Propagation Dynamics on Complex Networks Models, Methods and Stability Analysis

Xinchu Fu, Department of Mathematics, Shanghai  University, China

Michael Small, School of Mathematics and Statistics, The University of Western Australia, Australia

Guanrong Chen, Department of Electronic Engineering, City University of Hong Kong, China

Preface xi

Summary xiii

1 Introduction 1

1.1 Motivation and background 1

1.2 A brief history of mathematical epidemiology 2

1.3 Organization of the book 5

References 6

2 Various epidemic models on complex networks 10

2.1 Multiple stage models 10

2.2 Staged progression models 13

2.3 Stochastic SIS model 17

2.4 Models with population mobility 19

2.5 Models in meta-populations 22

2.6 Models with effective contacts 24

2.7 Models with two distinct routes 26

2.8 Models with competing strains 28

2.9 Models with competing strains and saturated infectivity 31

2.10 Models with birth and death of nodes and links 33

2.11 Models on weighted networks 34

2.12 Models on directed networks 38

2.13 Models on colored networks 40

2.14 Discrete epidemic models 44

References 47

3 Epidemic threshold analysis 53

3.1 Threshold analysis by the direct method 53

3.2 Epidemic spreading efficiency threshold and epidemic threshold 69

3.3 Epidemic thresholds and basic reproduction numbers 76

References 98

4 Networked models for SARS and avian influenza 101

4.1 Network models of real diseases 101

4.2 Plausible models for propagation of the SARS virus 102

4.3 Clustering model for SARS transmission: Application to epidemic control and risk assessment 108

4.4 Small-world and scale-free models for SARS transmission 114

4.5 Super-spreaders and the rate of transmission 118

4.6 Scale-free distribution of avian influenza outbreaks 124

4.7 Stratified model of ordinary influenza 130

References 136

5 Infectivity functions 139

5.1 A model with nontrivial infectivity function 140

5.2 Saturated infectivity 143

5.3 Nonlinear infectivity for SIS model on scale-free networks 143

References 148

6 SIS models with an infective medium 150

6.1 SIS model with an infective medium 150

6.2 A modified SIS model with an infective medium 159

6.3 Epidemic models with vectors between two separated networks 162

6.4 Epidemic transmission on interdependent networks 167

6.4.1 Theoretical modeling 168

6.5 Discussions and remarks 179

References 181

7 Epidemic control and awareness 184

7.1 SIS model with awareness 184

7.2 Discrete-time SIS model with awareness 192

7.3 Spreading dynamics of a disease-awareness SIS model on complex networks 198

7.4 Remarks and discussions 201

References 203

8 Adaptive mechanism between dynamics and epidemics 207

8.1 Adaptive mechanism between dynamical synchronization and epidemic behavior on complex networks 207

8.2 Interplay between collective behavior and spreading dynamics 216

References 228

9 Epidemic control and immunization 231

9.1 SIS model with immunization 231

9.2 Edge targeted strategy for controlling epidemic spreading on scale-free networks 235

9.3 Remarks and discussions 237

References 239

10 Global stability analysis 240

10.1 Global stability analysis of the modified model with an infective medium 240

10.2 Global dynamics of the model with vectors between two separated networks 241

10.3 Global behavior of disease transmission on interdependent networks 247

10.4 Global behavior of epidemic transmissions 250

10.5 Global attractivity of a network-based epidemic SIS model 260

10.6 Global stability of an epidemic model with birth and death and adaptive weights 264

10.7 Global dynamics of a generalized epidemic model 268

References 274

11 Information diffusion and pathogen propagation 277

11.1 Information diffusion and propagation on complex networks 277

11.2 Interplay between information of disease spreading and epidemic dynamics 281

11.3 Discussions and remarks 284

References 286

Appendix A Proofs of theorems 289

A.1 Transition from discrete-time linear system to continuous-time linear system 289

A.2 Proof of Lemma 6.1 291

A.3 Proof of Theorem 10.4 291

A.4 Proof of Theorem 10.3 292

A.5 Proof of Theorem 10.42 296

Appendix B Further proofs of results 302

B.1 Eigenvalues of the matrix ? F in (6.27) 302

B.2 The matrix 𝛤 in (6.32) 304

B.3 Proof of (7.6) in Chapter 7 305

B.4 The positiveness of 𝜎′: proof of 𝜎′ > 0 in Section 9.1.2 306

B.5 The relation between 𝛬 and 𝜅 in Section 9.1.3 308

Index 311

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