University of Southern California Dissertations and Theses

The spread of an epidemic on a dynamically evolving network

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The spread of an epidemic on a dynamically evolving network

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Content THE SPREAD OF AN EPIDEMIC ON A DYNAMICALLY EVOLVING NETWORK
by
Fuliang Lyu
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF ARTS
(APPLIED MATHEMATICS)
August 2023
Copyright 2023 Fuliang Lyu
Dedication
To my dearest father and mother.
ii
Acknowledgements
To my supervisor, Professor Cymra Haskell, I would like to express my deepest gratitude. Her insightful
comments and suggestions enable me to complete the entire thesis process, from topic selection to the
final draft.
I would also like to express my gratitude to the Mathematics Department faculty at the University of
Southern California. In two years, I gained a wealth of mathematical knowledge from them.
I am grateful for my family and friends. They consistently offer me emotional support whenever I require
it.
I express my thanks to the students I assisted as a Teaching Assistant. Their acknowledgment of my
abilities has been instrumental in bolstering my confidence.
In addition, I appreciate the University of Southern California. This thesis was completed in a conducive
academic environment created by the university.
iii
TableofContents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2: Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Gillespie Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Variability in the Evolution of the Disease in Network Models . . . . . . . . . . . . . . . . 5
Chapter 3: ODE Compartmental Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 The SIR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Connections with Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 4: Choice of Network in Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Watts-Strogatz Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Watts-Strogatz Network vs ODE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 Barabási-Albert Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.5 Barabási-Albert Network vs ODE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Chapter 5: Proposed Hybrid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.1 Ideas In Proposed Hybrid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 Proposed Method Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Chapter 6: Comparative Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.1 Incremental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2 Method 1 vs Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.3 Method 2 vs Method 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.4 Method 3 vs Method 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.5 Method 4 vs Method 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
iv
6.6 Method 5 vs Method 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.7 Optimal implementation of government policies . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter 7: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
v
ListofTables
4.1 Comparison of Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.1 Remaining Susceptible after 100 days (high beta, controlling on 80 percent remaining
susceptible) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Remaining Susceptible after 100 days (high beta, controlling on 85 percent remaining
susceptible) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3 Remaining Susceptible after 100 days (high beta, controlling on 90 percent remaining
susceptible) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.4 Remaining Susceptible after 100 days (high beta, controlling on 95 percent remaining
susceptible) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.5 Remaining Susceptible after 100 days (low beta, controlling on 80 percent remaining
susceptible) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.6 Remaining Susceptible after 100 days (low beta, controlling on 85 percent remaining
susceptible) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.7 Remaining Susceptible after 100 days (low beta, controlling on 90 percent remaining
susceptible) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.8 Remaining Susceptible after 100 days (low beta, controlling on 95 percent remaining
susceptible) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
vi
ListofFigures
2.1 Left: Gillespie simulation for susceptible individuals on Random Regular Network, total
population size is 20, degree per node is 8, the infection rate is 0.1, the recovery rate is 0.1,
the initial infected is 1; Right: Gillespie simulation for susceptible individuals on Random
Regular Network, total population size is 1000, degree per node is 8, the infection rate is
0.1, the recovery rate is 0.1, the initial infected is 1 . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Gillespie simulation on two Random Regular Networks, total population size is 1000, one
network degree per node is 8, the other network degree per node is 4, the infection rate is
0.1, the recovery rate is 0.1, the initial infected is 1 . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Same ODE method. Left: Gillespie simulation on Fully Connected Network, total
population size 100, degree per node is 99, the infection rate is 0.1, the recovery rate is
0.1, the initial infected is 1; Right: Gillespie simulation on Fully Connected Network, total
population size 100, degree per node is 99, the infection rate is
0.8
99
, the recovery rate is 0.1,
the initial infected is 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Same ODE method. Left: Gillespie simulation on Random Regular Network, total
population size 100, degree per node is 4, the infection rate is 0.1, the recovery rate is 0.1,
the initial infected is 1; Right: Gillespie simulation on Random Regular Network, total
population size 100, degree per node is 8, the infection rate is 0.1, the recovery rate is 0.1,
the initial infected is 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.1 Gillespie simulation on Fully Connected Network, total population size 200, degree per
node is 99, the infection rate is
0.8
99
, the recovery rate is 0.1, the initial infected is 1 . . . . . 16
4.2 Left: k = 2; Middle: k = 4; Right: k = 5 (fully connected) . . . . . . . . . . . . . . . . . . . . 18
4.3 Same ODE method. Left: Gillespie simulation on Watts-Strogatz Network, total population
size 100, degree per node is 4, the infection rate is 0.1, the recovery rate is 0.1, the initial
infected is 1; Right: Gillespie simulation on Watts-Strogatz Network, total population
size 100, degree per node is 8, the infection rate is 0.1, the recovery rate is 0.1, the initial
infected is 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 degree distribution in a Barabasi-Albert network with N = 10000 (log scales) . . . . . . . . 21
4.5 Right: degree distribution in a Watts-Strogatz network with N = 10000 (log scales) . . . . . 21
vii
4.6 Same ODE method. Left: Gillespie simulation on Barabási-Albert Network, total
population size 100, degree per node is 4, the infection rate is 0.1, the recovery rate is
0.1, the initial infected is 1; Right: Gillespie simulation on Barabási-Albert Network, total
population size 100, degree per node is 8, the infection rate is 0.1, the recovery rate is 0.1,
the initial infected is 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.1 Left: Population is 1000, the Infection rate is 0.1, the Recovery rate is 0.1, Degree per
node (Method 1) is 4, Degree per node (Method 2) is 4 (2 in fixed 2 in dynamics); Right:
Population is 1000, the Infection rate is 0.1, the Recovery rate is 0.1, Degree per node
(Method 1) is 8, Degree per node (Method 2) is 8 (2 in fixed 6 in dynamics) . . . . . . . . . 30
6.2 Left: Population is 10000, the Infection rate is 0.1, the Recovery rate is 0.1, Degree per
node (Method 1) is 4, Degree per node (Method 2) is 4 (2 in fixed 2 in dynamics); Right:
Population is 10000, the Infection rate is 0.1, the Recovery rate is 0.1, Degree per node
(Method 1) is 8, Degree per node (Method 2) is 8 (2 in fixed 6 in dynamics) . . . . . . . . . 30
6.3 Left: Population is 1000, the Infection rate is 0.08, the Recovery rate is 0.1, Degree per
node (Method 1) is 4, Degree per node (Method 2) is 4 (2 in fixed 2 in dynamics); Right:
Population is 1000, the Infection rate is 0.08, the Recovery rate is 0.1, Degree per node
(Method 1) is 8, Degree per node (Method 2) is 8 (2 in fixed 6 in dynamics) . . . . . . . . . 31
6.4 Left: Population is 10000, the Infection rate is 0.08, the Recovery rate is 0.1, Degree per
node (Method 1) is 4, Degree per node (Method 2) is 4 (2 in fixed 2 in dynamics); Right:
Population is 10000, the Infection rate is 0.08, the Recovery rate is 0.1, Degree per node
(Method 1) is 8, Degree per node (Method 2) is 8 (2 in fixed 6 in dynamics) . . . . . . . . . 31
6.5 population is 10000, the infection rate is 0.1, the recovery rate is 0.1, Degree per node is 8
(2 in fixed 6 in dynamics)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.6 Left: Population is 10000, the Infection rate is 0.1, the Recovery rate is 0.1, Degree per node
(Method 2) is 8, Degree per node (Method 3) in pre-pandemic period is 8 (2 in fixed 6 in
dynamics) Degree per node (Method 3) in pandemic period is 4 (2 in fixed 2 in dynamics),
pre-pandemic means less than 10 percent people infected; Right: Population is 10000, the
Infection rate is 0.05, the Recovery rate is 0.1, Degree per node (Method 2) is 8, Degree per
node (Method 3) in pre-pandemic period is 8 (2 in fixed 6 in dynamics) Degree per node
(Method 3) in pandemic period is 4 (2 in fixed 2 in dynamics), pre-pandemic means less
than 10 percent people infected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.7 population is 10000, the infection rate is 0.1, the recovery rate is 0.2, Degree per node
(pre-pandemic) is 8, Degree per node (pandemic) is 4, pre-pandemic criteria is 0.9 . . . . . 35
6.8 Left: Population is 1000, the Infection rate (Method 3) for all contacts is 0.1, the Infection
rate (Method 4) for the fixed network is 0.1, the Infection rate (Method 4) for dynamics
network is 0.05, the Recovery rate is 0.2, Degree per node in pre-pandemic period is 8,
Degree per node in pandemic period is 4; Right: Population is 1000, the Infection rate
(Method 3) for all contacts is 0.2, the Infection rate (Method 4) for the fixed network is 0.2,
the Infection rate (Method 4) for dynamics network is 0.1, the Recovery rate is 0.2, Degree
per node in pre-pandemic period is 8, Degree per node in pandemic period is 4; . . . . . . 36
viii
6.9 Left: Population is 1000, the Infection rate (Method 3) for all contacts is 0.05, the Infection
rate (Method 4) for the fixed network is 0.1, the Infection rate (Method 4) for dynamics
network is 0.05, the Recovery rate is 0.2, Degree per node in pre-pandemic period is 8,
Degree per node in pandemic period is 4; Right: Population is 1000, the Infection rate
(Method 3) for all contacts is 0.1, the Infection rate (Method 4) for the fixed network is 0.2,
the Infection rate (Method 4) for dynamics network is 0.1, the Recovery rate is 0.2, Degree
per node in pre-pandemic period is 8, Degree per node in pandemic period is 4; . . . . . . 37
6.10 population is 10000, the infection rate for the fixed network is 0.2, the infection rate for
the dynamics network is 0.05, the recovery rate is 0.2, Degree per node (pre-pandemic) is
8, Degree per node (pandemic) is 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.11 Left: Population is 1000, the Infection rate for the fixed network is 0.2, the Infection
rate for dynamics network is 0.05, the Recovery rate (Method 4) for all people is 0.2, the
Recovery rate (Method 5) for young people is 0.2, the Recovery rate (Method 5) for elder
people is 0.1, Degree per node in pre-pandemic period is 8, Degree per node in pandemic
period is 4; Right: Population is 1000, the Infection rate for the fixed network is 0.2, the
Infection rate for dynamics network is 0.05, the Recovery rate (Method 4) for all people is
0.16, the Recovery rate (Method 5) for young people is 0.16, the Recovery rate (Method 5)
for elder people is 0.08, Degree per node in pre-pandemic period is 8, Degree per node in
pandemic period is 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.12 Left: Population is 1000, the Infection rate for the fixed network is 0.2, the Infection
rate for dynamics network is 0.05, the Recovery rate (Method 4) for all people is 0.1, the
Recovery rate (Method 5) for young people is 0.2, the Recovery rate (Method 5) for elder
people is 0.1, Degree per node in pre-pandemic period is 8, Degree per node in pandemic
period is 4; Right: Population is 1000, the Infection rate for the fixed network is 0.2, the
Infection rate for dynamics network is 0.05, the Recovery rate (Method 4) for all people is
0.08, the Recovery rate (Method 5) for young people is 0.16, the Recovery rate (Method 5)
for elder people is 0.08, Degree per node in pre-pandemic period is 8, Degree per node in
pandemic period is 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.13 Population is 1000, the Infection rate for the fixed network is 0.2, the Infection rate for
dynamics network is 0.05, the Recovery rate (Method 4) for all people is 0.18, the Recovery
rate (Method 5) for young people is 0.2, the Recovery rate (Method 5) for elder people is
0.1, Degree per node in pre-pandemic period is 8, Degree per node in pandemic period is 4 42
6.14 population is 10000, the infection rate for the fixed network is 0.2, the infection rate for
the dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate
for elder people is 0.1, Degree per node (pre-pandemic) is 8, Degree per node (pandemic)
is 4, both networks are Barabasi-Albert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
ix
6.15 Left: Population is 1000, the Infection rate for the fixed network is 0.1, the Infection rate for
dynamics network is 0.05, both fixed and dynamics networks are Barabasi-Albert model
(Method 5), the fixed network is Watts-Strogatz model (Method 6), dynamics network is
Barabasi-Albert model (Method 6), the Recovery rate for young people is 0.2, the Recovery
rate for elder people is 0.1, Degree per node in pre-pandemic period is 10, Degree per
node in pandemic period is 6; Right: Population is 10000, the Infection rate for the fixed
network is 0.1, the Infection rate for dynamics network is 0.05, both fixed and dynamics
networks are Barabasi-Albert model (Method 5), the fixed network is Watts-Strogatz
model (Method 6), dynamics network is Barabasi-Albert model (Method 6), the Recovery
rate for young people is 0.2, the Recovery rate for elder people is 0.1, Degree per node in
pre-pandemic period is 10, Degree per node in pandemic period is 6 . . . . . . . . . . . . . 45
6.16 population is 10000, the infection rate for the fixed network is 0.1, the infection rate for
the dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate
for elder people is 0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic)
is 6, the fixed network is Watts-Strogatz, the rewiring probability is 0.2, the dynamic
network is Barabási-Albert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.17 population is 1000, the infection rate for the fixed network is 0.1, the infection rate for the
dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate for
elder people is 0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6,
the fixed network is Watts-Strogatz, the rewiring probability is 0.2, the dynamic network
is Barabási-Albert, controlling on 80 percent remaining susceptible . . . . . . . . . . . . . 49
6.18 population is 1000, the infection rate for the fixed network is 0.1, the infection rate for the
dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate for
elder people is 0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6,
the fixed network is Watts-Strogatz, the rewiring probability is 0.2, the dynamic network
is Barabási-Albert, controlling on 85 percents remaining susceptible . . . . . . . . . . . . . 50
6.19 population is 1000, the infection rate for the fixed network is 0.1, the infection rate for the
dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate for
elder people is 0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6,
the fixed network is Watts-Strogatz, the rewiring probability is 0.2, the dynamic network
is Barabási-Albert, controlling on 90 percents remaining susceptible . . . . . . . . . . . . . 51
6.20 population is 1000, the infection rate for the fixed network is 0.1, the infection rate for the
dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate for
elder people is 0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6,
the fixed network is Watts-Strogatz, the rewiring probability is 0.2, the dynamic network
is Barabási-Albert, controlling on 95 percents remaining susceptible . . . . . . . . . . . . . 52
6.21 population is 1000, the infection rate for the fixed network is 0.04, the infection rate for
the dynamics network is 0.02, the recovery rate for young people is 0.2, the recovery rate
for elder people is 0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic)
is 6, the fixed network is Watts-Strogatz, the rewiring probability is 0.2, the dynamic
network is Barabási-Albert, controlling on 80 percents remaining susceptible . . . . . . . . 53
x
6.22 population is 1000, the infection rate for the fixed network is 0.04, the infection rate for
the dynamics network is 0.02, the recovery rate for young people is 0.2, the recovery rate
for elder people is 0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic)
is 6, the fixed network is Watts-Strogatz, the rewiring probability is 0.2, the dynamic
network is Barabási-Albert, controlling on 85 percents remaining susceptible . . . . . . . . 54
6.23 population is 1000, the infection rate for the fixed network is 0.04, the infection rate for
the dynamics network is 0.02, the recovery rate for young people is 0.2, the recovery rate
for elder people is 0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic)
is 6, the fixed network is Watts-Strogatz, the rewiring probability is 0.2, the dynamic
network is Barabási-Albert, controlling on 90 percents remaining susceptible . . . . . . . . 55
6.24 population is 1000, the infection rate for the fixed network is 0.04, the infection rate for
the dynamics network is 0.02, the recovery rate for young people is 0.2, the recovery rate
for elder people is 0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic)
is 6, the fixed network is Watts-Strogatz, the rewiring probability is 0.2, the dynamic
network is Barabási-Albert, controlling on 95 percents remaining susceptible . . . . . . . . 56
xi
Abstract
The accurate modeling of social contacts plays a crucial role in predicting the dynamics of infectious
disease transmission. This study addresses the limitations of using a single fixed Barabási-Albert network
as a model of social contacts. In particular, a single fixed Barabási-Albert network does not accurately
reflect real-world social contacts since it does not differentiate between casual and close social contacts,
does not take into account people’s changing behavior upon recognition of the pandemic, assumes all
the individuals have the same infection rates and recovery rates and lacks high clustering properties. To
overcome these limitations, we propose a hybrid model, that adds a dynamic network to an existing fixed
network. The hybrid model is introduced as five incremental improvements. The implications of these
improvements are assessed using the control variables method. In addition, an exploration of suitable one-
to-one ordinary differential equations for each of the five network models is conducted. By integrating
the gap between fixed and dynamic networks, this research provides a more realistic and comprehensive
approach to simulating the infectious disease process, thus paving the way for further research in the field
of infectious disease modeling and control strategies.
xii
Chapter1
Introduction
In recent years, epidemics such as HIV/AIDS, severe acute respiratory syndrome (SIRS), and COVID-19
have posed enormous risks to society and wreaked devastation. The COVID-19 epidemic, which began
in late 2019, continues to affect every country in the world. According to the latest data from the World
Health Organization (2023) [42], it has caused more than 700 million infections and nearly 7 million deaths
worldwide. In order to effectively reduce the impact of epidemics, it is not sufficient to study them only
from biological perspectives. Mathematical modeling has proven to be an important tool for analyzing the
complex dynamics of infectious disease epidemics. Within this domain, ordinary differential equations
(ODE) models and network models emerge as the predominant modeling frameworks.
The ODE models generally permit rigorous analysis allowing us to compute crucial quantities, such as epi-
demic thresholds, disease peaks, and control times. They can be used to inform public health interventions
including vaccination strategies, social distancing measures, and isolation protocols. (Miller, 2017)[25](Jia
et al., 2020)[17](Singh and Adhikari, 2020)[34](Cao et.al, 2021)[9]
Network models explicitly incorporate a social network and they can be more realistic especially when
they include things such as real-world data, spatial dynamics, individual behavior, and other factors that
significantly influence disease spread. (Mikler et al. 2005)[23](White et al. 2007)[41](Volz, 2008)[38](Prette-
john et al., 2011)[29](Haw et al., 2020)[14](Biswas et al., 2020)[7](Rafiq et al., 2023) [31] However, network
1
models don’t generally permit rigorous analysis making them hard to use in specific cases.
This thesis refines a Barabási-Albert network model by combining the fixed network in the model with
a dynamically evolving network. This choice allows for more realistic simulations without the need for
dynamically altering edge weights, which would impose greater computational demands. We explore the
differences between the evolution of disease using this proposed hybrid network and the fixed Barabási-
Albert network.
We restrict our focus in this thesis to diseases in which individuals progress from susceptible, to infected,
and then to recovered, acquiring lifelong immunity. In addition, we assume a closed system with no birth,
death, or migration. We deliberately keep the scenarios simple to effectively demonstrate the differences
between the two network approaches.
Here is the structure of the thesis:
In Chapter 2, we carefully describe what is meant by a network model approach. We also discuss the
relationship between the total population size (N) and the level of determinism exhibited by the network
model.
In Chapter 3, we derive the ODE compartmental model as the average behavior of a network model on
a fully connected network. This is a deterministic model facilitating easy analysis. We show that with a
suitable choice of parameters, the ODE model can provide a good approximation of a network model on a
random regular network.
In Chapter 4, we introduce two additional network construction methods, namely the Watts-Strogatz net-
work and the Barabasi-Albert network. These networks possess unique characteristics making them more
akin to real-world human social networks. We show that the ODE can provide a relatively good approxi-
mation for the network model using these networks under certain circumstances.
In Chapter 5, we address the limitations of using a single fixed Barabási-Albert network as a model of social
contacts. In particular, a single fixed Barabási-Albert network does not accurately reflect real-world social
2
contacts since it does not differentiate between casual and close social contacts, does not take into account
people’s changing behavior upon recognition of the pandemic, assumes all the individuals have the same
infection rates and recovery rates and lacks high clustering properties. To overcome these limitations, we
propose a hybrid model, that adds a dynamic network to an existing fixed network. The hybrid model is
introduced as five incremental improvements.
In Chapter 6, we apply the control variables method to assess the implications of these improvements. We
investigate the discrepancies between simulations generated by each pair of network models. For each
network established, we also refine the corresponding ODE by rewriting it to better match the character-
istics of the network. we provide suggestions on how to reduce the number of infections.
In Chapter 7, we present a comprehensive summary of the advantages of the proposed model, potential
limitations, and suggestions for future research directions.
3
Chapter2
NetworkModels
2.1 GillespieSimulation
In a network model of an infectious disease, the disease evolves over an underlying graph or network. The
vertices of the graph represent individuals, and individuals who interact are connected by an edge. For
the simple disease considered in this thesis, at every point in time each individual is in one of three states:
susceptible (S), infected (I), or recovered (R). Initially, one randomly chosen individual is in state (I) and all
others are in state (S).
Individuals change state when one of the following types of events occurs: (1) an infection event, where
an infected individual infects a susceptible individual to whom he/she is connected by an edge, causing the
susceptible individual to pass from (S) to (I) and (2) a recovery event where an individual transitions from
infected (I) to recovered (R). In the model, these events are considered to occur as independent Poisson
processes with infection rate (β ) and recovery rate (γ ) respectively.
Given the network, the evolution of the disease can be easily simulated on a computer using the Gillespie
Simulation, also known as the Stochastic Simulation Algorithm (SSA). This is an event-driven method of
simulation developed by Gillespie (1976) [13]. It rests on the memoryless property of Poisson processes;
irrespective of what happened in the past, the time until the next event has an exponential distribution
whose mean is the reciprocal of the rate at which events occur. When independent Poisson processes are
4
combined, the resulting process is Poisson with a rate that is equal to the sum of the rates in the individ-
ual processes. Thus, infection events occur at rate λ 1
= β (# S-I edges), recovery events occur at rate
λ 2
=γ (# I-vertices), and the total overall rate at which events occur isλ =λ 1
+λ 2
. (Keeling and Eames,
2005)[18] Simulation proceeds by determining the time until the next event as an exponentially distributed
variable with mean
1
λ . Once the time until the next event is determined, the nature of that event can be de-
termined by generating a random number p, uniformly distributed on the interval [0,1]. Ifp<
λ 1
λ 1
+λ 2
, that
event is an infection event; otherwise, it is a recovery event. Once the nature of the event is determined, it
remains to determine the precise edge (in the case of an infection event) or the precise vertex (in the case
of a recovery event) where the event occurred. This is done by choosing the edge or vertex at random with
all S-I edges, respectively all I vertices, having the same probability of being chosen. The network is then
updated and the new infection rateλ 1
, the new recovery rateλ 2
, and the new event rateλ =λ 1
+λ 2
are
calculated. Utilizing the new exponential distribution, the next event can then be simulated. This iterative
process continues.
2.2 VariabilityintheEvolutionoftheDiseaseinNetworkModels
To understand the impact of an epidemic on a community, it is not necessary to know which individuals
are in which state at every point in time. Instead, it suffices to know how the total numbers of susceptible,
infected, and recovered individuals evolve over time. In a network model, this is random. However, when
the total population size (N) is large, most simulations are similar leading to increased predictability in the
outcomes.
To illustrate this, consider the simulations in Figure 2.2. The figure on the left shows 10 simulations on a
network with N = 20 individuals and the figure on the right shows 10 simulations with N = 1000 individuals.
5
The simulation has more variability with small N and less variability with large N. The two networks used
in these simulations were constructed as random regular networks (Steger, Wormald 1999) [35].
Figure 2.1: Left: Gillespie simulation for susceptible individuals on Random Regular Network, total popu-
lation size is 20, degree per node is 8, the infection rate is 0.1, the recovery rate is 0.1, the initial infected
is 1; Right: Gillespie simulation for susceptible individuals on Random Regular Network, total population
size is 1000, degree per node is 8, the infection rate is 0.1, the recovery rate is 0.1, the initial infected is 1
However, even though in a fixed network with large N the variability is small, there can be significant
variability between models depending on the nature of the network. To illustrate this, consider the simula-
tions in Figure 2.3. This figure shows how the numbers of susceptible, infected, and recovered individuals
evolve over time in a Random Regular network where the degree per node is 8 and in a Random Regular
network where the degree per node is 4. Notice the simulations are very different; in the network where
the degree per node is 8, all 1000 people are infected; while in the network where the degree per node is
4, only 800 people are infected.
6
Figure 2.2: Gillespie simulation on two Random Regular Networks, total population size is 1000, one net-
work degree per node is 8, the other network degree per node is 4, the infection rate is 0.1, the recovery
rate is 0.1, the initial infected is 1
7
Chapter3
ODECompartmentalModels
3.1 TheSIRModel
In an ode compartmental model, there is no attempt to model the states of all individuals in the population;
instead, we only track how the numbers of individuals in each of the three states S,I,R evolve over time.
The ode model can be derived from the Reed-Frost model. This is a discrete-time model in which the time
step is taken to be the amount of time it takes someone to recover from the disease. (Abbey, 1952)[1] The
population is assumed to be well-mixed; every infected person is as likely as every other infected person
to pass the disease on to a susceptible person. At each discrete time step, the whole of the I group passes to
the R group, i.e.R
n+1
=R
n
+I
n
, a random numberX
n
in the S group gets infected, i.e.S
n+1
=S
n
− X
n
,
and the newly infected individuals become part of the I group for the next time step, i.e. I
n+1
= X
n
.
Consider the event that a particular susceptible person meets a particular infected person in one time step,
causing the susceptible person to become infected. If we assume these events are independent and all
occur with probability p then the distribution ofX
n
is binomial where the probability of success each trial
is1− (1− p)
In
and the number of trials isS
n
(a "trial" is the observation of whether or not a particularly
susceptible person becomes infected and "success" is that the person becomes infected). When N is large,
the simulations of the Reed-Frost model exhibit little variability, suggesting a deterministic model might
8
be appropriate. If we replace the random variableX
n
with its expected valueS
n
(1− (1− p)
In
), this gives
us the model
S
n+1
= S
n
− S
n
((1− (1− p)
In
))=S
n
(1− p)
In
I
n+1
= S
n
(1− (1− p)
In
)
R
n+1
= R
n
+I
n
We can also let the time step vary by assuming each person recovers as a Poisson process so that the
number of recoveries in a small time step isbI
n
for some number of b that depends on the recovery rate
and the time step. If we simplify(1− p)
In
bye
− aIn
, this gives us the model
S
n+1
= S
n
e
− aIn
I
n+1
= I
n
+S
n
(1− e
− aIn
)− bI
n
=S
n
(1− e
− aIn
)+(1− b)I
n
R
n+1
= R
n
+bI
n
Taking a limit as the time step∆ t→0, we get the ode compartmental model (Kermack and McKendrick,
1927)[19]
dS
dt
= − αSI
dI
dt
= αSI − γI
dR
dt
= γI
Notice the main underlying assumption in the ode compartmental model is that the population is well-
mixed. So we expect it should provide a good approximation to a network model on a fully connected
network when N is large. However, we must pay attention to how the values of the parameters in the
9
network model compare with the values in the ode model. In particular, recall the parameterα came from
the parameter p which was the probability a particular susceptible and particular infected person met in
one time step causing the susceptible person to become infected. We would expect p to decrease as N
increases since people have a limited number of interactions per day. Indeed, dimension analysis shows
that[α ] =
1
(time)(people)
and[γ ] =
1
(time)
. (Thieme, 2018)[37] Hethcote (1976) [16] introduced a new term
σ called the "contact rate," which replaced the previously used term "infection rate" α . Additionally, he
normalized the population size N by introducing the parameterα , defined as α =
σ N
, where[σ ] =
1
(time)
.
With this formulation the contact rateσ is a characteristic of the disease and the culture of the population
(e.g. how close people stand to each other) and does not depend on the population size N. The following
ordinary differential equations, using this property, characterize the rates of change over time for each
compartment (Allen, 2008) [8]:
dS
dt
= − σ N
SI
dI
dt
=
σ N
SI− γI
dR
dt
= γI
The infection rate(− σ N
SI) can be rewritten as(− σS
I
N
). The term (σS ) represents the total number of
contacts all susceptible individuals combined have per day, and (
I
N
) represents the proportion of infected
people in the total population. So(− σS
I
N
) represents the expected number of contacts between suscepti-
ble and infected individuals per day. The term(γI ) corresponds to the rate at which infected individuals
recover and transition into the recovered compartment;(
1
γ ) is the average infection time.
Notice, as Hethcote (1976) [16] pointed out, the SIR model assumes a well-mixed homogeneous population,
where individuals have equal chances of interacting with one another, disregarding variations in location
10
or social connections. This corresponds to a network model on a fully connected network. So we expect
the ode model to be a good approximation to a network model on a fully connected network when N is
large (and there is little variation in simulations on the network model).
In the model of infectious disease transmission, the ODE Compartment model utilizes mathematical ex-
pressions to govern transitions between compartments. This approach offers a valuable simplification for
simulating the intricate dynamics of disease spread, making it easier to analyze and understand the impact
of individual parameters.
The simulations presented in Figure 3.1 demonstrate how well an ode model approximates a network
model on a fully connected network. In both cases N = 100. The parameterβ referred to in the figures (see
Section 3.2) isβ =
σ N− 1
=
σ 99
. Thus in the plot on the leftσ = 9.9 and in the plot on the rightσ = 0.8.
The ode does an excellent job in both cases though it is slightly better in the case of higherβ .
Figure 3.1: Same ODE method. Left: Gillespie simulation on Fully Connected Network, total population
size 100, degree per node is 99, the infection rate is 0.1, the recovery rate is 0.1, the initial infected is 1;
Right: Gillespie simulation on Fully Connected Network, total population size 100, degree per node is 99,
the infection rate is
0.8
99
, the recovery rate is 0.1, the initial infected is 1
11
3.2 ConnectionswithNetworkModels
The ode compartmental model assumes the population is well-mixed; all individuals interact with all other
individuals. The whole point of a network model is to not make this assumption but instead explicitly
model social interactions. However, if we choose the corresponding parameters with a little more care, the
ode model can still provide a good approximation to some network models.
In a network model, the recovery rate isγ (# I-vertices)=γI . This is the same as the recovery rate in the
ode compartmental model. Thus, theγ ’s in the two models should be the same. In a network model, the
infection rate is β (# S-I edges). Let k = average degree per node in the network. In a random network,
we would expect the total number of edges stemming from S-vertices to bekS of which a proportion
I
N
should connect to an infected individual. Thus, we would expect the infection rate to be β (kS)
I
N
. This
means, for the ode compartmental model to provide a good approximation for the network model, we
should expect to needσ =βk . The ODE compartmental model that we expect to provide a good average
of a network model is the following:
dS
dt
= − βk
I
N
S
dI
dt
= βk
I
N
S− γI
dR
dt
= γI
We utilize Random Regular networks, proposed by Steger and Wormald (1999) [35], to explore the appli-
cability of the ordinary differential equation (ODE) as a good approximation for the network model when
k is smaller than N. The simulations presented in Figure 3.2 demonstrate how well the ode compartmental
model approximates a network model on a Random Regular network. In both cases, N = 100. The simula-
tions are simulated with comparison conditions: (1) low-connected network; (2) high-connected network.
12
For the low-connected network, the network construction method is a Random Regular network with a de-
gree per node equal to 4. For the high-connected network, the network construction method is a Random
Regular network with a degree per node equal to 8.
Figure 3.2: Same ODE method. Left: Gillespie simulation on Random Regular Network, total population
size 100, degree per node is 4, the infection rate is 0.1, the recovery rate is 0.1, the initial infected is 1;
Right: Gillespie simulation on Random Regular Network, total population size 100, degree per node is 8,
the infection rate is 0.1, the recovery rate is 0.1, the initial infected is 1
In Fig 3.2, the plot generated with a high-connected network closely resembles the one produced by
the ODE method, whereas the plot drawn with a low-connected network shows more divergence with
fewer total people becoming infected in the network model compared to the ode model.
It is worth noting that a comparison between Fig 3.1 (Right) and Fig 3.2 (Right) is essential, as both have
βk =0.8. In other words, the ODE equations in Fig 3.1 (Right) and Fig 3.2 (Right) are identical. The simu-
lations in these two graphs show the approximation of the lines drawn by these two Gillespie simulations
closely matches the ODE equations, although the former uses a fully-connected network and the latter a
Random Regular network. In order words, the bigger the term (βk ) is, the closer the approximation be-
tween the ODE method and the Gillespie simulation. The infection rateβ is a property of the disease and
the culture of the population often assumes to be constant. In such cases, when the infection rate is the
13
same, increasing the average degree per node (k) leads to a closer alignment between the results obtained
from the Gillespie simulation and the ODE method.
14
Chapter4
ChoiceofNetworkinNetworkModels
4.1 Background
Although the ODE model provided a good approximation to the network model in the case when the
underlying network was a Random Regular network andβk was large, it does not provide a good approxi-
mation in the case of all networks even ifβk is large. Take an extreme network G as an example. G is built
by two separate fully connected networks with a population of 100. We use the same large value of (βk ) as
0.8 in Fig 3.2 (Right) and Fig 3.3 (Right). As the simulations presented in Figure 4.1 show the ODE model
does not provide a good approximation to the network model G even with this large value of βk . Since
the number of individuals initially infected is 1, only one of the two fully connected networks composing
G has an infected individual, so almost half of the whole population becomes infected.
This extreme example serves to demonstrate how variations in the network’s structure can impact the
precision of ODE model fitting. This begs the question - can the ode model provide a good approximation
for epidemics spreading on human networks?
15
Figure 4.1: Gillespie simulation on Fully Connected Network, total population size 200, degree per node is
99, the infection rate is
0.8
99
, the recovery rate is 0.1, the initial infected is 1
Random regular networks and Fully connected networks above have notable differences from realistic
human networks. Firstly, they oversimplify the population by homogenizing individuals, assuming each
node has the same degree. In reality, individuals have varying degrees of social contact. Lanning(2017) [20]
proposed a correlation between social contact and an individual’s personality and Bianchi and Vohs (2016)
[6] found that higher household income is associated with less social contact. Secondly, the regularity of
these networks fails to capture clustering properties present in real social contact networks. Szendroi and
Csányi (2004) [36] pointed out that social networks have clusters, such as families, where individuals tend
to have more connections with each other. Thirdly, these networks do not accurately model the average
shortest path length between vertices. Milgram (1967) [24] pointed out that human social networks exhibit
16
the small-world phenomenon, characterized by a relatively short average path between vertices of about
six.
As Volz et al. (2011) [39] summarized, a more realistic human network should exhibit clustering properties,
degree per node heterogeneity, and the small world phenomenon.
The following section introduces two more complex network models: the Watts-Strogatz model and the
Barabási-Albert model. These network models exhibit degree per node heterogeneity, clustering proper-
ties, and the small world phenomenon. The following section also shows how well the ODE model provides
an approximation to these networks.
4.2 Watts-StrogatzNetwork
Watts-Strogatz Network, proposed by Watts and Strogatz (1998) [40], proposed a method of constructing
networks that exhibit the small-world phenomenon and clustering. The construction depends on a param-
eterp∈[0,1] and a desired average degree per node k. It begins with the construction of a regular lattice,
where each node is connected to its ‘k’ nearest neighbors. Each edge is then rewired to a randomly chosen
new node with probability ‘p’. The rewiring creates the small-world phenomenon; Pastor et al. (2015)[28]
indicated that the larger p is the smaller the average path length between nodes is.
Watts-Strogatz networks can exhibit high clustering. The clustering coefficient of a vertex measures the
density of connections among its neighbors. Watts and Strogatz[40] defined the clustering coefficient of
vertex i as:C
i
=
number of triangles connected to vertex i
number of triples centered on vertex i
, and the clustering coefficient ‘C’ for the entire graph
as the average of all theC
i
’s. Clustering coefficients lie in the interval [0,1].
Fig 4.2 shows the clustering coefficient ‘C’ in a regular lattice for different values of k when N = 6.
17
Figure 4.2: Left: k = 2; Middle: k = 4; Right: k = 5 (fully connected)
In graphs, where C is close to 1, two vertices that are both connected to a vertex i have a high proba-
bility of being connected to each other. A fully connected graph hasC = 1 as do graphs that consist of
two or more disconnected fully connected sub-graphs (provided the fully connected sub-graphs have at
least 3 vertices). A regular lattice hasC =0 whenk =2 but otherwise has a relatively large value for C.
When N is large andk >2, the clustering coefficient for regular lattice without rewiring can be expressed
asC =
3(k− 2)
4(k− 1)
. Barrat and Weigt(2000)[5] showed that the average clustering coefficient in Watts-Strogatz
networks with parameters p and k isC =
3(k− 2)
4(k− 1)
(1− p)
3
.
The results reported by Liu et al. (2015) [22] align with the expression by Barrat and Weigt (2000)[5] illus-
trating the relationship between ’p’ and clustering property in the Watts-Strogatz model.
For low values of ‘p’, the network retains its regular lattice structure, promoting strong local clustering and
limiting the presence of shortcut connections. As ‘p’ increases, the model transitions to a more random
network with decreased clustering and increased presence of shortcut connections.
18
4.3 Watts-StrogatzNetworkvsODEModel
The simulation presented in Figure 4.2 demonstrates how well the ode model compares with the network
on a Watts-Strogatz network. The simulation on the left show the case whenβk =0.4 and the simulation
on the right show the case whenβk =0.8. In both cases, the Watts-Strogatz network rewiring probability
isp=0.2, so the network has a strong clustering property but limits the presence of shortcut connections.
Figure 4.3: Same ODE method. Left: Gillespie simulation on Watts-Strogatz Network, total population
size 100, degree per node is 4, the infection rate is 0.1, the recovery rate is 0.1, the initial infected is 1;
Right: Gillespie simulation on Watts-Strogatz Network, total population size 100, degree per node is 8, the
infection rate is 0.1, the recovery rate is 0.1, the initial infected is 1
When the value ofβk is low, the difference between the simulations of the ODE model and the Watts-
Strogatz model is large. When the value ofβk is high, the ODE model provides a good approximation to
the Watts-Strogatz model but the approximation is not so close compared with the same value ofβk in a
random regular network.
4.4 Barabási-AlbertModel
Scale-free models form a broad class of network models that encompass various mechanisms and pro-
cesses leading to networks with a power-law degree distribution. This power-law degree distribution
19
implies that the probability of a node having k connections to other nodes follows a power-law relation-
ship,P(k)∼ k
− γ , whereγ is a parameter determining the degree of heterogeneity in the network. Pastor
et al.(2015) [28] mentioned that the value ofγ in many real-world networks is between 2 and 3. Scale-free
networks are characterized by the presence of a few highly connected nodes, known as hubs, while the ma-
jority of nodes have relatively fewer connections. Networks can grow into scale-free networks by various
mechanisms. The Barabasi-Albert (BA) model uses preferential attachment as a mechanism for generating
a scale-free network. Pachon et al. (2018) [27] proposed another specific example of a scale-free model.
This model extends the preferential attachment mechanism by incorporating a uniform attachment. Scale-
free models capture the heterogeneous connectivity patterns observed in many real-world networks.
In the Barabási-Albert model, the construction of a random graph involves a preferential attachment mech-
anism, first proposed by Price (1976) [30]. It posits that new nodes joining the network tend to connect
preferentially to existing nodes based on their current degree, resulting in a rich-get-richer effect. In other
words, highly connected are more likely to receive new connections, while less connected nodes are less
likely to attract new links. This mechanism leads to the emergence of hubs.
Barabási and Albert (1999) [3] incorporated the preferential attachment mechanism as the probability that
a new node connecting to an existing node is proportional to its existing degree, which can be expressed
asP(k) =
k
P
k
i
, where k is the degree of the existing node and
P
k
i
represents the sum of degrees of all
existing nodes in the network.
Then, Barabási and Albert (1999) [3] showed that the degree distribution of the Barabási-Albert model
follows a power-law distribution, given byP(k)∼ k
− 3
, where k is the degree of a node.
The distributions of nodes in the Barabási-Albert Network and the Watts-Strogatz Network are shown in
Fig 4.3 and Fig 4.4. The Barabási-Albert Network and the Watts-Strogatz Network are set with the number
of nodes (N) as 10000, and the average degree per node as 8. It shows that the hubs only emerge in the
Barabási-Albert Network and not in the Watts-Strogatz Network.
20
Figure 4.4: degree distribution in a Barabasi-Albert network with N = 10000 (log scales)
Figure 4.5: Right: degree distribution in a Watts-Strogatz network with N = 10000 (log scales)
Furthermore, the Barabási-Albert model exhibits a small-world phenomenon, characterized by a rel-
atively short average path length between nodes. Chen, F., Chen, Z., Wang, and Yuan (2008) [10] proves
21
that the average path length ’L’ in the Barabási-Albert model can be approximated as L ≈ log(N)
log(log(N))
,
where N is the number of the nodes. Set N = 10000, the average path length of the Barabási-Albert model
is between 4 and 5.
The Barabási-Albert model exhibits disassortative mixing, where high-degree nodes tend to connect with
low-degree nodes. (Newman, 2002)[26] Unfortunately, this means that they have a low clustering coeffi-
cient.
4.5 Barabási-AlbertNetworkvsODEModel
The simulation presented in Figure 4.5 demonstrates how well an ode model approximates a network
model on a Barabási-Albert Network. The value ofβk on the left is 0.4 and the value on the right is 0.8.
Figure 4.6: Same ODE method. Left: Gillespie simulation on Barabási-Albert Network, total population
size 100, degree per node is 4, the infection rate is 0.1, the recovery rate is 0.1, the initial infected is 1;
Right: Gillespie simulation on Barabási-Albert Network, total population size 100, degree per node is 8,
the infection rate is 0.1, the recovery rate is 0.1, the initial infected is 1
When the value ofβk is high, the ODE model provides a good approximation to the Barabási-Albert
model. However, even with highβk , the ODE approximation to the Barabási-Albert model is still worse
than the previous fit to the fully connected and random regular networks; but is similar to the previous fit
22
to the Watts-Strogatz network. When the value ofβk is low, the ODE approximation to the Barabási-Albert
model is better than the previous fit to the Watts-Strogatz network.
4.6 Summary
Below is a summary table comparing the Watts-Strogatz Model and the Barabási-Albert Model (scale-free
network). This table provides an overview of the key characteristics of each model, following the individ-
ual introductions of the Watts-Strogatz Model and the Barabási-Albert Model in the previous sections.
Table 4.1: Comparison of Network Models
Model NetworkCreation DegreeDistribution KeyFeature
Watts-Strogatz
Model
Rewiring edges with a
certain probability
Degree distribution with
a peak around the aver-
age degree of the network
Captures small-world be-
havior and some cluster-
ing properties but lacks
hubs
Barabási-Albert
Model (scale-
free network)
New nodes connect to
existing nodes with a
preferential attachment
mechanism
Power-law distribution Exhibits hubs in net-
works and scale-free
behavior but lacks clus-
tering properties of
real-world networks
23
Chapter5
ProposedHybridModel
5.1 IdeasInProposedHybridModel
The goal of this paper is to find a more realistic network model to simulate the spread of an epidemic
among people.
Herrmann and Schwarz (2020) [15] and Saunders and Schwartz (2021) [33] mentioned that some individu-
als, such as doctors in public hospitals, tend to have a higher tendency to have more social contacts. They
called these high-connecting individuals "hubs" and used a Barabási-Albert network to model the spread
of an epidemic among people in their research because the Barabási-Albert model is a scale-free network
that is able to generate power law distribution and shows hubs. In my work, I address there are three
major limitations to their approach.
To illustrate the first limitation, consider again a doctor in a public hospital. When Barabási-Albert is gen-
erated, the doctor is modeled as a hub in the graph, which is a node that connects tens of other nodes.
But note that if only one fixed Barabási-Albert model is used, then the tens of points connecting to the
hub are fixed. In other words, a doctor sees more people daily than other people. This is not the desired
situation. The doctor should see approximately the same number of people as other people each day but
have the probability to meet different groups of patients each day. The second limitation is that there
are some factors, such as government policies or psychological factors, which influence the frequency of
24
individuals’ social contact. When the disease reaches the level of a pandemic, the government may force
people to quarantine and may close down some social places. People may go out less because of panic and
fear of getting sick. These factors will reduce the frequency of individuals’ social contact. The third limi-
tation stems from the inherent drawbacks of the Barabási-Albert model. While the Barabási-Albert model
reflects hubs in a network, it fails to reflect the high clustering property. In contrast, the Watts-Strogatz
model with a low value of the rewiring probability exhibits a high clustering property, shown by Barrat
and Weigt (2000) [5].
My Proposed Hybrid Model is dynamic instead of fixed; it consists of a fixed Watts-Strogatz model com-
bined with a dynamically changing Barabási-Albert model. Returning to the example of the doctor in the
public hospital, the fixed Watts-Strogatz network models the close social contacts people have with family
members, friends, and colleagues. In contrast, the dynamically changing Barabási-Albert network mod-
els social contacts of doctors with patients, as well as random encounters people have with, for example,
checkers at the supermarkets or couriers delivering items in a single day.
To simulate changes in social contact frequency over time, the model distinguishes between the pre-
pandemic and pandemic periods. During the pre-pandemic period, when the number of infected indi-
viduals is relatively small and the seriousness of the disease is not yet recognized by the government or
the public, social contact frequency remains unchanged. However, during the pandemic period, as aware-
ness of infectious diseases increases, people reduce their outdoor activities, leading to a further reduction
in social contact frequency.
25
5.2 ProposedMethodSetup
Using the fixed Barabási-Albert model method as a base, there are five progressive improvements.
(i) Improvement on Dynamicity: Instead of exclusively using a fixed Barabási-Albert network, the pro-
posed method incorporates both a fixed Barabási-Albert network ( G
1
) with a smaller average number of
edges per node (k
1
) and a dynamic Barabási-Albert network (G
2
) with a larger average number of edges
per node (k
2
). The resulting network (G) is a combination of G
1
and G
2
, where the average number of
edges per node is denoted as ’k’ and calculated as the sum ofk
1
andk
2
. The simulation time accumulates,
and when an integer value is exceeded, indicating a new day in the real world, the dynamic Barabási-Albert
network is updated by renewing the connecting edge relationships while preserving the recorded states
of the nodes.
(ii) Temporal Enhancement: To simulate the effect of government policies such as travel restrictions, re-
duced social interaction, and quarantine measures, the model distinguishes between the pre-pandemic
period and the pandemic period. The value of k, representing the average number of edges per node in
the dynamic network, is higher in the pre-pandemic period and lower in the pandemic period.
(iii) Enhancement of Infection Rate Differentiation: In contrast to the original model’s use of a single in-
fection rate β , the improved model differentiates the infection rate of intimate social contacts from that
of other social contacts. The fixed Barabási-Albert network covers intimate social contacts, while the dy-
namic Barabási-Albert network represents other social contacts. Rea et al. (2007) [32] pointed out that
close social contact and long exposure time increase the risk of disease transmission. Consequently, the
infection rate (β f
) in the fixed network is greater than the infection rate ( β d
) in the dynamic network. Now
there are two types of infection events: (1) infections that happen on the fixed network at rate λ 1f
= β f
(# S-I edges in the fixed network); (2) infections that happen on the dynamic network at rate λ 1d
= β d
(# S-I edges in the dynamics network).
(iv) Improvement on Age-based Recovery Rate Differentiation: Unlike the existing model, which assumes
26
all individuals have the same recovery rateγ , the improved model considers age as a differentiating fac-
tor. Specifically, individuals are divided into two age groups: those over 65 (older) and those not over 65
(younger). Bajaj et al. (2021) [2] indicates that older individuals have a lower recovery rate (γ o
) due to their
less capable immune system compared to younger individuals, who have a higher recovery rate (γ y
). The
initial population in the model comprises a combination of twenty percent older people and eighty percent
younger people, reflecting the age distribution in the U.S. population in 2030, predicted by Fulmer et al.
(2021)[12]. Now there are two types of recovery events: (1) recovery of older people at rateλ 2o
= γ o
(#
I-vertices among older individuals); (2) recovery of younger people at rateλ 2y
= γ y
(# I-vertices among
younger individuals).
(v) Improvement on Clustering: The original model employed Barabási-Albert networks due to its prefer-
ential attachment mechanism, which captures population hubs effectively but lacks high-clustering prop-
erties. To address this limitation, the improved model introduces a fixed Watts-Strogatz network alongside
the dynamic Barabási-Albert network. This modification enhances the model’s ability to exhibit high-
clustering properties of the Watts-Strogatz network proposed by Barrat et al. (2008) [4] while maintaining
the advantages of the Barabási-Albert network in capturing hubs within the population. Liu et al. (2015)
[22] displayed that while setting the rewiring probability as 0.2, the Watts-Strogatz network maintains the
small-world property.
27
Chapter6
ComparativeSimulations
6.1 Incremental
The analysis section of this paper adopts a control variables approach. As discussed in the "Proposed Model
Setup" section, a total of five improvements are employed to construct social networks and simulate SIR
epidemics. Each improvement is compared with the previous method. These six methods are progressive,
and the improvement resulting from each method is clearly discernible through a comparative simulation
of each pair of methods.
To ensure consistency, the initial number of infected individualsI
0
is set to 1. Notice there is a nonzero
probability that the disease dies out before a significant number of people become infected. In this Chapter,
we only show "valid" simulations where an epidemic occurs. This choice is crucial because our aim is to
compare the impact of different improvements on the evolution of the disease. In the plots below, each
method is only simulated one time with each set of parameter values. So the reader should bear in mind
that the models are random so simulations exhibit variability.
Liu et al. (2021) [21] and Drolet et al. (2022) [11] conducted studies on social patterning before and after
the COVID-19 outbreak, providing valuable insights for the convenience of network modeling. Liu et
al.(2021[21]) found that the number of social contacts per day ranges between 7 and 26, while in the
28
pandemic period, it decreases to between 2 and 5. In the simulations we show below, our choice of the
average degree per node on the underlying graphs is based on this research.
6.2 Method1vsMethod2
Method 1, known as the ’fixed Barabási model’, has been widely utilized in numerous studies.
Method 2, referred to as the ’fixed-dynamic combined Barabási model’, builds upon the first method and
provides a better representation of the stochastic nature of social contacts.
Each figure shows a comparison between one simulation of Method 1 and one simulation of Method 2. In
all figures, the average degree per node on the left is 4 and the average degree per node on the right is 8. In
Figure 6.1 N is 1000 and in figure 6.2 N is 10000. We can see the difference between Method 1 and Method
2 is more pronounced when N is larger. Figure 6.3 and Figure 6.4 are similar to Figure 6.1 and Figure 6.2
respectively with a lower valueβ . Figures 6.1 to 6.4 show that the total number of infected individuals in
Method 2 is larger than that in Method 1. In all figures, the difference between the two methods is more
pronounced on the left plots with a degree per node of 4, whereas the difference is smaller on the right
plots with a degree per node of 8. This suggests that when the degree per node is 8, the network is already
highly connected, which coincides with the results in Chapter 2. Method 2 using a combined fixed and
dynamic Barabási-Albert network allows for potential connections between points that are not connected
in a single fixed Barabási-Albert network of Method 1.
29
Figure 6.1: Left: Population is 1000, the Infection rate is 0.1, the Recovery rate is 0.1, Degree per node
(Method 1) is 4, Degree per node (Method 2) is 4 (2 in fixed 2 in dynamics); Right: Population is 1000, the
Infection rate is 0.1, the Recovery rate is 0.1, Degree per node (Method 1) is 8, Degree per node (Method
2) is 8 (2 in fixed 6 in dynamics)
Figure 6.2: Left: Population is 10000, the Infection rate is 0.1, the Recovery rate is 0.1, Degree per node
(Method 1) is 4, Degree per node (Method 2) is 4 (2 in fixed 2 in dynamics); Right: Population is 10000, the
Infection rate is 0.1, the Recovery rate is 0.1, Degree per node (Method 1) is 8, Degree per node (Method
2) is 8 (2 in fixed 6 in dynamics)
30
Figure 6.3: Left: Population is 1000, the Infection rate is 0.08, the Recovery rate is 0.1, Degree per node
(Method 1) is 4, Degree per node (Method 2) is 4 (2 in fixed 2 in dynamics); Right: Population is 1000, the
Infection rate is 0.08, the Recovery rate is 0.1, Degree per node (Method 1) is 8, Degree per node (Method
2) is 8 (2 in fixed 6 in dynamics)
Figure 6.4: Left: Population is 10000, the Infection rate is 0.08, the Recovery rate is 0.1, Degree per node
(Method 1) is 4, Degree per node (Method 2) is 4 (2 in fixed 2 in dynamics); Right: Population is 10000, the
Infection rate is 0.08, the Recovery rate is 0.1, Degree per node (Method 1) is 8, Degree per node (Method
2) is 8 (2 in fixed 6 in dynamics)
In Method 2, the average degree per node (k) is the sum of the degree per node in the fixed network
(k
f
) and the degree per node in the dynamic network (k
d
). The modified ODE can be expressed as (shown
31
in Fig 6.5):
dS
dt
= − β N
(k
f
+k
d
)SI
dI
dt
=
β N
(k
f
+k
d
)SI− γI
dR
dt
= γI
whereβ is the infection rate,γ is the recovery rate,k
f
is the degree per node in the fixed network, and k
d
is the degree per node in the dynamics network.
Figure 6.5: population is 10000, the infection rate is 0.1, the recovery rate is 0.1, Degree per node is 8 (2 in
fixed 6 in dynamics))
32
6.3 Method2vsMethod3
Method 2, referred to as the ’fixed-dynamic combined Barabási model’, builds upon the first method and
provides a better representation of the stochastic nature of social contacts.
Method 3, referred to as the ’pre-pandemic pandemic fixed-dynamic combined Barabási model,’ incorpo-
rates the temporal variation in the number of social contacts. Specifically, we have a pre-pandemic and
a pandemic period. In the pre-pandemic period, people do their normal activities; in the pandemic pe-
riod, the government imposes reduced social contacts. In this section, the pandemic period starts when
10 percent of the population has contracted the disease. The impact of this choice of percentage level is
thoroughly discussed in the section titled "Optimal implementation of government policies".
We set the degree per node for Method 2 to be 8, which corresponds to the pre-pandemic period in Method
3. We set the degree per node for the pandemic period in Method 3 to be 4. Fig 6.6 shows the total number
of infected individuals in Method 3 is smaller than the total number of infected individuals in Method
2. This suggests that the number of infections can be reduced if the government detects the disease and
implements effective control police to reduce social contact.
33
Figure 6.6: Left: Population is 10000, the Infection rate is 0.1, the Recovery rate is 0.1, Degree per node
(Method 2) is 8, Degree per node (Method 3) in pre-pandemic period is 8 (2 in fixed 6 in dynamics) Degree
per node (Method 3) in pandemic period is 4 (2 in fixed 2 in dynamics), pre-pandemic means less than
10 percent people infected; Right: Population is 10000, the Infection rate is 0.05, the Recovery rate is 0.1,
Degree per node (Method 2) is 8, Degree per node (Method 3) in pre-pandemic period is 8 (2 in fixed 6 in
dynamics) Degree per node (Method 3) in pandemic period is 4 (2 in fixed 2 in dynamics), pre-pandemic
means less than 10 percent people infected
Hence, the modified ODE can be expressed as shown in Fig 6.7:
when S ≥ αN :
dS
dt
= − β N
(k
f
+k
d1
)SI
dI
dt
=
β N
(k
f
+k
d1
)SI− γI
dR
dt
= γI
when S < αN :
dS
dt
= − β N
(k
f
+k
d2
)SI
dI
dt
=
β N
(k
f
+k
d2
)SI− γI
dR
dt
= γI
34
where α is the pre-pandemic criteria, β is the infection rate, γ is the recovery rate, k
f
is the degree per
node in the fixed network, k
d1
is the degree per node in the dynamics network in the pre-pandemic period,
andk
d2
is the degree per node in the dynamics network in the pandemic period.
Figure 6.7: population is 10000, the infection rate is 0.1, the recovery rate is 0.2, Degree per node (pre-
pandemic) is 8, Degree per node (pandemic) is 4, pre-pandemic criteria is 0.9
6.4 Method3vsMethod4
Method 3, referred to as the ’pre-pandemic pandemic fixed-dynamic combined Barabási model,’ incorpo-
rates the temporal variation in the number of social contacts.
Method 4, denoted as the ’pre-pandemic pandemic fixed-dynamic two-infection rate combined Barabási
model’, incorporates the consideration that different social contacts may have distinct infection rates.
35
Figure 6.8 shows that whenβ in Method 3 is equal toβ f
in Method 4, the total number of infected indi-
viduals in Method 3 is larger than that in Method 4. In the plot on the leftβ = β f
= 0.1 andβ d
= 0.05;
in the plot on the rightβ =β f
=0.2 andβ d
=0.1.
Figure 6.8: Left: Population is 1000, the Infection rate (Method 3) for all contacts is 0.1, the Infection rate
(Method 4) for the fixed network is 0.1, the Infection rate (Method 4) for dynamics network is 0.05, the
Recovery rate is 0.2, Degree per node in pre-pandemic period is 8, Degree per node in pandemic period is
4; Right: Population is 1000, the Infection rate (Method 3) for all contacts is 0.2, the Infection rate (Method
4) for the fixed network is 0.2, the Infection rate (Method 4) for dynamics network is 0.1, the Recovery rate
is 0.2, Degree per node in pre-pandemic period is 8, Degree per node in pandemic period is 4;
Figure 6.9 shows that when β in Method 3 is equal to β d
in Method 4, the total number of infected
individuals in Method 3 is smaller than that in Method 4. In the plot on the left β = β d
= 0.05 and
β f
=0.1; in the plot on the rightβ =β d
=0.1 andβ f
=0.2.
36
Figure 6.9: Left: Population is 1000, the Infection rate (Method 3) for all contacts is 0.05, the Infection rate
(Method 4) for the fixed network is 0.1, the Infection rate (Method 4) for dynamics network is 0.05, the
Recovery rate is 0.2, Degree per node in pre-pandemic period is 8, Degree per node in pandemic period is
4; Right: Population is 1000, the Infection rate (Method 3) for all contacts is 0.1, the Infection rate (Method
4) for the fixed network is 0.2, the Infection rate (Method 4) for dynamics network is 0.1, the Recovery rate
is 0.2, Degree per node in pre-pandemic period is 8, Degree per node in pandemic period is 4;
We can roughly conclude that if we want the total number of infected people in Method 4 to remain the
same as that in Method 3, the infection rate (Method 4) for close social contact should be greater than the
infection rate (Method 3) and the infection rate (Method 4) for not close social contact should be smaller
37
than the infection rate (Method 3).
Hence, the modified ODE can be expressed as shown in Fig 6.10:
when S ≥ αN :
dS
dt
= − 1
N
(β f
k
f
+β d
k
d1
)SI
dI
dt
=
1
N
(β f
k
f
+β d
k
d1
)SI− γI
dR
dt
= γI
when S < αN :
dS
dt
= − 1
N
(β f
k
f
+β d
k
d2
)SI
dI
dt
=
1
N
(β f
k
f
+β d
k
d2
)SI− γI
dR
dt
= γI
where α is the pre-pandemic criteria, β f
is the close infection rate in the fixed network, β d
is the not
close infection rate in the dynamic network,γ is the recovery rate,k
f
is the degree per node in the fixed
network,k
d1
is the degree per node in the dynamics network in the pre-pandemic period, andk
d2
is the
degree per node in the dynamics network in the pandemic period.
38
Figure 6.10: population is 10000, the infection rate for the fixed network is 0.2, the infection rate for the
dynamics network is 0.05, the recovery rate is 0.2, Degree per node (pre-pandemic) is 8, Degree per node
(pandemic) is 4
6.5 Method4vsMethod5
Method 4, denoted as the ’pre-pandemic pandemic fixed-dynamic two-infection rate combined Barabási
model’, incorporates the consideration that different social contacts may have distinct infection rates.
Method 5, known as the ’pre-pandemic pandemic fixed-dynamic two-infection rate two-recovery rate com-
bined Barabási model’, accounts for the differential recovery rates between elderly and young individuals.
To begin, based on the prediction by Fulmer et al. (2021) [12], 20 percent of the population is designated
as elderly individuals aged above 65 years. Based on the conclusion by Bajaj et.al (2021) [2], we assume
that the recovery rate (γ 1
) of elder people is half of that of young people.
39
Figure 6.11 shows that whenγ in Method 4 is equal toγ y
in Method 5, the total number of infected indi-
viduals in Method 4 is lower than that in Method 5. Specifically, the rate of decrease on the slope of the
recovery line of Method 4 is faster than that of Method 5. In the plot on the leftγ = γ y
0.2 andγ o
= 0.1;
in the plot on the rightγ =γ y
=0.16 andγ o
=0.08.
Figure 6.11: Left: Population is 1000, the Infection rate for the fixed network is 0.2, the Infection rate for
dynamics network is 0.05, the Recovery rate (Method 4) for all people is 0.2, the Recovery rate (Method
5) for young people is 0.2, the Recovery rate (Method 5) for elder people is 0.1, Degree per node in pre-
pandemic period is 8, Degree per node in pandemic period is 4; Right: Population is 1000, the Infection rate
for the fixed network is 0.2, the Infection rate for dynamics network is 0.05, the Recovery rate (Method 4)
for all people is 0.16, the Recovery rate (Method 5) for young people is 0.16, the Recovery rate (Method 5)
for elder people is 0.08, Degree per node in pre-pandemic period is 8, Degree per node in pandemic period
is 4
Figure 6.12 shows that when γ in Method 4 is equal to γ o
in Method 5, the total number of infected
individuals in Method 4 is higher than that in Method 5. Specifically, the rate of decrease on the slope of
the recovery line of Method 4 is faster than that of Method 5. In the plot on the left γ = γ o
= 0.1 and
γ y
=0.2; in the plot on the rightγ =γ o
=0.08 andγ y
=0.16.
40
Figure 6.12: Left: Population is 1000, the Infection rate for the fixed network is 0.2, the Infection rate for
dynamics network is 0.05, the Recovery rate (Method 4) for all people is 0.1, the Recovery rate (Method
5) for young people is 0.2, the Recovery rate (Method 5) for elder people is 0.1, Degree per node in pre-
pandemic period is 8, Degree per node in pandemic period is 4; Right: Population is 1000, the Infection rate
for the fixed network is 0.2, the Infection rate for dynamics network is 0.05, the Recovery rate (Method 4)
for all people is 0.08, the Recovery rate (Method 5) for young people is 0.16, the Recovery rate (Method 5)
for elder people is 0.08, Degree per node in pre-pandemic period is 8, Degree per node in pandemic period
is 4
In Figure 6.13, the recovery event rates for Method 4 and Method 5 remain consistent. As discussed in
the previous chapter, the rate of recovery events (λ 2
) is calculated by multiplying the recovery rate (γ ) by
the number of ’I’ nodes in the network. For Method 5, the recovery rate is set as 0.2 for young people and
0.1 for elderly people.
The recovery rate for Method 4 is calculated by: Recovery rate (Method 4)=
0.2× 0.8N+0.1× 0.2N
N
=0.18.
In Figure 6.13, the curves representing Method 4 and Method 5 are closely aligned. The total number of
infected individuals in Method 4 is comparable to that in Method 5 while the recovery event rates are
consistent.
41
Figure 6.13: Population is 1000, the Infection rate for the fixed network is 0.2, the Infection rate for dynam-
ics network is 0.05, the Recovery rate (Method 4) for all people is 0.18, the Recovery rate (Method 5) for
young people is 0.2, the Recovery rate (Method 5) for elder people is 0.1, Degree per node in pre-pandemic
period is 8, Degree per node in pandemic period is 4
In the ODE method, individuals are divided into young and old categories. Young susceptible individ-
uals can be infected by both young and elderly infected individuals, while elderly susceptible individuals
42
can also be infected by both young and elderly infected individuals. Hence, the modified ODE can be
expressed as shown in Fig 6.14:
when S
young
+S
old
≥ αN :
dS
young
dt
= − 1
N
(β f
k
f
+β d
k
d1
)S
young
(I
young
+I
old
)
dS
old
dt
= − 1
N
(β f
k
f
+β d
k
d1
)S
old
(I
young
+I
old
)
dI
young
dt
=
1
N
(β f
k
f
+β d
k
d1
)S
young
(I
young
+I
old
)− γ y
I
young
dI
old
dt
=
1
N
(β f
k
f
+β d
k
d1
)S
old
(I
young
+I
old
)− γ o
I
old
dR
young
dt
= γ y
I
young
dR
old
dt
= γ o
I
old
when S
young
+S
old
< αN :
dS
young
dt
= − 1
N
(β f
k
f
+β d
k
d2
)S
young
(I
young
+I
old
)
dS
old
dt
= − 1
N
(β f
k
f
+β d
k
d2
)S
old
(I
young
+I
old
)
dI
young
dt
=
1
N
(β f
k
f
+β d
k
d2
)S
young
(I
young
+I
old
)− γ y
I
young
dI
old
dt
=
1
N
(β f
k
f
+β d
k
d2
)S
old
(I
young
+I
old
)− γ o
I
old
dR
young
dt
= γ y
I
young
dR
old
dt
= γ o
I
old
where α is the pre-pandemic criteria, β f
is the close infection rate in the fixed network, β d
is the not
close infection rate in the dynamic network,γ y
is the recovery rate for young people,γ o
is the recovery
rate for elder people, k
f
is the degree per node in the fixed network, k
d1
is the degree per node in the
dynamics network in the pre-pandemic period, andk
d2
is the degree per node in the dynamics network in
the pandemic period.
43
Figure 6.14: population is 10000, the infection rate for the fixed network is 0.2, the infection rate for the
dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate for elder people is 0.1,
Degree per node (pre-pandemic) is 8, Degree per node (pandemic) is 4, both networks are Barabasi-Albert
6.6 Method5vsMethod6
Method 5, known as the ’pre-pandemic pandemic fixed-dynamic two-infection rate two-recovery rate com-
bined Barabási model’, accounts for the differential recovery rates between elderly and young individuals.
Method 6, labeled the ’pre-pandemic pandemic fixed-dynamic two-infection rate two-recovery rate com-
bined Watts-Strogatz Barabási model’, incorporates the high-clustering property among individuals by
incorporating a Watts-Strogatz network alongside the Barabási model.
Using the results from Liu et al. (2015) [22], the rewiring probability of the fixed Watts-Strogatz network
in Method 6 sets as 0.2.
44
The plots in Figure 6.15 illustrate that the total number of infected people in Method 6 is greater than that
in Method 5. The peak of infected people in Method 6 comes later than that of infected people in Method 5.
Figure 6.15: Left: Population is 1000, the Infection rate for the fixed network is 0.1, the Infection rate for
dynamics network is 0.05, both fixed and dynamics networks are Barabasi-Albert model (Method 5), the
fixed network is Watts-Strogatz model (Method 6), dynamics network is Barabasi-Albert model (Method 6),
the Recovery rate for young people is 0.2, the Recovery rate for elder people is 0.1, Degree per node in pre-
pandemic period is 10, Degree per node in pandemic period is 6; Right: Population is 10000, the Infection
rate for the fixed network is 0.1, the Infection rate for dynamics network is 0.05, both fixed and dynamics
networks are Barabasi-Albert model (Method 5), the fixed network is Watts-Strogatz model (Method 6),
dynamics network is Barabasi-Albert model (Method 6), the Recovery rate for young people is 0.2, the
Recovery rate for elder people is 0.1, Degree per node in pre-pandemic period is 10, Degree per node in
pandemic period is 6
45
The modified ODE is the same as the above one, shown in Fig 6.16:
when S
young
+S
old
≥ αN :
dS
young
dt
= − 1
N
(β f
k
f
+β d
k
d1
)S
young
(I
young
+I
old
)
dS
old
dt
= − 1
N
(β f
k
f
+β d
k
d1
)S
old
(I
young
+I
old
)
dI
young
dt
=
1
N
(β f
k
f
+β d
k
d1
)S
young
(I
young
+I
old
)− γ y
I
young
dI
old
dt
=
1
N
(β f
k
f
+β d
k
d1
)S
old
(I
young
+I
old
)− γ o
I
old
dR
young
dt
= γ y
I
young
dR
old
dt
= γ o
I
old
when S
young
+S
old
< αN :
dS
young
dt
= − 1
N
(β f
k
f
+β d
k
d2
)S
young
(I
young
+I
old
)
dS
old
dt
= − 1
N
(β f
k
f
+β d
k
d2
)S
old
(I
young
+I
old
)
dI
young
dt
=
1
N
(β f
k
f
+β d
k
d2
)S
young
(I
young
+I
old
)− γ y
I
young
dI
old
dt
=
1
N
(β f
k
f
+β d
k
d2
)S
old
(I
young
+I
old
)− γ o
I
old
dR
young
dt
= γ y
I
young
dR
old
dt
= γ o
I
old
whereα is the pre-pandemic criteria,β f
is the close infection rate in the fixed Watts-Strogatz network, β d
is the not close infection rate in the dynamic Barabási-Albert network,γ y
is the recovery rate for young
people,γ o
is the recovery rate for elder people,k
f
is the degree per node in the fixed Watts-Strogatz net-
work, k
d1
is the degree per node in the dynamics Barabási-Albert network in the pre-pandemic period,
andk
d2
is the degree per node in the dynamics Barabási-Albert network in the pandemic period.
46
Figure 6.16: population is 10000, the infection rate for the fixed network is 0.1, the infection rate for the
dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate for elder people is
0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6, the fixed network is Watts-
Strogatz, the rewiring probability is 0.2, the dynamic network is Barabási-Albert
6.7 Optimalimplementationofgovernmentpolicies
When examining the issue of epidemic infectious diseases, the crucial question of how to reduce the
number of infected individuals becomes inevitable. Travel restrictions, reduced social interactions, and
quarantine measures are commonly employed as mainstream methods to control the spread of infectious
diseases. In reality, only a few diseases are effectively controlled at their source before reaching epidemic
levels. Most infectious diseases spread rapidly and reach a certain number of infections before appropriate
measures like quarantine or reduced social interactions are implemented.
47
In this section, we explore how the duration of this rapid transmission phase impacts the final total number
of infections.
We employ the "pre-pandemic pandemic fixed-dynamic two-infection rate two-recovery rate combined
Watts-Strogatz Barabási model." This network not only exhibits high clustering and power-law distribu-
tion but is also dynamic, allowing it to simulate the length of the rapid transmission period accurately.
The pre-pandemic period is defined as the time of the rapid transmission phase.
Figures 6.17 to 6.20 below show how the total number of people who become infected depends on the du-
ration of the pre-pandemic period when the value ofβ is high. We consider four scenarios corresponding
to the pandemic period starting when 20%, 15%, 10%, and 5% of the total population has been infected,
i.e. 80%, 85%, 90%, and 95% of the total population remains susceptible. In each case, we show ten valid
simulations. Tables 6.1 to 6.4 record the total number of infected people for each case and the average
across the ten simulations.
48
Figure 6.17: population is 1000, the infection rate for the fixed network is 0.1, the infection rate for the
dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate for elder people is
0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6, the fixed network is Watts-
Strogatz, the rewiring probability is 0.2, the dynamic network is Barabási-Albert, controlling on 80 percent
remaining susceptible
Table 6.1: Remaining Susceptible after 100 days (high beta, controlling on 80 percent remaining susceptible)
1 2 3 4 5 6 7 8 9 10 average
78 100 77 160 78 48 91 91 72 51 84.6
49
Figure 6.18: population is 1000, the infection rate for the fixed network is 0.1, the infection rate for the
dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate for elder people is
0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6, the fixed network is Watts-
Strogatz, the rewiring probability is 0.2, the dynamic network is Barabási-Albert, controlling on 85 percents
remaining susceptible
Table 6.2: Remaining Susceptible after 100 days (high beta, controlling on 85 percent remaining susceptible)
1 2 3 4 5 6 7 8 9 10 average
72 57 84 73 124 81 97 198 80 79 94.5
50
Figure 6.19: population is 1000, the infection rate for the fixed network is 0.1, the infection rate for the
dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate for elder people is
0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6, the fixed network is Watts-
Strogatz, the rewiring probability is 0.2, the dynamic network is Barabási-Albert, controlling on 90 percents
remaining susceptible
Table 6.3: Remaining Susceptible after 100 days (high beta, controlling on 90 percent remaining susceptible)
1 2 3 4 5 6 7 8 9 10 average
188 85 109 81 73 68 70 67 89 65 89.5
51
Figure 6.20: population is 1000, the infection rate for the fixed network is 0.1, the infection rate for the
dynamics network is 0.05, the recovery rate for young people is 0.2, the recovery rate for elder people is
0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6, the fixed network is Watts-
Strogatz, the rewiring probability is 0.2, the dynamic network is Barabási-Albert, controlling on 95 percents
remaining susceptible
Table 6.4: Remaining Susceptible after 100 days (high beta, controlling on 95 percent remaining susceptible)
1 2 3 4 5 6 7 8 9 10 average
79 94 74 77 78 76 205 111 70 88 95.2
Surprisingly, when the value ofβ is high, the total number of infections for the four scenarios is very
similar. Thus, a shorter pre-pandemic period does not necessarily result in a lower total number of infec-
tions under these conditions.
Conversely, figures 6.21 to 6.24 below show how the total number of people who become infected depends
on the duration of the pre-pandemic period when the value ofβ is low. The same four cases are considered,
52
and again, for each case, ten valid simulations are shown. The results are recorded in Table 6.5 to Table 6.8.
Figure 6.21: population is 1000, the infection rate for the fixed network is 0.04, the infection rate for the
dynamics network is 0.02, the recovery rate for young people is 0.2, the recovery rate for elder people is
0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6, the fixed network is Watts-
Strogatz, the rewiring probability is 0.2, the dynamic network is Barabási-Albert, controlling on 80 percents
remaining susceptible
Table 6.5: Remaining Susceptible after 100 days (low beta, controlling on 80 percent remaining susceptible)
1 2 3 4 5 6 7 8 9 10 average
585 411 430 607 523 458 469 446 483 481 489.3
53
Figure 6.22: population is 1000, the infection rate for the fixed network is 0.04, the infection rate for the
dynamics network is 0.02, the recovery rate for young people is 0.2, the recovery rate for elder people is
0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6, the fixed network is Watts-
Strogatz, the rewiring probability is 0.2, the dynamic network is Barabási-Albert, controlling on 85 percents
remaining susceptible
Table 6.6: Remaining Susceptible after 100 days (low beta, controlling on 85 percent remaining susceptible)
1 2 3 4 5 6 7 8 9 10 average
563 519 510 779 510 527 447 535 536 488 541.4
54
Figure 6.23: population is 1000, the infection rate for the fixed network is 0.04, the infection rate for the
dynamics network is 0.02, the recovery rate for young people is 0.2, the recovery rate for elder people is
0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6, the fixed network is Watts-
Strogatz, the rewiring probability is 0.2, the dynamic network is Barabási-Albert, controlling on 90 percents
remaining susceptible
Table 6.7: Remaining Susceptible after 100 days (low beta, controlling on 90 percent remaining susceptible)
1 2 3 4 5 6 7 8 9 10 average
537 623 477 503 572 542 799 548 483 626 571.0
55
Figure 6.24: population is 1000, the infection rate for the fixed network is 0.04, the infection rate for the
dynamics network is 0.02, the recovery rate for young people is 0.2, the recovery rate for elder people is
0.1, Degree per node (pre-pandemic) is 10, Degree per node (pandemic) is 6, the fixed network is Watts-
Strogatz, the rewiring probability is 0.2, the dynamic network is Barabási-Albert, controlling on 95 percents
remaining susceptible
Table 6.8: Remaining Susceptible after 100 days (low beta, controlling on 95 percent remaining susceptible)
1 2 3 4 5 6 7 8 9 10 average
525 884 769 528 610 491 616 659 556 616 625.4
The results reveal that when theβ value is low, a shorter pre-pandemic period leads to a decrease in
the total number of infections. In this scenario, the timing of infection detection during the rapid transmis-
sion phase holds significant importance since the earlier the infection is detected and controlled through
reduced social interactions or quarantine, the lower the total number of infections.
56
Chapter7
Conclusions
In this study, a series of incremental improvements were proposed to enhance the simulation of infectious
disease transmission using the Barabási-Albert network model. The introduced methods, starting from
Method 2 and extending to Method 6, aimed to address limitations related to randomness, temporality,
parameter differentiation, and clustering within the network.
The incorporation of a dynamic Barabási-Albert network updated daily, introduced greater randomness in
the network’s edge connections (Method 2). Additionally, the distinction between pre-pandemic and pan-
demic conditions improved the temporality of the network (Method 3). Methods 4 and 5 further enhanced
the model by differentiating infection and recovery rates based on social proximity and population age,
respectively. Method 6 introduced the Watts-Strogatz model to replace the Barabási-Albert model, adding
a clustering property to the network. Through the analysis of simulation results, it became evident that
these incremental improvements contributed to a more comprehensive model compared to the existing
fixed Barabási-Albert model (Method 1). Furthermore, for infectious diseases with low infection rates, we
observed that earlier detection, coupled with swift containment measures, leads to a significant reduction
in the total number of infections.
However, it is important to acknowledge the limitations imposed by the computing power of personal com-
puters in this study. Consequently, the upper limit of the population has been set to 10,000 individuals,
57
and larger populations have not been explored. This serves as a direction for future research, as inves-
tigating the model’s behavior with larger populations, such as 100,000 or one million individuals, would
provide valuable insights. Future research should explore the model’s applicability to larger populations
and address the trade-off between considering more network features and longer running times.
Moreover, the current analysis focused on individual simulation runs, and there is space for improvement
by incorporating multiple simulations and averaging the results for more robust comparisons. However,
due to computational limitations, this step was not pursued in the present study, but it remains a valuable
direction for future research.
This study contributes to the field of infectious disease modeling and control strategies by providing an im-
proved approach that considers randomness, temporality, parameter differentiation, and clustering within
the network. It serves as a foundation for further investigations and opens avenues for refining and ex-
panding the understanding of infectious disease dynamics.
58
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Creator Lyu, Fuliang (author)

Core Title The spread of an epidemic on a dynamically evolving network

School College of Letters, Arts and Sciences

Degree Master of Arts

Degree Program Applied Mathematics

Degree Conferral Date 2023-08

Publication Date 08/04/2023

Defense Date 08/03/2023

Publisher University of Southern California. Libraries (digital)

Tag Barabási-Albert network,COVID-19,dynamic network,Gillespie simulation,infectious disease modeling,network model,OAI-PMH Harvest,ODE compartmental model,SIR model,Watts-Strogatz network

Language English

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Advisor Haskell, Cymra (committee chair), Zhang, Jianfeng (committee member), Mancera, Ricardo (committee member)

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