on disease transmission is important for
understanding the effectiveness of vaccination programs and designing effective vaccine
distribution policies.
Although the primary goal of vaccination is to provide immunity to individuals against a specific
infectious agent, it is crucial to ensure that the vaccination campaign is designed to maximize the
benefits of vaccination while minimizing the risks and limitations. Thus, designing effective
vaccine distribution policies is essential for successful vaccination programs in controlling the
transmission of an epidemic. One crucial factor that must be considered is the age of the population
targeted for vaccination. Vaccines typically have different efficacy rates and side effects in
different age groups, and this must be taken into account when designing a distribution policy. For
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example, vaccines such as the Pfizer-BioNTech and Moderna vaccines have been found to be
highly effective in preventing infection and severe disease in all age groups, including adolescents,
adults, and elders. On the other hand, the Johnson & Johnson vaccine has shown slightly lower
efficacy rates in preventing infection, but still high efficacy in preventing severe disease and death.
When it comes to vaccine distribution, it is essential to prioritize those who are at the highest risk
of developing severe disease or dying from the virus. This includes the elderly, those with
underlying medical conditions, and healthcare workers. By vaccinating these high-risk groups first,
the transmission of the virus can be significantly reduced. The Centers for Disease Control and
Prevention (CDC) has recommended that frontline essential workers, including those in education,
transportation, and food service, also be prioritized for vaccination due to their increased risk of
exposure to the virus.
Another critical factor in vaccine distribution policy is the availability of vaccines. Limited vaccine
supply can make it difficult to prioritize groups effectively, and it may be necessary to implement
a phased approach. In such a situation, it may be appropriate to prioritize those at the highest risk
of severe disease, including the elderly and those with underlying medical conditions, followed by
essential workers and then the general population.
In this chapter, we will delve deeper into the development of a transmission model that takes into
account the varying age groups and vaccination statuses of individuals, utilizing dynamic modeling
with time-varying transmission parameters. This approach enables us to capture the complex
interactions between different age groups and vaccination coverage, providing a more accurate
representation of disease transmission dynamics.
To validate the accuracy and reliability of our model, we will compare its predictions against
historical case and death data from all 50 states. This comparison serves as an essential benchmark,
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ensuring that our model is capable of capturing the true dynamics of disease transmission and
providing reliable insights for public health decision-making.
Building upon the age-structured dynamic model that incorporates vaccination, we will then
propose a novel method for optimizing vaccine allocation across different regions. This method
takes into consideration the varying transmission severity and population structures among
different areas, as well as the limited resources available for vaccine distribution. By dynamically
optimizing the allocation of vaccines, we aim to minimize the overall impact of the disease while
maximizing the efficient use of available resources.
Finally, we will analyze the implications of different vaccine allocation policies under various
scenarios of vaccine availability. This analysis will provide valuable insights into how different
strategies perform under a range of circumstances, informing decision-makers on the most
effective approaches for managing the disease and mitigating its impact on public health. Through
the development and application of our age-structured dynamic model with vaccination, we hope
to contribute to a better understanding of disease transmission dynamics and inform evidence-
based decision-making for vaccine allocation and distribution strategies. This, in turn, will
ultimately help minimize the adverse effects of infectious diseases on populations and ensure a
more efficient and equitable use of limited resources.
6.2 Age-Structured Dynamic Modeling with Vaccination
6.2.1 Age-Structured Transmission Data and Vaccine Data
Age-structured case and death data provide crucial insights into the COVID-19 pandemic's impact
on different age groups. The age-specific case data reveals how the infection rate varies among
different age brackets. Early in the pandemic, it became clear that older populations were more
susceptible to severe illness and death from COVID-19. According to the CDC, individuals aged
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65 years and older accounted for the majority of COVID-19-related deaths, while younger age
groups experienced significantly fewer fatalities. This age distribution also influenced the policy
recommendations for vaccination priority, with older individuals and those with underlying health
conditions receiving vaccinations first.
Vaccine data is another critical component in understanding the progression of the pandemic and
the effectiveness of public health interventions. The CDC tracks vaccine distribution,
administration, and coverage across different age groups, geographic regions, and demographic
categories. This information allows for the identification of disparities in vaccine access and
uptake, as well as areas that may require targeted outreach and education efforts. Age-stratified
vaccine data also allows for the assessment of vaccine effectiveness in preventing severe illness
and death, especially among high-risk age groups.
In our research about vaccine distribution, we will use CDC as the primary source of COVID-19
data in the United States. They collect and publish data on cases, deaths, and vaccinations through
various channels, including state and local health departments, laboratories, and healthcare
providers. One key resource provided by the CDC is the COVID Data Tracker
(https://covid.cdc.gov/covid-data-tracker), an interactive web-based platform that presents up-to-
date information on cases, deaths, and vaccinations, as well as other relevant metrics such as testing,
hospitalizations, and variant tracking. The CDC continually updates these data sources to provide
the most accurate and comprehensive information possible, helping inform policy decisions and
public health interventions.
Although the CDC has been diligent in tracking and reporting COVID-19 vaccination data, there
was a period between December 16, 2020, and March 4, 2021, when age-structured vaccination
data was not readily available. In order to estimate the daily administered vaccinations in each age
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group for every state during this period, we utilized a data-driven approach. First, the ratio of
eligible persons in each age group to the total number of eligible persons was calculated based on
the vaccination policies in place at the time, which primarily prioritized older adults and
individuals with underlying health conditions. Next, the daily administered vaccinations for each
age group were estimated by multiplying the total amount of daily administered vaccines by the
calculated ratio for each age group.
This approach provides a valuable approximation of age-specific vaccination trends during this
critical period when vaccine distribution was in its early stages. It is essential to acknowledge that
these estimations come with a degree of uncertainty, as they are based on the assumption that the
proportion of eligible individuals among different age groups directly corresponds to the actual
vaccine uptake in each age group. Nevertheless, this method offers a useful framework for
analyzing vaccination patterns in the absence of complete age-structured data from the CDC during
this specific time frame.
6.2.2 Model Structure and Parameter Estimation
In order to better understand the impact of vaccination on COVID-19 transmission and mortality,
we have expanded upon dynamic modeling with time-varying transmission and fatality rates in
Chapter 4 while taking into account different age groups and the effect of vaccination. We divided
the whole population into three age groups: 0-17, 18-64, and 65 plus. Each age group has different
transmission and death rates, reflecting the observed disparities in the susceptibility to the virus
and the severity of the disease.
To capture these differences, we introduced two scalars, 𝜔 𝑖 and 𝜏 𝑖 , for each age group 𝑖 . The scalar
𝜔 𝑖 represents the difference of transmission rate between age group i and the reference age group
2 (18-64 years), while 𝜏 𝑖 denotes the difference in fatality rate between age group 𝑖 and the
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reference age group 3 (65+ years). These scalars allow us to quantify the relative transmission and
mortality risk for each age group in comparison to the reference groups.
Next, we incorporated the effect of vaccination by introducing a vaccine compartment (𝑉 𝑖 ) for each
age group. The vaccination process reduces the number of susceptible individuals in each age
group, with the reduction being proportional to the number of vaccinated individuals and the
effectiveness of the vaccine (𝜃 ). By considering the vaccine's effectiveness, we can account for
the fact that vaccinated individuals may still be at risk of infection, albeit at a lower level than
those who are unvaccinated. In our modeling approach that incorporates the effect of vaccines, we
have disregarded the vaccine's impact on reducing the death rate among vaccinated individuals.
This is primarily due to the absence of specific mortality data tracking the number of vaccinated
individuals who succumbed to the disease across different age groups. Additionally, we have not
factored in inter-state transportation within our model due to the unavailability of age-specific
mobility data. Taking these factors into account, the system of equations of the proposed SEIRD_V
model is given by Equation (16):
𝜕 𝑆 𝑖 (𝑡 )
𝜕𝑡
= −𝜔 𝑖 ∙ 𝛽 (𝑡 ) ∙ 𝐼 𝑖 (𝑡 ) ∙
𝑆 𝑖 (𝑡 ) − 𝜃 ∙ 𝑉 𝑖 (𝑡 )
𝑁
𝜕 𝐸 𝑖 (𝑡 )
𝜕𝑡
= 𝜔 𝑖 ∙ 𝛽 (𝑡 ) ∙ 𝐼 𝑖 (𝑡 ) ∙
𝑆 𝑖 (𝑡 ) − 𝜃 ∙ 𝑉 𝑖 (𝑡 )
𝑁 − 𝜎 ∙ 𝐸 𝑖 (𝑡 )
𝜕 𝐼 𝑖 (𝑡 )
𝜕𝑡
= 𝜎 ∙ 𝐸 𝑖 (𝑡 ) − (1 − 𝜏 𝑖 ∙ 𝛼 (𝑡 )) ∙ 𝛾 𝐼 𝑖 (𝑡 ) − 𝜏 𝑖 ∙ 𝛼 (𝑡 ) ∙ 𝜌 ∙ 𝐼 𝑖 (𝑡 )
𝜕 𝑅 𝑖 (𝑡 )
𝜕𝑡
= 𝜏 𝑖 ∙ (1 − 𝛼 (𝑡 )) ∙ 𝛾 ∙ 𝐼 𝑖 (𝑡 )
𝜕 𝐷 𝑖 (𝑡 )
𝜕𝑡
= 𝜏 𝑖 ∙ 𝛼 (𝑡 ) ∙ 𝜌 ∙ 𝐼 𝑖 (𝑡 )
(16)
where:
𝑆 𝑖 (𝑡 ): the number of susceptible individuals in age group 𝑖 over time.
𝐸 𝑖 (𝑡 ): the number of exposed individuals in age group 𝑖 over time.
𝐼 𝑖 (𝑡 ): the number of infectious individuals in age group 𝑖 over time.
𝑅 𝑖 (𝑡 ): the number of recovered individuals in age group 𝑖 over time.
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𝐷 𝑖 (𝑡 ): the number of dead individuals in age group 𝑖 over time.
𝑉 𝑖 (𝑡 ): the number of individuals vaccinated in age group 𝑖 over time.
N: the total population size.
β(t): the effective contact rate, a measure of how many people to whom an infected person
can transmit the disease at time t.
α(t): the fraction of infectious individuals detected and isolated at time t.
γ: the recovery rate of infected individuals.
δ: the rate at which exposed individuals become infectious.
ρ: the fatality rate among infected individuals.
𝜔 𝑖 : scalars representing the difference in transmission rate between age groups 𝑖 with
respect to age group 2.
𝜏 𝑖 : scalars representing the difference in fatality rate between age groups 𝑖 with respect to
age group 3.
The model we developed aims to provide a more comprehensive understanding of the interplay
between age-specific transmission dynamics, vaccine rollout, and disease outcomes. By
considering the heterogeneous nature of COVID-19 transmission and death rates across different
age groups and accounting for the impact of vaccination, we can generate more accurate and
nuanced predictions about the pandemic's progression. This, in turn, can help inform public health
policies and interventions tailored to the specific needs of each age group, ultimately contributing
to more effective control of the pandemic.
6.2.3 Fitting Results of Age-Structured Dynamic Modeling with Vaccination
Figure 20 and Figure 21 summarize the model fitting accuracy of the transmission model with
vaccination for three age groups (0-17, 18-64, and 65+) across all 50 states in the United States
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from December 16, 2020, to June 30, 2021. The model's accuracy is evaluated by the relative root
mean square error (RRMSE) defined in Equation (12) for both the number of COVID-19 cases
and deaths within each age group for each state. The results provide valuable insights into the
model's performance and the effectiveness of incorporating vaccination data into the transmission
model.
Figure 20: Model fitting accuracy across age groups for covid-19 cases in all 50 states
Figure 21: Model fitting accuracy across age groups for covid-19 deaths in all 50 states
The model fitting accuracy varies across states and age groups, which may be attributed to factors
such as differences in state-level vaccination rates, adherence to public health guidelines,
population density, and other regional factors influencing transmission and death rates. On average,
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the fitting accuracy for the number of cases is 0.092 for the 0-17 age group, 0.080 for the 18-64
age group, and 0.078 for the 65+ age group. The average fitting accuracy for the number of deaths
is 0.009 for the 0-17 age group, 0.073 for the 18-64 age group, and 0.038 for the 65+ age group.
The model generally exhibits higher fitting accuracy for the number of cases than for the number
of deaths. This may be because the number of cases is typically higher and more consistently
reported than the number of deaths, making it easier for the model to fit case data. Furthermore,
the relatively low fitting accuracy for the number of deaths in the 0-17 age group could be due to
the rarity of COVID-19-related deaths in this demographic, resulting in fewer data points and
greater uncertainty in the model.
It is notable that some states, such as Mississippi, have zero fitting accuracy for all age groups in
both cases and deaths. This may be caused by a lack of available data, inconsistencies in reporting,
or potential issues with the model's assumptions for that specific state. Further investigation would
be required to determine the cause of these discrepancies and improve the model's performance.
Moreover, the lower fitting accuracy for certain states may be attributed to the discrete daily
variation of administered vaccines. Daily fluctuations in vaccination numbers can add complexity
to the modeling process, making it more challenging for the model to generate smooth transmission
rate and death rate functions that accurately capture historical trends. This issue can be particularly
pronounced in states with inconsistent vaccination rollouts or disruptions due to supply chain
issues, logistical challenges, or changes in vaccine eligibility criteria. In such cases, the model may
struggle to accurately account for the impacts of these fluctuations on overall transmission and
death rates. The daily variation in administered vaccines can lead to inconsistencies in the model's
predictions, which may contribute to lower fitting accuracy observed in some states.
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In summary, the model fitting results demonstrate that incorporating vaccination data into a
transmission model can provide reasonably accurate estimates of COVID-19 cases and deaths
across different age groups and states. The model's varying accuracy across states highlights the
importance of considering regional factors when evaluating its performance and potential
improvements. This analysis also emphasizes the need for continued data collection and reporting
to refine the model and better understand the impact of vaccination on the pandemic's trajectory.
Despite its limitations, the transmission model with vaccination data offers valuable insights into
the progression of the COVID-19 pandemic in the United States, particularly in terms of age-
specific trends. By accounting for the effects of vaccination and different age group transmission
dynamics, this model can help inform public health policies and interventions that are tailored to
the specific needs of each age group. This, in turn, can contribute to more effective control of the
pandemic and better health outcomes for all.
6.3 Vaccine Allocation Optimization with Dynamic Transmission
Pattern
The COVID-19 pandemic has demonstrated the importance of efficient and equitable vaccine
allocation strategies to control the spread of the virus and reduce morbidity and mortality rates. In
this section, we propose a dynamic optimization framework to allocate vaccines among 50 states
in the United States, taking into account the transmission patterns and the impact of vaccination
on disease transmission and death rates.
Previously, we developed a transmission model incorporating vaccination data to provide
reasonably accurate estimates of COVID-19 cases and deaths across different age groups and states.
The model considers age-structured case and death data, vaccine data, and time-varying
transmission and death rates, accounting for the effects of vaccination on susceptible populations
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in each age group. The fitting results demonstrated that the model effectively captured the
historical trends of COVID-19 cases and deaths and the impact of vaccination on these trends.
Building upon this transmission model, we now aim to optimize vaccine allocation among the 50
states by solving an optimization problem. The objective function for the optimization problem is
defined as the sum of the weighted case numbers and death numbers. The constraints are the bi-
weekly available amount of vaccination for each state. For the vaccine allocation optimization
problem, we consider the objective function and constraints in Equation (17):
min
V
𝑖 ,𝑡
∑ 𝑤 1
∙ 𝐶𝑎𝑠𝑒 𝑠 𝑖 ,𝑡 + 𝑤 2
∙ 𝐷𝑒𝑎𝑡 ℎ𝑠 𝑖 ,𝑡 𝑖 ,𝑡
s. t. ∑ 𝑉 𝑖 ,𝑡 𝑖 ≤ 𝑄 𝑡 ∀𝑡
0 ≤ 𝑉 𝑖 ,𝑡 ≤ 𝑁 𝑖
𝜕 𝑆 𝑖 (𝑡 )
𝜕𝑡
= −𝜔 𝑖 ∙ 𝛽 (𝑡 ) ∙ 𝐼 𝑖 (𝑡 ) ∙
𝑆 𝑖 (𝑡 ) − 𝜃 ∙ 𝑉 𝑖 (𝑡 )
𝑁
𝜕 𝐸 𝑖 (𝑡 )
𝜕𝑡
= 𝜔 𝑖 ∙ 𝛽 (𝑡 ) ∙ 𝐼 𝑖 (𝑡 ) ∙
𝑆 𝑖 (𝑡 ) − 𝜃 ∙ 𝑉 𝑖 (𝑡 )
𝑁 − 𝜎 ∙ 𝐸 𝑖 (𝑡 )
𝜕 𝐼 𝑖 (𝑡 )
𝜕𝑡
= 𝜎 ∙ 𝐸 𝑖 (𝑡 ) − (1 − 𝜏 𝑖 ∙ 𝛼 (𝑡 )) ∙ 𝛾 𝐼 𝑖 (𝑡 ) − 𝜏 𝑖 ∙ 𝛼 (𝑡 ) ∙ 𝜌 ∙ 𝐼 𝑖 (𝑡 )
𝜕 𝑅 𝑖 (𝑡 )
𝜕𝑡
= 𝜏 𝑖 ∙ (1 − 𝛼 (𝑡 )) ∙ 𝛾 ∙ 𝐼 𝑖 (𝑡 )
𝜕 𝐷 𝑖 (𝑡 )
𝜕𝑡
= 𝜏 𝑖 ∙ 𝛼 (𝑡 ) ∙ 𝜌 ∙ 𝐼 𝑖 (𝑡 )
𝐶𝑎𝑠𝑒 𝑠 𝑖 ,𝑡 = 𝐼 𝑖 (𝑡 ) + 𝑅 𝑖 (𝑡 ) + 𝐷 𝑖 (𝑡 )
𝐷𝑒𝑎𝑡 ℎ𝑠 𝑖 ,𝑡 = 𝐷 𝑖 (𝑡 )
(17)
Where 𝑉 𝑖 ,𝑡 refers to the vaccine number in region 𝑖 on day 𝑡 , 𝑄 𝑡 refers to the total amount of
available vaccine on day 𝑡 , 𝑤 1
and 𝑤 2
are the weight the policymaker put on the case number and
death number. The dynamic nature of this optimization problem lies in the fact that it considers
the evolving transmission patterns and vaccination rates over time. By accounting for these
dynamics, the optimization process can be adjusted as new data on the pandemic's progression and
vaccination efforts becomes available, allowing for a more adaptive and responsive allocation
strategy.
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The optimization problem is highly nonlinear since the case number and death number are solved
using implicit ordinary differential equations (ODEs) with time-varying transmission parameters
and discrete vaccine numbers. The decision variables in the functions are the bi-weekly vaccine
numbers allocated to each state. To solve the optimization problem, we employ the Sequential
Least Squares Quadratic Programming (SLSQP) method [129], a gradient-based optimization
algorithm. The SLSQP algorithm is well-suited for this problem because it can handle both
equality and inequality constraints and is capable of solving nonlinear optimization problems with
a large number of variables.
To illustrate the SLSQP solving process for vaccine allocation optimization, let's first consider the
optimization problem, which aims to minimize the weighted sum of cases and deaths over a
specific time horizon. The objective function consists of two components: the number of cases and
the number of deaths. The decision variables in the optimization problem are the bi-weekly vaccine
allocations for each state, subject to constraints on the total available vaccines and the maximum
vaccination capacity of each state.
The SLSQP algorithm starts with an initial guess for the decision variables (i.e., the bi-weekly
vaccine allocations) and iteratively updates these values to minimize the objective function. At
each iteration, the algorithm computes the gradient of the objective function with respect to the
decision variables, which is essential for updating the decision variables in the right direction.
In this context, the gradient computation is challenging due to the implicit nature of the objective
function, which depends on the solution of ordinary differential equations (ODEs) describing the
transmission dynamics. To calculate the case/death number in the objective functions and the
gradient with respect to the decision variables, we must first solve the ODEs. Traditional third-
party ODE solvers, such as the odeint function provided by the SciPy library, utilize the fourth-
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order Runge-Kutta method to achieve a higher level of accuracy. This method approximates the
daily increments with a sufficiently small step size. However, the transmission rate and death rate
of the disease will remain constant within the same day, and the implicit formulation of the fourth-
order Runge-Kutta method makes it challenging for the algorithm to find the derivatives with
respect to the decision variables, potentially leading to gradient vanishment.
To address this issue, we utilize Euler's method, a first-order numerical method for solving ODEs,
instead of the fourth-order Runge-Kutta method. By employing Euler's method, we can calculate
the weekly case and death numbers with fixed decision variables (i.e. bi-weekly allocated
vaccination for each state). This approach allows for a more straightforward computation of the
gradient, avoiding the complexities associated with higher-order ODE solvers like the Runge-
Kutta method.
Once the gradient is computed, the SLSQP algorithm updates the decision variables by moving in
the direction of the negative gradient, which corresponds to the steepest descent in the objective
function. The algorithm also takes into account the constraints on vaccine availability and state
capacities, ensuring that the updated decision variables are feasible. This iterative process
continues until the algorithm converges to a solution that minimizes the objective function, subject
to the constraints.
The SLSQP-based optimization framework offers a systematic approach to determining the
optimal distribution of vaccines among the 50 states, taking into account the dynamic nature of the
transmission patterns, vaccine availability, and state capacities. By continuously updating the
decision variables, the algorithm ensures that the vaccine allocation strategy remains aligned with
the evolving pandemic landscape, ultimately leading to a more efficient and effective allocation of
resources and better public health outcomes. The dynamic vaccine allocation optimization
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framework proposed in this study allows for a more effective and equitable distribution of vaccines
among the 50 states, considering the regional transmission patterns and the impact of vaccination
on disease transmission and death rates. By dynamically adjusting the vaccine allocation based on
the latest available data and evolving transmission patterns, this framework can help inform public
health decision-making and guide effective pandemic response efforts at the national and state
levels.
In conclusion, the dynamic optimization of vaccine allocation with transmission patterns provides
a valuable tool for public health authorities and policymakers to make data-driven decisions on
vaccine distribution. The use of Euler's method to solve the highly nonlinear optimization problem
with implicit ODEs and the application of the SLSQP method for solving the optimization problem
ensures that the algorithm can find the derivatives with respect to the decision variables, enabling
an effective solution to the problem. By incorporating regional transmission patterns and the
impact of vaccination on disease transmission and death rates, the proposed framework enables a
more targeted and efficient allocation of vaccines among the 50 states.
6.4 Vaccine Allocation Policy under Different Scenarios
In this section, we will discuss the best vaccine distribution policy under different scenarios of
vaccine availability. A thorough analysis of various vaccine distribution strategies is essential to
identify the most effective approaches for allocating limited resources to minimize the impact of
COVID-19 on public health. By studying the best vaccine distribution policy under different
scenarios of vaccine availability, we aim to understand the implications of alternative vaccine
allocation strategies, as well as to inform and improve future vaccine distribution efforts.
Studying the best vaccine distribution policy under different scenarios is essential to understand
the implications of various allocation strategies and to identify the most effective approach to
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controlling the spread of COVID-19. Considering different vaccine availability scenarios allows
us to account for uncertainties in vaccine production and distribution, as well as potential changes
in demand due to factors such as vaccine hesitancy or new variants. The objective of this analysis
is to ensure that vaccines are allocated in a manner that minimizes the number of cases and deaths,
while also maximizing the overall public health benefits.
By examining different vaccine distribution scenarios, we can explore the impact of prioritizing
certain age groups or geographical regions, and assess the potential trade-offs between focusing
on high-risk populations versus wider coverage. This information is invaluable for policymakers,
public health officials, and other stakeholders involved in the decision-making process, as it
enables them to make informed choices about vaccine distribution strategies that optimize resource
allocation and ultimately save lives.
Furthermore, understanding the best vaccine distribution policy under various scenarios can
inform future vaccination campaigns, not only for COVID-19 but also for other infectious diseases.
Lessons learned from this analysis can contribute to the development of more robust vaccination
strategies that can be adapted to different contexts and changing circumstances, thereby improving
the overall effectiveness of public health interventions.
In the following part, we will consider several scenarios of vaccine availability, starting with the
hypothetical situation of zero vaccine availability. This baseline scenario will enable us to evaluate
the effectiveness of historical vaccine distribution efforts and inform our understanding of the
potential benefits of optimized vaccine distribution policies. We will then explore other scenarios
with varying levels of vaccine availability to identify the optimal vaccine distribution policy for
each situation, ultimately leading to more efficient and effective distribution strategies that
minimize the number of cases and deaths.
116
6.4.1 Healthcare Outcomes without Vaccination
First, let's consider the scenario with zero vaccine availability. This hypothetical situation allows
us to understand the effectiveness of the historical vaccine distribution by comparing the case and
death numbers under this scenario to the real historical data. By simulating the age-structured
dynamic model with no vaccination, we can estimate the number of cases and deaths that would
have occurred if no vaccines were distributed.
The results from this analysis show that the historical vaccine distribution has had a significant
impact on reducing the spread of the virus and saving lives. In the absence of any vaccine
distribution, the model estimates that there would have been an additional 1,827,631 cases and
9,180 deaths. These findings highlight the crucial role that vaccines have played in mitigating the
severity of the pandemic and demonstrate the importance of an effective vaccine distribution
strategy.
In order to further explore the best vaccine distribution policy, we can consider different scenarios
of vaccine availability. For each scenario, the age-structured dynamic model with vaccination can
be used to simulate the impact of various distribution strategies on the number of cases and deaths.
By comparing the outcomes under different strategies, we can identify the optimal vaccine
distribution policy for each level of vaccine availability.
The zero vaccine availability scenario serves as a baseline for understanding the effectiveness of
historical vaccine distribution efforts. As we analyze other scenarios with varying vaccine
availability, we can gain insights into how to optimize the allocation of vaccines across states and
age groups, ultimately leading to more efficient and effective distribution strategies that minimize
the number of cases and deaths. This valuable information can inform future vaccine distribution
117
policies, ensuring that limited resources are allocated in the most impactful way possible,
contributing to better public health outcomes and the eventual control of the pandemic.
6.4.2 Vaccine Allocation Policy with Original Vaccine Availability
In the second scenario, we explore the optimal vaccine allocation policy among the 50 states to
achieve better healthcare outcomes while considering the dynamic transmission patterns in each
state. The current government strategy of distributing vaccines proportional to the eligible
population size in each state can lead to an imbalance in vaccine allocation, with some states
receiving a surplus while others face a shortage of this vital resource. To address this issue, we
analyzed the best vaccine allocation policy under two different prioritizations: reducing the
number of cases as much as possible and reducing the number of deaths as much as possible. These
two priorities encompass the primary concerns of policymakers when managing the pandemic.
When focusing on case reduction, the optimal vaccine allocation policy suggests that a larger share
of vaccines should be distributed to the younger age group (0-17 years). This is because this
population is generally more active and has more frequent social interactions, leading to a higher
potential for spreading the virus. Additionally, younger individuals may exhibit milder symptoms
or be asymptomatic, making them more likely to unknowingly transmit the virus to others. By
vaccinating this age group, the overall transmission rate within the population can be significantly
reduced, ultimately lowering the total number of cases. In this prioritization, the focus is on
reducing the spread of the virus, leading to an overall decrease in cases and, consequently, a lower
number of associated deaths. According to the model results, this optimal allocation strategy could
potentially reduce 2,042,312 cases and 1,796 deaths.
118
Figure 22: Training process of case-prioritized vaccine optimization
with original vaccine availability
Figure 23: Vaccine allocation comparison for case-prioritized vaccine optimization
with original vaccine availability
On the other hand, when prioritizing the reduction of deaths, the optimal vaccine allocation policy
retains an emphasis on vaccinating the younger age group (0-17 years) due to their role in driving
overall case numbers. However, this strategy also allocates more vaccines to the older population
(65+ years), who face a higher risk of severe illness and death from COVID-19. Older individuals
typically have weaker immune systems and may suffer from comorbidities, making them more
vulnerable to the virus's severe effects. By prioritizing the vaccination of the older population, the
119
policy aims to protect those at the highest risk of death, leading to a substantial reduction in
fatalities. This allocation strategy strikes a balance between limiting virus transmission by
vaccinating the younger age group and protecting the most vulnerable members of society,
resulting in a decrease in both case numbers and deaths. According to the model results, this
optimal allocation strategy could reduce 220,010 cases and 6,319 deaths.
The results of the second scenario demonstrate that a more targeted vaccine allocation strategy,
considering the dynamic transmission patterns and the specific priorities of policymakers, can lead
to significantly better healthcare outcomes. By shifting the vaccine distribution towards age groups
that have the most significant impact on transmission and death rates, it is possible to achieve
substantial reductions in both cases and fatalities. This analysis highlights the importance of data-
driven and adaptable vaccine allocation policies to effectively manage the ongoing pandemic and
safeguard public health.
Figure 24: Training process of death-prioritized vaccine optimization
with original vaccine availability
120
Figure 25: Vaccine allocation comparison for death-prioritized vaccine optimization
with original vaccine availability
Figure 26 and Figure 27 show the redistribution of original amount of vaccine among 50 states for
case/death-prioritized scenario. According to the comparison, some states, like Maine, Vermont,
Montana, South Dakota, and New Hampshire, need more vaccines than distributed. There are
several factors contribute to their increased need for vaccines to effectively reduce the number of
cases and deaths. These states generally have smaller populations, fewer resources, and limited
healthcare infrastructure, particularly in rural areas, which can affect their ability to quickly
identify, treat, and manage cases. An increased allocation of vaccines could help to compensate
for these limitations by reducing the number of severe cases that require hospitalization and
specialized care. Moreover, the low population density and sparse distribution of the population in
these states create challenges in vaccine distribution and administration, leading to slower
immunization rates. Increasing the allocation of vaccines to these states can help overcome
logistical challenges and ensure that more people receive the vaccine. Additionally, the age
distribution of the population in these states may play a role in the increased need for vaccines, as
121
some have a higher proportion of older adults who are at greater risk of severe illness and death
due to COVID-19. Prioritizing vaccine allocation to these states can help protect their most
vulnerable citizens and reduce fatalities. Lastly, the effectiveness of public health policies and their
implementation varies between states, and those with a higher need for vaccines may have less
stringent public health measures or lower compliance. By increasing the vaccine allocation, these
states can mitigate the impact of less effective public health policies on case numbers and deaths.
In summary, a targeted approach to vaccine allocation that considers the unique needs and
challenges faced by states like Maine, Vermont, Montana, South Dakota, and New Hampshire can
lead to a more effective reduction in both cases and deaths through a data-driven and adaptable
vaccine allocation strategy that addresses disparities in vaccine distribution and helps to better
manage the ongoing pandemic.
Figure 26: Redistribution of original amount of vaccine among 50 states for case-prioritized scenario
(Change of the vaccine distribution divided by original vaccine number)
122
Figure 27: Redistribution of original amount of vaccine among 50 states for death-prioritized scenario
(Change of the vaccine distribution divided by original vaccine number)
6.4.3 Vaccine Allocation Policy with 10 Times of Original Vaccine Availability
In the third scenario, we explore the impact of a substantial increase in vaccine availability,
specifically 10 times the original weekly availability. This scenario aims to understand the optimal
vaccine allocation policy and the corresponding healthcare outcomes, given this significant
increase in resources. Similar to the second scenario, we analyze the best vaccine allocation policy
under two different prioritizations: reducing the number of cases as much as possible and reducing
the number of deaths as much as possible.
The vaccine allocation results for both prioritizations remain consistent with the second scenario.
They maintain an emphasis on vaccinating the younger age group (0-17 years) due to their role in
driving overall case numbers. Additionally, prioritizing the vaccination of the older population is
123
recommended if policymakers emphasize the reduction in fatalities. This demonstrates the
robustness of the allocation strategies across different levels of vaccine availability.
According to the model results, the optimal allocation strategy could potentially reduce 2,561,885
cases and 6,735 deaths when prioritizing the reduction of cases. Although this represents a
significant reduction in case numbers, the increment in vaccination resources only leads to an
additional reduction of 519,573 cases. This is primarily because, even with 10 times the vaccine
availability for each week, the available amount of vaccine is still too small to control the epidemic
during the first several months. The limited vaccination resources at the beginning of a new wave
of transmission make it difficult to substantially reduce the number of cases once the disease has
spread widely throughout the population.
Figure 28: Training process of case-prioritized vaccine optimization
with 10 times vaccine availability
124
Figure 29: Vaccine allocation comparison for case-prioritized vaccine optimization
with 10 times vaccine availability
In contrast, the optimal allocation strategy could potentially reduce 1,537,008 cases and 16,014
deaths when prioritizing the reduction of deaths, leading to an additional 9,695 lives saved. These
finding highlights that, even though a 10-fold increase in vaccination resources may not result in
a dramatic decrease in case numbers, it can have a substantial impact on saving lives. The
allocation of sufficient vaccination resources can protect vulnerable populations, particularly the
older population, who are at the highest risk of severe illness and death from COVID-19.
When analyzing the third scenario, it is crucial to acknowledge the inherent limitations of
increasing vaccine availability. A 10-fold increase in weekly availability may not be feasible due
to production constraints, logistical challenges, and the need for a rapid and efficient rollout.
Nonetheless, this scenario provides valuable insights into the potential impact of increased
vaccination resources on the overall healthcare outcomes during the pandemic.
In conclusion, the third scenario demonstrates the importance of strategic vaccine allocation,
particularly when resources are limited. Although a substantial increase in vaccine availability can
lead to significant reductions in both case numbers and deaths, it is essential to prioritize the most
vulnerable populations and target age groups that play a significant role in virus transmission. By
125
understanding the potential outcomes under different prioritizations and levels of vaccine
availability, policymakers can make informed decisions to optimize healthcare outcomes and
mitigate the impact of the COVID-19 pandemic.
Figure 30: Training process of death-prioritized vaccine optimization
with 10 times vaccine availability
Figure 31: Vaccine allocation comparison for death-prioritized vaccine optimization
with 10 times vaccine availability
6.5 Conclusion
In this chapter, we have presented an age-structured dynamic model that incorporates vaccination
data, providing a more accurate and nuanced understanding of disease transmission dynamics
126
among different age groups. We have also proposed a novel method for optimizing vaccine
allocation across regions, taking into consideration the varying transmission severity, population
structures, and limited resources for vaccine distribution. By analyzing the implications of
different vaccine allocation policies under various scenarios of vaccine availability, we have
provided valuable insights into the most effective strategies for managing the disease and
mitigating its impact on public health.
Our work has demonstrated the importance of data-driven and adaptable vaccine allocation
policies in managing the ongoing pandemic and safeguarding public health. By incorporating
regional transmission patterns, population structures, and the impact of vaccination on disease
transmission and death rates, our age-structured dynamic model with vaccination and optimization
framework offers a valuable tool for public health authorities and policymakers to make informed
decisions on vaccine distribution.
The analysis of different vaccine allocation policies under various scenarios of vaccine availability
highlights the need for targeted and strategic vaccine distribution, particularly when resources are
limited. By prioritizing the most vulnerable populations and targeting age groups that play a
significant role in virus transmission, it is possible to achieve substantial reductions in both cases
and fatalities. Our findings emphasize the crucial role that vaccines play in mitigating the severity
of the pandemic, and the importance of an effective vaccine distribution strategy in controlling the
spread of the virus and reducing morbidity and mortality rates.
As the COVID-19 pandemic continues to evolve, it is essential for public health authorities and
policymakers to remain vigilant and adapt to changing circumstances. Our age-structured dynamic
model with vaccination and optimization framework can serve as a valuable foundation for future
research and decision-making, as new data becomes available and new challenges arise. The
127
ongoing refinement and application of our model will help inform evidence-based decision-
making for vaccine allocation and distribution strategies, contributing to a better understanding of
disease transmission dynamics and ensuring a more efficient and equitable use of limited resources.
In summary, our work contributes to a deeper understanding of the complex interplay between
age-specific transmission dynamics, vaccine rollout, and disease outcomes. By developing and
applying our age-structured dynamic model with vaccination, we hope to inform evidence-based
decision-making for vaccine allocation and distribution strategies, ultimately helping to minimize
the adverse effects of infectious diseases on populations and ensure a more efficient and equitable
use of limited resources.
128
Chapter 7 Conclusion & Discussion
7.1 Conclusion and Discussion
The present thesis has focused on advancing the understanding of infectious disease dynamics and
providing tools for effective intervention and optimization strategies. By integrating dynamic
modeling with time-varying transmission and fatality rates, spatial interaction analysis, and
vaccination reallocation, we have presented a comprehensive and adaptable framework for
studying the complex interplay of factors that influence the spread and control of infectious
diseases.
7.1.1 Dynamic Modeling with Time-Varying Transmission and Fatality Rates
Our work on dynamic modeling with time-varying transmission and fatality rates highlights the
importance of employing flexible and data-driven approaches to understanding and managing
infectious disease outbreaks. The SEIRD model with time-dependent transmission and death rates
offers a robust framework for analyzing the historical disease transmission and informing
evidence-based public health decision-making. The insights gained from this research can help
researchers and policymakers develop targeted and effective interventions to mitigate the
devastating impacts of infectious diseases on global health and wellbeing.
In particular, our research demonstrates the effectiveness of the SEIRD model with time-dependent
transmission and death rates in capturing the transmission dynamics of COVID-19 across the
United States. The model's fitting accuracy is relatively high for both the number of cases and
deaths, with a majority of states exhibiting an RRMSE below 2%. This demonstrates the value of
incorporating time-varying transmission and fatality rates into epidemic modeling efforts.
Moreover, the multi-phase model introduced to account for locations exhibiting multiple waves of
129
infection further improves the model's accuracy, allowing for better prediction and management
of the ongoing pandemic.
Moreover, our sensitivity analysis highlights the significance of certain parameters in influencing
the predicted number of cases and deaths. Understanding the impact of changes in these key
parameters can help researchers and policymakers anticipate potential shifts in disease
transmission and develop appropriate interventions. Furthermore, our Monte Carlo simulations
demonstrate the extent to which uncertainties in parameter values can impact model outcomes,
emphasizing the need for robust sensitivity analyses in infectious disease modeling.
7.1.2 Integration of Dynamic Modeling with Spatial Interaction and Effect
Analysis
By incorporating spatial interaction analysis, we have demonstrated the critical role of
understanding complex disease transmission dynamics across regions for effective prevention and
control efforts. Our multi-regional dynamic model with spatial interaction provides a valuable tool
for identifying high-risk areas and assessing the impact of various interventions, such as travel
restrictions or quarantine measures. Furthermore, our research on the transmission export index
and the interplay between various factors influencing disease transmission can help guide future
decision-making processes and inform public health policy.
The investigation of disease transmission through spatial interaction, particularly state-level travel,
has provided valuable insights into the complex dynamics that govern the spread of infectious
diseases. By developing a multi-regional dynamic model with spatial interaction, we have been
able to accurately capture historical transmission patterns and evaluate the impact of long-distance
travel on disease transmission. This understanding is crucial for developing effective prevention
and control strategies for future outbreaks.
130
The introduction of the transmission export index, which combines local transmissibility and the
potential for infectious travelers to spread diseases to new regions, has proven to be an important
tool for assessing the risk of disease transmission between regions. By identifying high-risk areas,
appropriate interventions, such as travel restrictions or quarantine measures, can be put in place to
mitigate disease spread and protect public health.
Furthermore, our causal analysis of factors influencing the transmission export index has
highlighted the importance of considering a wide range of influences, including state-level actions,
political events, and festivals/entertainment events. By examining the interplay between these
factors and their potential impact on disease transmission, we can better inform future decision-
making processes and guide public health policy.
Additionally, our analysis of state-level policies aimed at reducing the transmission export index
has demonstrated the effectiveness of various measures, such as promoting public health guidance,
implementing social distancing measures, introducing phased plans for economic reopening, and
imposing travel restrictions. These strategies play a vital role in protecting residents and limiting
the spread of infections both within individual states and across the nation.
In conclusion, this chapter has shed light on the critical role of spatial interaction in disease
transmission and the importance of understanding these dynamics for effective prevention and
control efforts. The methods and findings presented here can serve as a foundation for future
research, policy development, and public health interventions aimed at mitigating the impact of
infectious diseases on a regional, national, and global scale.
7.1.3 Integration of Dynamic Modeling with Vaccination Reallocation
The integration of an age-structured dynamic model with vaccination data and optimization
techniques allows us to better understand age-specific transmission dynamics and develop efficient
131
strategies for vaccine allocation and distribution. Our analysis demonstrates the importance of
targeted and strategic vaccine distribution, especially when resources are limited. By prioritizing
vulnerable populations and targeting age groups that play a significant role in virus transmission,
it is possible to achieve substantial reductions in both cases and fatalities.
In this chapter, we have presented an age-structured dynamic model that incorporates vaccination
data, providing a more accurate and nuanced understanding of disease transmission dynamics
among different age groups. We have also proposed a novel method for optimizing vaccine
allocation across regions, taking into consideration the varying transmission severity, population
structures, and limited resources for vaccine distribution. By analyzing the implications of
different vaccine allocation policies under various scenarios of vaccine availability, we have
provided valuable insights into the most effective strategies for managing the disease and
mitigating its impact on public health.
Our work has demonstrated the importance of data-driven and adaptable vaccine allocation
policies in managing the ongoing pandemic and safeguarding public health. By incorporating
regional transmission patterns, population structures, and the impact of vaccination on disease
transmission and death rates, our age-structured dynamic model with vaccination and optimization
framework offers a valuable tool for public health authorities and policymakers to make informed
decisions on vaccine distribution.
The analysis of different vaccine allocation policies under various scenarios of vaccine availability
highlights the need for targeted and strategic vaccine distribution, particularly when resources are
limited. By prioritizing the most vulnerable populations and targeting age groups that play a
significant role in virus transmission, it is possible to achieve substantial reductions in both cases
and fatalities. Our findings emphasize the crucial role that vaccines play in mitigating the severity
132
of the pandemic, and the importance of an effective vaccine distribution strategy in controlling the
spread of the virus and reducing morbidity and mortality rates.
7.1.4 Overall Summary
Our work contributes to a deeper understanding of the complex interplay between disease
transmission dynamics, intervention strategies, and optimization techniques. The frameworks and
models developed in this thesis can serve as a foundation for future research, policy development,
and public health interventions aimed at mitigating the impact of infectious diseases on a regional,
national, and global scale.
As the COVID-19 pandemic continues to evolve, and new diseases arise, it is essential for public
health authorities and policymakers to remain vigilant and adapt to changing circumstances. The
ongoing refinement and application of the models and approaches presented in this thesis will help
inform evidence-based decision-making for intervention strategies and resource allocation,
ultimately helping to minimize the adverse effects of infectious diseases on populations and ensure
a more efficient and equitable use of limited resources.
In summary, our work highlights the potential of dynamic modeling with time-varying
transmission and fatality rates, spatial interaction analysis, and vaccination reallocation in
understanding and managing infectious disease outbreaks. By leveraging the insights gained from
our models and approaches, researchers and policymakers can work together to develop targeted
and effective interventions to address infectious diseases and safeguard public health.
133
7.2 Future Study
The findings from the main chapters of this thesis have provided valuable insights into disease
transmission dynamics and informed public health decision-making. By integrating dynamic
modeling with time-varying transmission and fatality rates, spatial interaction and effect analysis,
and vaccination reallocation, the research has demonstrated the potential for significant
improvements in our understanding of disease transmission and the development of targeted
intervention strategies. The following future study outlines several key areas for further
exploration, building on the foundation established by the current research.
(1) Developing a Unified Modeling Framework
An important direction for future research is the development of a unified modeling framework
that integrates the components from the three chapters. This would involve incorporating time-
varying transmission and fatality rates, spatial interaction and effect analysis, and vaccination
reallocation into a single, comprehensive model. Such a framework could provide a more holistic
understanding of disease transmission dynamics and support the development of more effective
and targeted public health policies. Simultaneously, as part of our sensitivity analysis, we intend
to scrutinize the impact of fixed parameters on the model outputs. These fixed parameters include
key elements such as the incubation period and recovery time. By analyzing these factors, we aim
to understand their influence on the model and gain insights into how variations in these parameters
might affect the model's predictions. This will also help us assess the robustness of our model
under different scenarios and conditions.
(2) Incorporating Additional Factors and Granularity
Expanding the model to include additional factors influencing disease transmission, such as
population density, age distribution, socioeconomic status, cultural factors, climate, and healthcare
134
infrastructure, could provide a more comprehensive understanding of disease transmission
dynamics. Moreover, enhancing the granularity of the model to operate at finer spatial resolutions,
such as county or city-level, would help identify localized transmission hotspots and inform
targeted intervention strategies.
(3) Investigating the Impact of Emerging Variants, Waning Immunity, and Vaccine
Hesitancy
Future research should explore the impact of emerging virus variants, waning immunity, and
vaccine hesitancy on transmission and fatality rates, as well as the effectiveness of various
intervention strategies. This could involve modeling the potential influence of emerging variants,
assessing the impact of public health interventions aimed at addressing vaccine hesitancy, and
exploring the use of booster shots in response to waning immunity.
(4) Analyzing and Comparing the Effects of Different Intervention Strategies
Future studies could adapt the dynamic model to assess the impact of various intervention
strategies, such as vaccination campaigns, social distancing measures, and testing, tracing, and
isolation protocols. This would involve comparing the effectiveness of these strategies under
different scenarios and providing valuable insights for policymakers and public health officials.
(5) Assessing the Impact of Public Health Policies on Transmission Dynamics
Systematically evaluating the effectiveness of various public health policies in reducing
transmission risk and optimizing resource allocation can help inform policy decisions. Future
research could develop computational models to simulate the impact of different policy scenarios
or conduct comparative analyses of regions with varying policy approaches.
By pursuing these future research directions, the scientific community can continue to refine and
expand upon the insights gained from integrating dynamic modeling with time-varying
135
transmission and fatality rates, spatial interaction and effect analysis, and vaccination reallocation.
This ongoing research will contribute to a more comprehensive understanding of disease
transmission dynamics and support the development of evidence-based public health policies and
interventions in the face of emerging infectious diseases.
136
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Pandemic prediction and control with integrated dynamic modeling of disease transmission and healthcare resource optimization
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