網路導向式計算流行病學：整合疾病動態與人類社會網路的多層次傳染病模型架構

國

立

交

通

大

學

資訊科學與工程研究所

博

士

論

文

網路導向式計算流行病學：

整合疾病動態與人類社會網路的多層次

傳染病模型架構

Network-based Computational Epidemiology:

A Multilayer Framework Integrating Social Networks with Epidemic Dynamics

研 究 生：蔡宇軒

指導教授：孫春在 教授

網路導向式計算流行病學：

整合疾病動態與人類社會網路的多層次傳染病模型架構

Network-based Computational Epidemiology:

A Multilayer Framework Integrating Social Networks with Epidemic Dynamics

研 究 生：蔡宇軒 Student：Yu-Shiuan Tsai

指導教授：孫春在 Advisor：Chuen-Tsai Sun

國 立 交 通 大 學

資 訊 科 學 與 工 程 研 究 所

博 士 論 文

A Dissertation Submitted to

Institute of Computer Science and Engineering College of Computer Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Computer Science January 2011

Hsinchu, Taiwan, Republic of China

網路導向式計算流行病學：

整合疾病動態與人類社會網路的多層次傳染病模型架構

學生：蔡宇軒

指導教授：孫春在博士

國立交通大學資訊科學與工程研究所博士班

摘

摘

摘

摘

要

要

要

要

網路導向式計算流行病學利用電腦與理論或真實網路拓樸結構研究

人類疾病動態和社會趨勢。本論文的主旨在於探討網路導向式計算流

行病的重要性、研究現況、優勢與建模過程，並詳述三項原創研究。

首先，第一項研究以理論探討無尺度網路下個體資源和疾病傳播成本

對於疾病傳播關鍵門檻值的影響，並於流行病模型的基礎上提出解析

方程式來解釋關鍵門檻值在無尺度網路下的存在性。該研究指出個體

資源和疾病傳播成本的控管對於在無尺度網路下疫情擴散防治的可

行性。其次，第二項研究提出整合真實社會網路、個體觀點、國家觀

點的多層流行病學架構－多層流病動態模擬器（MEDSim），並以該

架構模擬 2009 年 A 型 H1N1 流感疫情在台灣爆發的情形，測試該架

構對於不同爆發地點和傳染阻絕方案的靈活性，希望藉此釐清複雜的

個體接觸行為對於疾病傳播動態的影響。最後，在第三項研究中分析

網路導向式計算流行病學的潛在優勢，並針對網路導向式計算流行病

學初學者給予建立網路導向式流行病模型的方法。該研究的目標在於

協助擁有較低電腦技能者建立流行病學模型、決定合適的模擬參數與

建立操作流程。本論文期望透過上述三項研究，利用電腦模擬來分析

多層次的個體互動行為，進而協助傳染阻絕政策的制定。

Network-based Computational Epidemiology:

A Multilayer Framework Integrating Social Networks with

Epidemic Dynamics

Student：Yu-Shiuan Tsai

Advisor：Dr. Chuen-Tsai Sun

Institute of Computer Science and Engineering

National Chiao Tung University

ABSTRACT

Network-based computational epidemiologists use computers and

either theoretical or actual network topologies to study the transmission

dynamics of human diseases and social trends. In this dissertation I

discuss the importance, current status, advantages, and modeling

procedures of network-based computational epidemiology, specifically

presenting three original studies in detail. The first study is an

investigation of how resources and transmission costs influence diffusion

dynamics and tipping points in scale-free networks. An epidemic model

based on an analytic equation is proposed to explain the existence of

epidemic critical thresholds in scale-free networks. Study results suggest

the possibility of controlling the spread of epidemics in scale-free

networks by manipulating resources and costs associated with an

infection event. In the second study, a proposal for a multilayer

epidemiological framework that integrates realistic social networks,

called the Multilayer Epidemic Dynamics Simulator (MEDSim), is

described from individual and national perspectives. Model flexibility

and generalizability are tested using outbreak locations and intervention

scenarios for the 2009 A/H1N1 influenza epidemic in Taiwan. The results

coincide with the dynamic processes of epidemics under different

intervention scenarios, thus clarifying the effects of complex contact

structures on disease transmission dynamics. In the third study, the

potential benefits of epidemic simulations and instructions for building

network-based epidemic models by novices learning network-based

computational epidemiology approaches is investigated. The goal is to

help individuals with less advanced computing skills build

epidemiological models, determine appropriate simulation parameters,

and construct operational procedures. It is my hope that the studies

presented in this dissertation can assist in efforts by public health

organizations to correctly implement intervention strategies by using

simulations to analyze multilayer interactions.

Acknowledgements

I am greatly thankful to my advisor, Professor Chuen-Tsai Sun,

whose encouragement, suggestions and support from the proposal to the

concluding level enabled me to develop an understanding of the

dissertation. I am deeply indebted to my admirable friends, Dr.

Chung-Yuan Huang and Dr. Tzai-Hung Wen, for their continuous

support and intellectual suggestions. I would like to thank my family, and

my lovely girlfriend, Pin-Han Wang. For their love and encouragement,

I can express only an inadequate acknowledgement of my appreciation. I

dedicate this dissertation to them. I also dedicate this dissertation to my

many friends who have supported me throughout the process.

Contents

Abstract (in Chinese)

i

Abstract (in English)

iii

Acknowledgments

v

Contents

vi

List of Tables

ix

List of Figures

x

Chapter 1. Introduction... 1

1.1.

Computational and Network-based Computational

Epidemiology ... 2

1.2.

Computational Epidemiology History ... 4

1.3.

Current Status of Computational Epidemiology ... 8

1.4.

Trends in Social Network Integration in Computational

Epidemiology ... 11

1.5.

Advantages of Network-based Computational Epidemiology 14

1.6.

Dissertation Overview ... 17

Chapter 2. Preliminaries ... 19

2.1.

Epidemiological Approaches Overview ... 20

2.2.

Compartmental Models ... 23

2.3.

Social Network Models and Network-based Epidemiology ... 27

Chapter 3. Analysis of Epidemiological Transmission in Theoretical

Complex Networks ... 34

Limitations and Transmission Cost Considerations ... 38

3.3.

Epidemic Effect of Limited Resources/Transmission Cost

Ratio ... 43

Chapter 4. Effects of Individual Diversity on Epidemic Modeling in

Realistic Social Networks ... 46

4.1

Motivation... 48

4.2

A Multilayer Epidemiological Model Integrating Human

Commuting Networks ... 51

Layer 1: Within an age group ... 53

Layer 2: Among age groups ... 56

Layer 3: Commuting ... 58

Layer 4: Nationwide interactions ... 62

Technological Framework ... 65

Statistical Analysis for Model Validation ... 67

4.3

Simulating the 2009 Novel H1N1 Influenza ... 69

Parameterization ... 69

Intervention Policy Evaluation ... 72

Chapter 5. Simulation Architecture for Studying Network-based

Computational Epidemiology Issues and for Public Health Education

Purposes ... 85

5.1

Motivation... 87

5.2

Potential Benefits in Learning Through Epidemic Simulations ..

... 90

5.3

Teaching Computational Modeling and Simulation ... 97

Chapter 6. Conclusions... 101

List of Tables

T

ABLE

2.1.

T

WO

C

OMPLEX

N

ETWORKS

C

ATEGORIES

... 28

T

ABLE

4.1.

MEDS

IM

P

ARAMETERS

... 64

T

ABLE

4.2.

MEDS

IM PARAMETERS USED FOR FITTING SIMULATION CURVES

WITH ACTUAL SEASONAL INFLUENZA

A

CURVES IN

T

AIWAN BETWEEN

S

EPTEMBER

2008

AND

A

PRIL

2009 ... 71

T

ABLE

4.3.

MEDS

IM PARAMETERS USED FOR FITTING SIMULATION CURVES

TO ACTUAL SWINE

-

ORIGIN INFLUENZA

A

(H1N1)

CURVES IN

T

AIWAN

FROM WEEK

25

TO WEEK

52 ... 71

T

ABLE

4.4

O

BSERVATION INDEX VALUES ACCORDING TO DIFFERENT

TRANSMISSION RATES

. ... 78

T

ABLE

4.5.

O

BSERVATION INDEX VALUES ACCORDING TO DIFFERENT POLICY

ACTIVATION SCENARIOS DURING SWINE

-

ORIGIN INFLUENZA

A

(H1N1)

OUTBREAK IN

T

AIPEI

. ... 81

T

ABLE

4.6.

O

BSERVATION INDEX VALUES ACCORDING TO DIFFERENT POLICY

ACTIVATION SCENARIOS DURING SWINE

-

ORIGIN INFLUENZA

A

(H1N1)

List of Figures

F

IGURE

2.1.

F

LOWCHART OF THE

SIR

EPIDEMIOLOGIC MODEL

. ... 23

F

IGURE

2.2.

T

HE COMPARTMENT STATES

,

S,

I,

AND

R,

AS A FUNCTION OF T

. 24

F

IGURE

2.3.

P

HASE TRANSITION DIAGRAM FOR EPIDEMIC SIMULATIONS IN

HOMOGENEOUS NETWORKS

. ... 26

F

IGURE

2.4.

T

HREE TYPES OF COMPLEX NETWORKS

. ... 27

F

IGURE

2.5.

(

A

)

O

NE

-

DIMENSIONAL ORDERED NETWORK WITH EACH NODE

CONNECTED TO FOUR ADJACENT NODES

.

(

B

)

W

ATTS AND

S

TROGATZ

’

S

(D

UNCAN

J.

W

ATTS

&

S

TROGATZ

,

1998)

SMALL WORLD NETWORK WITH

FOUR REWIRED SHORTCUTS

.

(

C

)

N

EWMAN AND

W

ATTS

’

(M.

E.

J.

N

EWMAN

&

D.

J.

W

ATTS

,

1999)

IMPROVED SMALL WORLD NETWORK

WITH FIVE ADDITIONAL SHORTCUTS

.

(

D

)

E

XAMPLE OF A BROKEN

NETWORK IN

W

ATTS AND

S

TROGATZ

’

S

(D

UNCAN

J.

W

ATTS

&

S

TROGATZ

,

1998)

SMALL WORLD NETWORK

. ... 29

F

IGURE

2.6.

C

OMPARISON OF NODE DEGREE DISTRIBUTIONS AND NETWORK

STRUCTURES BETWEEN

D

AVIDSEN ET AL

.’

S TWO

-

RULE MODEL

(

A AND B

)

AND OUR PROPOSED THREE

-

RULE MODEL

(

C AND D

). ... 31

F

IGURE

3.1.

C

RITICAL

T

HRESHOLD

λ

_c IS A FUNCTION OF THE RATIO OF

TRANSMISSION COSTS TO INDIVIDUAL RESOURCES

(

c R

/

)

IN SCALE

-

FREE

NETWORKS

.

W

E USED IT TO ANALYZE RESULTS FROM OUR SIMULATION

EXPERIMENTS AND THREE MATHEMATICAL ANALYSES

. ... 41

F

IGURE

4.1.

M

ULTILAYER

E

PIDEMIC

D

YNAMICS

S

IMULATOR

(MEDS

IM

)

F

IGURE

4.2.

MEDS

IM FRAMEWORK

. ... 53

F

IGURE

4.3.

(

A

)

M

ODIFIED

SLIR

MODEL LAYER

1

CONCEPT

.

(

B

)

M

ODIFIED

SLIR

MODEL LAYER

1

FLOWCHART

. ... 55

F

IGURE

4.4.

MEDS

IM LAYER

2

ARCHITECTURE FLOWCHART

.

T

HICK SOLID LINES INDICATE PARAMETERS FOR OTHER

(

NON

-

P AND NON

-

Q

)

AGE GROUPS

.

T

HICK DASHED CURVES INDICATE RELATIVE PERCENTAGES OF EACH AGE GROUP WITHIN THE TOTAL POPULATION OF EACH LOCATION

. 58

F

IGURE

4.5.

P

OTENTIAL MOVEMENT OF INFECTIVITY BETWEEN LOCATIONS

i

AND

j

. ... 60

F

IGURE

4.6.

MEDS

IM LAYER

3

ARCHITECTURE FLOWCHART

.

P

ROPERTIES ASSOCIATED WITH COMMUTING BETWEEN TWO LOCATIONS ARE INDICATED BY THICK SOLID LINES

.

A

DDITIONAL LOCATION PROPERTIES ARE INDICATED BY THICK DASHED LINES

. ... 61

F

IGURE

4.7.

T

AIWAN

’

S NATIONWIDE COMMUTING NETWORK

. ... 63

F

IGURE

4.8.

MEDS

IM SIMULATION TOOL FRAMEWORK

. ... 66

F

IGURE

4.9.

MEDS

IM IMPLEMENTATION

GUI. ... 67

F

IGURE

4.10.

C

OMPARISON OF WEEKLY NEW INFECTED CASES BETWEEN ACTUAL AND SIMULATED RESULTS NORMALIZED FOR

(

A

)

SEASONAL INFLUENZA

A

AND

(

B

)

SWINE

-

ORIGIN

H1N1

INFLUENZA

A. ... 72

F

IGURE

4.11.

N

EW INFECTED CASES PER WEEK AT DIFFERENT TRANSMISSION RATES

. ... 74

F

IGURE

4.12.

C

UMULATIVE NEW INFECTED CASES AT DIFFERENT TRANSMISSION RATES

. ... 74

F

IGURE

4.13.

B

ASIC EPIDEMIC CURVE AT

A

0%

REDUCED TRANSMISSION RATE EXPRESSED ACCORDING TO TWO OBSERVATION INDEXES

. ... 75

F

IGURE

4.14.

C

OMPARISON OF NEW INFECTED CASES AT EPIDEMIC CURVE

PEAK AT DIFFERENT TRANSMISSION RATES

. ... 75

F

IGURE

4.15.

W

EEKLY NEW CASES AT CURVE PEAK AT DIFFERENT

TRANSMISSION RATES

. ... 76

F

IGURE

4.16.

N

EW INFECTED CASES AT EPIDEMIC CURVE PEAK ACCORDING

TO VARIOUS INTERVENTION POLICY SCENARIOS

. ... 76

F

IGURE

4.17.

N

UMBERS OF INFECTED CASES ACCORDING TO VARIOUS

INTERVENTION POLICY SCENARIOS

. ... 77

F

IGURE

4.18.

W

EEK NUMBERS OF EPIDEMIC CURVE PEAKS ACCORDING TO

VARIOUS INTERVENTION POLICY SCENARIOS

. ... 77

F

IGURE

4.19.

E

PIDEMIC PEAK WEEK NUMBERS FOR URBAN AND RURAL

Chapter 1.

Introduction

Network-based computational epidemiologists use computers and either

theoretical or realistic network topologies to study the reasons, conditions, and

transmission dynamics of human diseases and social trends. In this chapter I will

summarize several network-based computational epidemiological issues, and

introduce some of the details of both computational and network-based computational

epidemiology. After reviewing the history, current status, and importance of the

1.1.

Computational and Network-based

Computational Epidemiology

Epidemiologists study the distribution of individuals who are healthy or infected

during a contagious disease outbreak, as well as conditions and factors supporting the

spread of a disease (Lloyd & May, 2001). The two most common research approaches

are observation and experimentation. In the first, epidemic diseases are analyzed

using empirical data collected from clinical cases, epidemic monitoring surveys, and

other investigation tools to determine contagious patterns or disease properties. In the

second, subjects are randomly divided into two groups, members of one group are

treated with the experimental variable, and a comparison of the two groups

determines the positive, negative, or null effects of the variable.

Computational epidemiologists construct mathematical models and use

computing techniques to obtain epidemic results. Researchers validate results by

comparing them with observable empirical data, or use their results to explain

experimental variable characteristics. The most commonly used mathematical tool for

computational epidemiology studies is differential equations, in which individuals in a

population are divided into finite states representing different health statuses.

symbols. A major advantage of a differential equation system is that the number of

each state can be easily computed; a disadvantage is that they are weak in terms of

describing social properties such as social distance.

To compensate for the lack of social properties, epidemiologists are

incorporating social networks into their mathematical models, reflecting the idea that

there is no distance between individuals with the same health status—that is, there are

no restrictions on any two individuals being in contact with each other. Social network

structures consist of nodes (objects) and links (social relations). For example, in a

friendship network, nodes represent individuals and links represent whether or not

two nodes are friends. Due to its ability to represent social relations, network-based

computational epidemiology has grown in popularity. However, since individual

characteristics and their corresponding integrated mathematical models are so

complex, powerful computers are required to solve equation systems. As the size and

detail of a social network increases, so does the need for increased computation time

1.2.

Computational Epidemiology History

The first approaches used in computational epidemiology were based on

differential equation systems. One of the first contagious epidemiology models,

proposed by Kermack and McKendrick (1927), is known as the compartmental SIR

model. In this model, all individuals in a population are classified as Susceptible

(vulnerable to infection but not yet infected), Infected (and capable of infecting others),

or Removed (recovered, dead, or otherwise not posing any further threat). Differential

equations mark the progress of each state. In the past 80 years, numerous

compartmental models have been created and improved for research purposes (for

examples, see Bailey, 1950; Bartlett, 1956; Diekmann, Heesterbeek & Metz, 1990;

Hyman & Stanley, 1988; Rollett, 1945). Major progress was made in the 1990s, with

the addition of other states and model revisions to emphasize cyclic characteristics

(Ahmed & Agiza, 1998; Anderson & May, 1991; Wang, 2006).

Describing epidemic dynamics using compartmental models based on

differential equation systems is an easy method for representing the time dimension,

but such approaches lack a spatial dimension. Individuals in the same compartment

are modeled as one group, implying that any two group members are directly

differences among individuals. To overcome this flaw, Von Neumann (1966)

introduced his cellular automata model (which considers spatial differences and the

movement of individuals) to epidemic propagation research (Fuentes & Kuperman,

1999; Sirakoulis, Karafyllidis & Thanailakis, 2000). Other researchers focused on

integrating compartmental and cellular automata models to support epidemiological

models (Liu & Jin, 2005; Mikler, Venkatachalam & Abbas, 2005; White, del Rey &

Sánchez, 2007).

In addition to using the cellular automata model to consider spatial effects, social

network models are increasingly being used by mathematical epidemiologists. Watts

and Strogatz (1998) have proposed the concept of a “small-world” phenomenon to

explain why any two individuals in the world can be contacted via a small number of

connecting individuals. Barabasi and Albert (1999) then proposed a “scale-free

network” algorithm to explain the phenomenon of “the rich becoming richer.” Unlike

theoretical random networks (Erdos & Renyi, 1960), social networks are much closer

to the real world situation, and hence can be used to depict individual contacts in

network-based epidemic model studies (Dezső & Barabási, 2002; Grais, Hugh Ellis &

Glass, 2003; Meyers, Newman, Martin & Schrag, 2003; Newman, 2002; Pourbohloul

et al., 2005; Parham & Ferguson, 2006; Handcock & Jones, 2006). Other researchers

(Barthelemy, Barrat, Pastor-Satorras & Vespignani, 2004, 2005; Draief, 2006;

Pastor-Satorras & Vespignani, 2001b; Shirley & Rushton, 2005; Silva, Ferreira &

Martins, 2007; Wang, 2002; Yang et al., 2007; Zhou, Yan & Wang, 2005;).

Agent-based differential equation system approaches emphasize heterogeneity

and interactions among individuals. In these approaches, individuals are represented

as agents whose interactions can be modeled in the form of rules (Boguñá &

Pastor-Satorras, 2002; Huang, Sun, Hsieh & Lin, 2004). The advantage of such an

approach is that it supports simulations of the movement of individuals, which in turn

supports an understanding of epidemic contagion routes. Using this kind of approach,

Barrett et al. (2005) constructed a society of 1.6 million agents to simulate the daily

behaviors of individuals in Portland, Oregon, and Epstein (2009) studied the 2009

influenza A (H1N1) epidemic by constructing a model containing 6.5 billion agents to

simulate international human contact and daily movement. Unlike compartmental

models that focus on the behaviors of whole populations, agent-based models focus

on individual behaviors.

A geometric structure has recently been integrated into epidemic models. Due to

the limitations of standard cellular automata, in this study geographical cellular

automata are used to simulate an environment (Liu, Xia, Yeh, Qiang & Jia, 2007; Zhou,

dynamics in social and geometric transformations (Flache & Hegselmann, 2001;

Menard, 2008). Other kinds of cellular automata have been tied to epidemic

contagious behaviors via network-based compartmental models (Zhong, Huang &

Song, 2009). To visualize the dynamics of a regional epidemic, at least two research

teams have integrated a geometric information system (GIS) into a mathematical

1.3.

Current Status of Computational

Epidemiology

Epidemiologists are currently emphasizing temporal and spatial depictions of

infectious disease occurrences and pathogenic mechanisms. Regarding the temporal

aspect, researchers have focused on understanding spreading trends and dynamic

changes in infectious diseases. The most common approach is to construct analytically

systematic epidemiological models with differential equations, and then derive stable

solutions (Feng, Huang & Castillo-Chavez, 2005; Inaba, 2007; Langlais & Naulin,

2003; Li & Jin, 2005; Shim, Feng, Martcheva & Castillo-Chavez, 2006; Supriatna,

Soewono & Van Gils, 2008; Wang & Zhao, 2005). Populations can be broken down

into infection stages such as Susceptible, Latent, Infectious and Recovered, and changes

in subpopulations over time can be modeled using system dynamic differential

equations (Feng et al., 2005; Inaba, 2007; Langlais & Naulin, 2003; Li & Jin, 2005;

Shim et al., 2006; Supriatna et al., 2008; Wang & Zhao, 2005). Using suitable

parameter values (e.g., transmission rate, recovery rate), infectious dynamics and

transmission thresholds that become endemic above and vanish below those values can

be derived to acquire analytic solutions from equations (Huang, Tsai & Sun, 2009;

Tsai, Sun & Huang, 2008). According to the different transmission capabilities of

epidemic diseases, basic reproduction numbers can be derived to estimate how many

individuals will be infected from the first infected individual (Hethcote, 2000; Keeling

& Grenfell, 2000).

Regarding the spatial aspect, researchers have focused on understanding the

distribution of infected individuals (which can be determined from medical case reports)

to help in monitoring and immunization efforts. Because of the advantages of computer

technology, GIS data on absolute distance and the properties of geographical regions

are now commonly applied in research (Rae, 2009; Wylie, Shah & Jolly, 2007). Many

researchers are also integrating GIS into epidemic disease monitoring and prevention

efforts (Edwards & Clarke, 2009; Jeger, Pautasso, Holdenrieder & Shaw, 2007; Mao &

Bian, 2010; Thakur & Sharma, 2009). By analyzing medical cases and collecting data

on environmental factors, geographic spatial distribution information can be

determined, and epidemic pathogenic mechanisms can be analyzed. For example, using

spatial clustering analysis, it is possible to analyze abnormal clusters that exceed an

expected number of infected cases, thus supporting efforts to understand the extent of

disease clustering relative to increases in disease vectors (Kan et al., 2008). Kan et al.

have used this approach to explain the smaller number of cases of dengue hemorrhagic

take advantage of both temporal and spatial aspects when analyzing infectious disease

propagation, therefore many researchers are trying to integrate both temporal and

spatial factors into their epidemiological models (Barrett, Eubank & Marathe, 2006;

1.4.

Trends in Social Network Integration in

Computational Epidemiology

The past decade has witnessed significant advancements in social network

research, ever since Watts and Strogatz (1998) first described small-world networks

characterized by highly clustered connections and short paths between node pairs.

Their work represents a fundamental change in our knowledge of human relationships,

which has influenced research avenues in a wide range of disciplines such as

epidemiology. (Diosan & Dumitrescu, 2007; Montoya & Solé, 2002; Vázquez,

Flammini, Maritan & Vespignani, 2003).

Complex networks can be used to model real-world complexity. A complex

network is a structure containing numerous nodes and edges. Nodes can represent

objects such as individuals, locations, organisms, or World Wide Web pages.

Depending on node type, edges can represent relationships such as human friendships,

food chains for non-human organisms, or links between web pages. Several network

indexes have been developed to measure relationships (Boccaletti, Latora, Moreno,

Chavez & Hwang, 2006). For example, degree of clustering has been used to determine

why our friend’s friend is often also our friend, degree of separation has been used to

has been used to explain the existence of super nodes(Huang, Tsai & Sun, 2010). Such

topological characteristics have also been used as epidemiological indexes to measure

the spreading speed of an epidemic disease (Edmunds, O'Callaghan & Nokes, 1997;

Estrada & Hatano, 2008; Hwang, Kim, Ramanathan & Zhang, 2008).

Infectious diseases spread through individual contact, and many epidemiologists

are using social networks to model individual contact behavior. Social networks, one

type of complex network that is also considered a social structure model, emphasize

individual heterogeneity, individual interaction, and network topological structure

(Boguñá & Pastor-Satorras, 2002; Huang et al., 2004). They are often used to model

populations, with nodes representing individuals and links representing contacts. Social

network topological structures have been used in many epidemic studies over the past

decade. Based on human epidemic disease or computer virus features, different social

network structures have been proposed to analyze epidemic spreading dynamics and

transmission rate thresholds (see, for example, Huang, Sun, Hsieh, Chen & Lin, 2005;

Langlais & Naulin, 2003; May & Lloyd, 2001; Pastor-Satorras & Vespignani, 2001b).

In addition, traffic networks such as daily commuting routes have been used to analyze

the spread of diseases via human transportation networks (Barrett et al., 2005, 2006).

Social network studies comparing the efficiencies of various public health policies have

and Vespignani (2001b, 2002).

New epidemiological models integrate spatial and social network factors. The

most commonly used approach adds various network topologies (e.g., small-world

network, scale-free network) to determine different epidemic spatial distributions

(Huang et al., 2004; Pastor-Satorras & Vespignani, 2001b, 2002). After building social

network models, parameters such as initial infected agent, and epidemic attributes such

as transmission and recovery rates, are manipulated to calculate disease propagation

within the defined network (Huang et al., 2004; Wang & Ruan, 2004). According to

epidemic properties, different simulation scenarios (e.g., network topologies, contact

patterns, agent attributes such as age or gender) can be studied using simulations in

order to develop effective public health policies. For example, HIV research entails

looking at how heterosexual sexual contact, homosexual sexual contact, or illegal drug

use affects virus transmission and propagation in a social network (Morris, 1997;

1.5.

Advantages of Network-based

Computational Epidemiology

Understanding the spreading dynamics of infectious diseases and the spatial

distribution of infected individuals is the primary concern of agencies involved in

infectious disease control and prevention (Hethcote, 2000; Moore & Newman, 2000;

Pastor-Satorras & Vespignani, 2002). Efforts to understand social network

associations among geographical characteristics such as coordinates, population size,

and census data represent a current trend in computation epidemiology. The

advantages of understanding these associations are as follows:

1. Epidemic disease properties such as the transmission capability of a virus and

recovery days among individuals are connected to geographical location (Barrett et al.,

2005; Larsen, Axhausen & Urry, 2006). For example, the transmission capability of

influenza in urban areas is greater than in rural areas because of population density

differences, therefore when setting epidemic parameters, transmission rate should be

higher in urban areas. Network integration into compartmental models can be used to

represent individual heterogeneity. Associating social networks with geographical

characteristics has the advantage of accurately describing the topology of individual

2. Public transportation systems such as aircraft, subways, commuter trains, and

buses support the spreading of a virus (Colizza, Barrat, Barthélemy & Vespignani,

2006; Grais et al., 2003; Kaza, Xu, Marshall & Chen, 2009). Modern public

transportation systems make it easy to move between distant locations, and pathogens

can be carried long distance within a matter of hours or days. In 2009, the

swine-origin H1N1 virus emerged in Mexico and rapidly spread throughout South

America, Europe, and Asia within a few weeks; by mid-November of that year, 6,770

deaths were reported in 206 geographic locations (Smith et al., 2009). This

underscores the importance of considering such factors as the location of public

transportation systems in epidemiological studies.

3. During a contagious disease outbreak, medical officials and/or public health

experts must consider balances among many factors when determining how to best

use medical resources and enact prevention policies (Riley et al., 2003; Molinari et al.,

2007). In addition, differences in resources and population densities among

administrative and geographical divisions must be considered (Sypsa, Pavlopoulou &

Hatzakis, 2009; Wylie et al., 2007). From the perspective of medical system utility, a

suitable mix of intervention policies is required to efficiently control a disease

outbreak according to limitations of medical resources (Tsai & Huang, 2010; Huang

division must be considered when planning the timing of interventions across

administrative divisions.

4. GIS is a suitable tool for graphically representing epidemics. By using

visualization tools, large bodies of complex data can be analyzed spatially. Based on

experience with newly emerging viruses such as SARS, avian influenza (H5N1), and

swine-adapted influenza (H1N1), public health officials must deal with the potential

of one such virus becoming pandemic (Fraser et al., 2009; Kuiken, Rimmelzwaan, Van

Amerongen & Osterhaus, 2003; Tomlinson & Cockram, 2003). However, traditional

epidemic models cannot adequately work with geographic information due to

limitations associated with equation size, therefore geographic network-based

computation epidemiology with GIS has value in terms of studying virus spreading

1.6.

Dissertation Overview

The rest of this dissertation is organized as follows: in Chapter 2 I will present a

brief overview of a preliminary study involving network-based computational

epidemiology, especially a network topology proposal from an original study

conducted by Huang, Tsai and Sun (2010).

In Chapter 3 I will present details from an original research project conducted by

Huang et al. (2010), Tsai & Huang (2010), and Tsai, Sun & Huang (2010) that used

network-based computational epidemiology with a theoretically complex network

topology. Based on considerations of resource limitations and transmission costs, I

will propose an epidemic model that uses analytic equations to identify critical

epidemic thresholds in scale-free networks.

In Chapter 4 I will discuss the details of an original research project by Tsai et al.

(2010) to integrate realistic social networks with standard epidemiological models,

and then describe a multilayer epidemiological framework—Multilayer Epidemic

Dynamics Simulator, or MEDSim—from national and individual perspectives. The

framework was used to compute outbreak locations and intervention scenarios for the

2009 A/H1N1 influenza epidemic as a means of testing model flexibility and

In Chapter 5 I will present details of an original research project on the potential

benefits of epidemic simulations, and describe the building of a network-based

epidemic model for epidemiology students with little computing experience who are

interested in studying computational epidemiology and public health education (Hsieh,

Huang, Sun & Tsai, 2009; Huang, Tsai & Wen, 2010a, 2010b). In Chapter 6 I will

Chapter 2.

Preliminaries

In this chapter, I will first introduce the most commonly used epidemiological

models for network-based computational epidemiological studies, and then briefly

2.1.

Epidemiological Approaches Overview

The two most commonly used approaches to modeling epidemic spreading

dynamics are population-based and network-oriented. In population-based approaches,

hosts that share the same symptoms are modeled or grouped in terms of a limited

numbers of classes (also known as compartments); the main task of researchers is to

study and compare their various dynamics (Feng et al., 2005; Inaba, 2007; Langlais &

Naulin, 2003; Shim et al., 2006; Supriatna et al., 2008; Wang & Zhao, 2005).

Combinations of classes are used to model and analyze population dynamics. For

example, the SLIR model puts individuals into one of four infection

statuses—Susceptible, Latent, Infectious, or Recovered (Li & Jin, 2005)—and

differential equations are used to determine transitions between epidemiological phases.

Depending on whether removed individuals can become susceptible a second time,

diseases can be modeled as SLIR or SLIRS cycles.

Population-based and network-oriented approaches respectively emphasize

large-scale population-level and individual-level perspectives. Population-based

approaches are suitable for discussing dynamic variation across individuals in the same

compartment, but they are weak in terms of modeling individual heterogeneity and

individuals are modeled as groups, any two members of the same group are assumed as

having a direct connection, which is not true in the real world. Furthermore, individual

movement and activity are location-dependent, therefore phenomena cannot be

simulated by a population-based approach that assumes a homogeneous population

distribution. In contrast, network-oriented approaches may be appropriate for

introducing individual heterogeneity, but they are computation-intensive and

time-consuming when simulating the behaviors of individuals with multiple attributes

in large-scale social environments (Barrett et al., 2005; Epstein, 2009). Many efforts

have been made to match individual and population behaviors with heterogeneity and

computation requirements when studying epidemic dynamics (Davis et al., 2003; Levin

& Durrett, 1996; Sawyer, 2003).

In contrast, network-oriented approaches emphasize individual heterogeneity,

interactions among individuals, and network structure (Boguñá & Pastor-Satorras,

2002; Huang et al., 2004). Individuals in a network are represented as nodes, and

interactions between them as links. Network nodes can be used to represent the

characteristics of individuals, locations, neighborhoods, or cities, and models can

incorporate the temporal dynamics of these features. Time frames for links between two

nodes can be preferentially defined (Ortiz-Pelaez, Pfeiffer, Soares-Magalhães &

individuals exhibiting interaction or relationship patterns (Barabási & Albert, 1999;

Erdos & Renyi, 1960; Newman, 2003; Watts & Strogatz, 1998). Network-oriented

approaches are suitable for capturing complex contact patterns among individuals,

exploring epidemic dynamics, and assessing the efficacies of public health policies

(Pastor-Satorras & Vespignani, 2001b, 2002; Huang et al., 2004, 2005). Lattice

networks have been used to determine distance relationships between individuals. In

contrast, random networks support features associated with casual contacts among

mobile individuals and the low degree of separation commonly observed in social

networks (Barrett et al., 2005). These approaches are viewed as reliable for

investigating epidemics, with the transmission dynamics of specific network models

being manipulated to investigate the spread of emerging infectious diseases (Liu, Lai &

Ye, 2003; May & Lloyd, 2001). The topological features of social networks have

recently been found to exert considerable influence on the transmission dynamics and

critical thresholds of infectious diseases, thus supporting subtle analyses that

network-oriented models are incapable of (Draief, Ganesh & Massoulié, 2008; Huang

2.2.

Compartmental Models

In standard epidemiological models, all individuals (nodes) in a population

(complex network) can be roughly classified into a limited number of states, including

Susceptible, Infected and Removed, as defined in Chapter 1. Epidemiologists use

combinations of these states to represent orders of transition between different

epidemiological phases, giving names such as “SIR” and “SIS” to their models. The

most commonly used model is the SIR (Susceptible→Infected→Recovery) (Figure 2.1),

which can be formulated using ordinary differential equations as follows:

-dS SI dt = β -dI SI I dt = β α dR I dt = α

β, a constant transmission rate, represents the speed at which Susceptible individuals become infected, and α is a constant recovery rate used to determine transformation speed from Infected to Recovered.

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% N um b e r ( % ) Time Susceptible Infected Recovery

Figure 2.2. The compartment states, S, I, and R, as a function of t.

When simulating epidemic dynamics in complex networks, epidemiologists

usually assume that nodes run stochastically through an SIS cycle, which does not take

into account the possibility of an individual’s removal due to death or acquired

immunization. The SIS model has been widely adopted to study contagious diseases

leading to endemic states with a stationary average density of infected individuals. It is

worth noting that for many contagious diseases, analyses derived from the SIS model

can be readily extended to the SIR and SIRS models (Pastor-Satorras & Vespignani,

2002). During each time step, each susceptible node is subject to a ν probability contagion rate if it is connected to one or more infected nodes. Infected nodes recover at

λ

is defined as λ ν δ= . Recovery rate δ can be assigned a value of 1, since it only affects the definition of the time scale of contagious disease propagation

(Pastor-Satorras & Vespignani, 2003). Pastor-Satorras and Vespignani (2002) define

( )

ρ t as the density of infected nodes at time step t. When time step t becomes

infinitely large, ρ can be represented as a steady-state density of infected nodes. Using these definitions, they applied mean-field theory to a SIS epidemiological model,

and used Anderson and May’s (1991) homogeneous mixing hypothesis according to the

topological features of homogeneous networks to obtain (a) a steady-state density ρ

of infected nodes during long time periods (Eq. 2.1), and (b) the critical threshold λ_c

(Eq. 2.2): 0 _{λ λ} ρ _{λ λ λ λ} λ  _<  = _{− ≥}  c c c (2.1) 1 λ = < > c k (2.2)

where k =

∑

_kkp is the average vertex degree of the network, and k p the fraction k

of nodes that have vertex degree k in the network. According to Eqs. 2.1 and 2.2, a

positive and nonzero critical threshold λ_c exists in a homogeneous network based on

the SIS epidemiological model. The contagion spreads and becomes epidemic if the

effective spreading rate exceeds the critical threshold (λ ≥λ_c); otherwise, the contagion

dies out. As shown in Figure 2.3, the SIS epidemiological model separates an infected

an SIS epidemiological model in a homogeneous network is the presence of a positive

critical threshold, proportional to the inverse of the average number of neighbors of

each node, below which epidemics die and endemic states are impossible.

0% 10% 20% 30% 40% 50% 0 0.1 0.2 0.3 0.4 0.5 Spreading Rate λ λ λ λ D e n s it y o f In fe c te d N o d e sρρρρ

Healthy state

Infected state

λλλλc

2.3.

Social Network Models and

Network-based Epidemiology

Complex networks are commonly used to represent structures for groups of

individuals who exhibit interaction or relationship patterns (Barabási & Albert, 1999;

Erdos & Renyi, 1960; Newman, 2003; Watts, 2003; Watts & Strogatz, 1998). As

shown in Figure 2.4 and Table 2.1, complex networks can be categorized as small

world, scale-free, or random according to basic statistical properties such as local

clustering, the small world phenomenon, or power-law connectivity distribution. They

are popular among researchers who construct computational simulations of virtual

societies, contagious diseases, Internet viruses, and the spread of cultural beliefs and

influences—all of which are affected by transmission routes.

Table 2.1. Two Complex Networks Categories Category Network Type Model Clustering Coefficient Degree of Separation Connectivity Distribution Homogeneous networks

Small-world Watts and Strogatz high low Normal

Random Erdös and Renyi very low low Normal

Heterogeneous

network Scale-free Barabási and Albert very low low Power-law

Generating a Watts and Strogatz (1998) small-world network begins with an

n-dimension ordered network with periodic boundary conditions, in which each node is

connected to a z quantity of neighbors, usually z≥2n (Figure 2.5a) (Watts &

Strogatz, 1998; Newman, 2003). Each link is randomly rewired to a new node with

probability p (Figure 2.5b). Under adverse circumstances, this construction method

can break the original ordered network into several isolated subgraphs (Figure 2.5d).

Newman and Watts (1999) introduced a variation of the original construction method

that emphasizes the insertion of long-range shortcuts instead of rewiring links. In their

version, two previously unconnected nodes are randomly selected and connected via a

newly added link, with users determining the number of links to be added (Figure 2.5c).

Newman and Watts’ small-world network thus avoids the problem of network breakage,

while preserving the positive characteristic of connecting each node in an

n-dimensional ordered network with 2n neighboring nodes. Since both the original

and new versions (Newman, 2003) exhibit small world and local clustering properties,

Figure 2.5. (a) One-dimensional ordered network with each node connected to

four adjacent nodes. (b) Watts and Strogatz’s (Duncan J. Watts & Strogatz, 1998)

small world network with four rewired shortcuts. (c) Newman and Watts’ (M. E.

J. Newman & D. J. Watts, 1999) improved small world network with five

additional shortcuts. (d) Example of a broken network in Watts and Strogatz’s

(Duncan J. Watts & Strogatz, 1998) small world network.

Generating a Barabási and Albert (1999) scale-free network begins with a small

number of nodes designated as z₀ (Newman, 2003). During each iteration, a new node

is introduced and connected to z≤z₀ pre-existing nodes according to a probability

based on each node’s vertex degree. New nodes are preferentially attached to existing

nodes that have large numbers of connections. This type of network exhibits

small-world and power-law connectivity distribution properties, implying the existence

of a small number of nodes with very large vertex degrees—similar to World Wide Web

hyperlinks and human sexual contact webs.

Erdös and Renyi’s (1960; Newman, 2003) random networks are generated by

are capable of exhibiting small-world properties if sufficient numbers of links are added,

but with little or no local clustering—an unusual situation in the real world.

Huang, Tsai and Sun (2010) used three rules to generate friend-making

networks—friend making, joining and leaving, and friendship updates—until each

network reached a statistically stationary state. Taking a bottom-up, network-oriented

simulation approach to modeling reflects the evolutionary mechanism of real-world

social networks. They built on insights from previous studies (e.g., Davidsen, Ebel &

Bornholdt, 2002) to apply local and interactive rules to acquaintance network

evolution. Findings from this approach can be used to explore human activity in

specific social networks—for example, rumor propagation and disease outbreaks

( )a ( )b

( )c ( )d

Figure 2.6. Comparison of node degree distributions and network

structures between Davidsen et al.’s two-rule model (a and b) and our proposed

three-rule model (c and d).

Communities, cities, and countries—even the entire planet—can be defined in

represents one individual with status-determining attributes (often referred to as

node-related local information) such as epidemiological progress, contagiousness, and

immunization (Huang et al., 2005; Xu et al., 2007). Connections between individuals

are referred to as links, with different links representing different interpersonal

relationships (Pastor-Satorras & Vespignani, 2001b). In HIV/AIDS epidemic

simulations they represent sexual relationships, and in SARS epidemic simulations

they represent close physical proximity (Huang et al., 2004, 2005). The states of all

network nodes change simultaneously during each time step. The state of an individual

node is determined by its original state, its linked neighbor’s state, and a set of

interaction rules.

Past epidemiological research has focused on the transmission dynamics and

spreading situations of biologically contagious diseases. A growing number of research

efforts are focusing on non-biological and intangible concepts such as computer viruses,

cultural influences, rumors, ideas, and beliefs that exist in social networks and on the

Internet. In these kinds of spreading scenarios, cultural influences move ideas and

beliefs between transmitters and receivers, eventually making the majority of receivers

behave in the same manner as transmitters (Huang et al., 2005; Lynch, 1996; Rogers,

2003). Researchers have recently looked at epidemic dynamics and critical thresholds

Chapter 3.

Analysis of Epidemiological

Transmission in Theoretical Complex

Networks

Avian influenza, a flu that originally spread only among birds but is now found

among birds and humans, is a likely candidate to become an epidemic or pandemic

disease. Another epidemic, the 1918 influenza outbreak in North America, is one of the

most studied by epidemiologists. Nine decades later, Watts (1998) described his

proposed small-world property in complex networks, which has strongly influenced

research involving human networks. Later, Pastor-Satorras and Vespignani (2001a)

combined epidemic dynamics and complex networks to propose an epidemic model

indicating that according to a scale-free network created by Barabási and Albert (1999),

an epidemic threshold tends toward 0 as long as the network is sufficiently large. Based

on their model (which I will refer to as the P-V model in this dissertation), Huang and

Tsai proposed a modified model containing resource limitations and transmission costs

for analyzing epidemic thresholds (Huang et al., 2010; Tsai and Huang, 2010). We

used computer simulations to verify the model, as well as to show its practical

3.1.

Motivation

Researchers who take network-oriented approaches to analyzing contagious

disease diffusion processes note that the topological features of social networks exert

considerable influence on transmission dynamics and spreading situations associated

with epidemics (Newman, 2003; Newman & Watts, 1999). Unlike non-network

approaches, they support subtle analyses of epidemic dynamics (Pastor-Satorras &

Vespignani, 2001a, 2001b, 2002, 2003; Huang et al., 2004, 2005). Researchers of

epidemic dynamics and critical thresholds in scale-free networks consistently conclude

that regardless of transmission capability, all contagious diseases have high

probabilities of stable spreading and survival in scale-free networks(Xu et al., 2007).

According to Pastor-Satorras and Vespignani (2001b), a positive critical

transmission threshold does not exist for the spreading of contagious diseases in

scale-free social networks. In other words, even contagious diseases with tiny

transmission capabilities survive in such networks. Pastor-Satorras and Vespignani’s

proposed spreading dynamic is expressed as follows:

[

]

{

}

( ) ( ) 1 ( ) ( ) k k k k d t t k t t dt ρ _{= −ρ +λ −ρ θ ρ }   (3.1)

where ρ_k( )t is the density of infected nodes with k connections, λ a constant

linked to an infected individual, with θ _{assumed to be a function of the partial} densities of infected individuals

{

ρ_k( )t

}

. Eq. (3.1) states that during each time step, infected individuals who have k connections will recover, yet continue to infect other

individuals according to four parameters: infection rate, connectivity, number of

healthy individuals, and probability θ ρ

[

{ _k( )}t

]

. Pastor-Satorras and Vespignani defined ρ_{k as the steady state of} ρ_k( )t , and observed that ρ_k is a function of λ _in

a steady state, therefore θ is a function of λ, such that ( ) 1 ( ) _k

k

kP k k

θ λ =

_∑

ρ , with ( )

P k representing connectivity distribution. Furthermore, when considering the

stationary condition dρ_k( )t dt=0 within a scale-free network in which

2 3

( ) 2

P k = m k− with minimum degree m, the critical epidemic threshold λ_c has the

property 2

0

c k k

λ = → as k→ ∞. Accordingly, for infinite size networks, either

no epidemic threshold exists, or the threshold approaches 0.

New contagious diseases are constantly emerging in different parts of the world,

but very few reach epidemic proportions or even survive in social networks; the

majority of diseases die almost immediately following their appearance. This

observation serves as our motivation to take a more detailed look at limitations in

transmission and interaction processes rather than the topological features of social

networks—the focus of many epidemiological studies published in the past decade.

been understudied: resource limitations and transmission costs. The term resource in

this situation is defined as what is consumed by individuals during the spreading

process of a contagious disease. There are five properties associated with resources: (a)

they can be visible (e.g., seminal fluid, physical power) or invisible (e.g., time, energy);

(b) individual resources are finite and can be temporarily exhausted; (c) the use of one

type of resource entails the consumption of smaller quantities of other types of

resources, thereby reducing the total available resource amount; (d) individual

resources can recover or regenerate after a period of time; and (e) they are

non-reproducible. Contagious carriers who apply resources to specific recipients

cannot reuse the same resources on other recipients; conversely, recipients cannot reuse

resources spent on individual carriers. We acknowledge the importance of

Pastor-Satorras and Vespignani’s (2001a) work on the topological power-law features

of scale-free social networks, especially since their ideas have inspired numerous

studies on critical thresholds and immunization strategies. However, such assumptions

may be unrealistic and inaccurate when applied to biologically contagious diseases

spread via face-to-face interactions and daily contacts. A closer inspection of their

mathematical analyses and numerical simulations reveal what we believe to be

incorrect assumptions that daily interaction processes are cost-free, and that the impacts

3.2.

A Contagious Epidemiological Model

under Resource Limitations and

Transmission Cost Considerations

Our mathematical model is based on the epidemic simulation model shown in Eq.

(3.1) as proposed by Pastor-Satorras and Vespignani (2001b). However, this model

neglects individual access to energy, time, and other finite resources. Therefore, we

propose a model under different infection rate-to-link degree assumptions.

To incorporate individual access to energy, time, and other finite resources, we

modified the model to consider resource limitations and transmission costs using two

different approaches, as shown in Eq. (3.2).

[

]

{

}

( ) ( ) 1 ( ) ( ) , where min( , ). k k k k k k d t R t S t t S k dt c ρ _{= −ρ +λ −ρ θ ρ  =}   (3.2)

According to the term _{S (with k} R representing average resources and c

transmission costs), the spreading of each infection is proportional to the minimum

value of each active node’s available resources (R/c) and number of links.

Using the mean field method, we let the stationary condition dρ_k( )t dt=0,

obtaining ( ) 1 ( ) k k k S S λ θ λ ρ λ θ λ = + (3.3)

1 ( ) 1 k k k S kP k k S λ θ θ λ θ = +

∑

(3.4)

Note that the right side of Eq. (3.4) is concave at about θ _{(i.e., the second} derivative is no larger than zero), and that θ =0 _{is considered a trivial solution. Since}

it is possible for θ _{to have a non-singular solution, we derived the inequality}

0 1 ( ) 1. 1 k k k S d kP k d k S θ λ θ θ λ θ ₌   ≥    ₊  

∑

 (3.5)

Differentiating Eq. (3.5) and substituting 0 for θ _{we get} 1 ( ) 1 or . ( ) k k k k k kP k S k

∑

λ ≥ λ≤

∑

kP k S (3.6)

Accordingly, critical threshold λ_c is defined as the maximal λ, resulting in

( ) c k k k kP k S λ =

∑

(3.7)

Since S_k =min(R c k, ), the denominator can be divided into two parts, obtaining

2 . ( ) ( ) c R R k k c c k R k P k kP k c λ ≤ > = +

∑

∑

(3.8)

According to the first term in the Eq. (3.8) denominator, k is smaller than R/c,

therefore substituting R/c for k makes the first term larger. Similarly, according to the

second term, the summation is smaller than the entire scope of k, therefore substituting

2 ( ) ( ) c R k k c k R R P k kP k c c λ ≤ ≥   +    

∑

∑

(3.9)

Using the same method, another substitution on the left side of the Eq. (3.9)

denominator results in 2 . ( ) ( ) c k k k R R P k kP k c c λ ≥   +    

∑

∑

(3.10) Since ( ) 1 k P k =

∑

, we arrive at 2 2 1 . c k R R R k c c c R k c λ ≥ =     ₊           _  _+       (3.11)

and observe that as k → ∞, λ_{c is at minimum equal to} c R. 0 0.1 0.2 0.3 0.4 0.5 0% 5% 10% 15% 20% 25% % = (Transmission Costs / Individual Usable Resources)

E p id em ic T h re sh o ld λλλλc

Figure 3.1. Critical Threshold λ_c is a function of the ratio of transmission costs to individual resources (c R/ ) in scale-free networks. We used it to analyze results from

our simulation experiments and three mathematical analyses.

As shown in Figure 3.1, the mathematical results are consistent with the

simulation result. The results indicate that when resources and transmission costs are

taken into consideration, a significant critical threshold (above which a contagious

disease exceeds control and becomes epidemic, and below which a contagious disease

disappears) exists when a contagion event occurs in a scale-free network—in short, a

non-zero critical threshold exists in scale-free networks. Our results also indicate that

the appearance of a critical threshold is tied to a ratio of transmission costs to available

resources. In summary, the lower bound of λ_{c becomes larger whether transmission} cost c increases or the average resource R decreases. Accordingly, an individual’s

available resources expand when c/R is large, thereby decreasing that individual’s

ability to contact almost all other personal network nodes. Since λ_{c represents the}

threshold at which a contagious disease exceeds control and become epidemic,

managing the value of λ_{c is the primary concern of epidemiologists and public health}

officials. The result supports what we know about immunization: appropriately

restricting one’s resources increases the critical threshold. Neglecting one’s resources

that the critical threshold λ_c will approach 0 as long as the size of the average number of links is large enough. The model thus becomes identical to Pastor-Satorras and

Vespignani’s model (Eq. 3.1), in which a disease has the potential to become epidemic

3.3.

Epidemic Effect of Limited

Resources/Transmission Cost Ratio

One scenario to which Eq. (3.11) can be applied is a network attack spread via the

Internet—an example of a scale-free network (Barabási et al., 2002). Although

spreading time is short, affected areas can be very large, with disastrous results in terms

of lost data, work time, and money. One suggested strategy for controlling computer

network attacks is placing restrictions on downloads from web services (e.g., a

maximum of one gigabyte per day)—in other words, a time resource limitation to raise

the outbreak critical threshold λ_c. Another potential strategy is charging downloading fees—that is, raising transmission costs to increase outbreak critical thresholds. The

algorithm Barabási and Albert (1999) introduced to build their model (which I will

refer to as the BA model in this dissertation) is based on a concept common to networks

such as the Internet, the World Wide Web, and social networks—that is, for each node

there is a large probability of connecting to other nodes that are already linked to still

other nodes. According to the algorithm, we take m₀

disconnected nodes, steadily add new nodes, and connect the new nodes to existing m nodes at a probability of

( ) i i j j k P k k =

∑

, where ki is the degree of the i-th node. The algorithm stops and a graph