Mathematical Epidemiology ....................................... 1
F. Brauer, P. van den Driessche and J. Wu, editors
Part I Introduction and General Framework
1 A Light Introduction to Modelling Recurrent Epidemics ........ 3
David J.D. Earn
1.1 Introduction ............................................ 3
1.2 Plague .................................................. 4
1.3 Measles ................................................. 5
1.4 The SIR Model ........................................... 6
1.5 Solving the Basic SIR Equations ......................... 8
1.6 SIR with Vital Dynamics ................................ 11
1.7 Demographic Stochasticity .............................. 13
1.8 Seasonal Forcing ....................................... 13
1.9 Slow Changes in Susceptible Recruitment ................ 14
1.10 Not the Whole Story .................................... 15
1.11 Take Home Message ...................................... 16
References .................................................. 16
2 Compartmental Models in Epidemiology ........................ 19
Fred Brauer
2.1 Introduction ........................................... 19
2.1.1 Simple Epidemic Models .......................... 22
2.1.2 The Kermack-McKendrick Model .................... 24
2.1.3 Kermack-McKendrick Models with General
Contact Rates ................................... 32
2.1.4 Exposed Periods ................................. 36
2.1.5 Treatment Models ................................ 38
2.1.6 An Epidemic Management (Quarantine-Isolation)
Model ........................................... 40
2.1.7 Stochastic Models for Disease Outbreaks ......... 45
2.2 Models with Demographic Effects ........................ 45
2.2.1 The SIR Model ................................... 45
2.2.2 The SIS Model ................................... 52
2.3 Some Applications ...................................... 55
2.3.1 Herd Immunity ................................... 55
2.3.2 Age at Infection ................................ 56
2.3.3 The Interepidemic Period ........................ 57
2.3.4 "Epidemic" Approach to the Endemic
Equilibrium ..................................... 59
2.3.5 Disease as Population Control ................... 60
2.4 Age of Infection Models ................................ 66
2.4.1 The Basic SPR Model ............................. 66
2.4.2 Equilibria ...................................... 69
2.4.3 The Characteristic Equation ..................... 70
2.4.4 The Endemic Equilibrium ......................... 72
2.4.5 An STS Model .................................... 74
2.4.6 An Age of Infection Epidemic Model .............. 76
References .................................................. 78
3 An Introduction to Stochastic Epidemic Models ............... 81
Linda J.S. Allen
3.1 Introduction ........................................... 81
3.2 Review of Deterministic SIS and SIR Epidemic Models .... 82
3.3 Formulation of DTMC Epidemic Models .................... 85
3.3.1 SIS Epidemic Model .............................. 85
3.3.2 Numerical Example ............................... 90
3.3.3 SIR Epidemic Model .............................. 90
3.3.4 Numerical Example ............................... 93
3.4 Formulation of CTMC Epidemic Models .................... 93
3.4.1 SIS Epidemic Model .............................. 93
3.4.2 Numerical Example ............................... 97
3.4.3 SIR Epidemic Model .............................. 98
3.5 Formulation of SDE Epidemic Models .................... 100
3.5.1 SIS Epidemic Model ............................. 100
3.5.2 Numerical Example .............................. 103
3.5.3 SIR Epidemic Model ............................. 103
3.5.4 Numerical Example .............................. 105
3.6 Properties of Stochastic SIS and SIR Epidemic Models .. 105
3.6.1 Probability of an Outbreak ..................... 105
3.6.2 Quasistationary Probability Distribution ....... 108
3.6.3 Final Size of an Epidemic ...................... 112
3.6.4 Expected Duration of an Epidemic ............... 115
3.7 Epidemic Models with Variable Population Size ......... 117
3.7.1 Numerical Example .............................. 119
3.8 Other Types of DTMC Epidemic Models ................... 121
3.8.1 Chain Binomial Epidemic Models ................. 121
3.8.2 Epidemic Branching Processes ................... 124
3.9 MatLab Programs ....................................... 125
References ................................................. 128
Part II Advanced Modeling and Heterogeneities
4 An Introduction to Networks in Epidemic Modeling ........... 133
Fred Brauer
4.1 Introduction .......................................... 133
4.2 The Probability of a Disease Outbreak ................. 134
4.3 Transmissibility ...................................... 138
4.4 The Distribution of Disease Outbreak and Epidemic
Sizes ................................................. 140
4.5 Some Examples of Contact Networks ..................... 142
4.6 Conclusions ........................................... 145
References ................................................. 145
5 Deterministic Compartmental Models: Extensions of Basic
Models ..................................................... 147
P. van den Driessche
5.1 Introduction .......................................... 147
5.2 Vertical Transmission ................................. 148
5.2.1 Kermack-McKendrick SIR Model ................... 148
5.2.2 SEIR Model ..................................... 150
5.3 Immigration of Infectives ............................. 152
5.4 General Temporary Immunity ............................ 154
References ................................................. 157
6 Further Notes on the Basic Reproduction Number ............. 159
P. van den Driessche and James Watmough
6.1 Introduction .......................................... 159
6.2 Compartmental Disease Transmission Models ............. 160
6.3 The Basic Reproduction Number ......................... 162
6.4 Examples .............................................. 163
6.4.1 The SEIR Model ................................. 163
6.4.2 A Variation on the Basic SEIR Model ............ 165
6.4.3 A Simple Treatment Model ....................... 166
6.4.4 A Vaccination Model ............................ 168
6.4.5 A Vector-Host Model ............................ 170
6.4.6 A Model with Two Strains ....................... 171
6.5 TZ0 and the Local Stability of the Disease-Free
Equilibrium ........................................... 173
6.6 1Z0 and Global Stability of the Disease-Free
Equilibrium ........................................... 175
References ................................................. 177
7 Spatial Structure: Patch Models ............................ 179
P. van den Driessche
7.1 Introduction .......................................... 179
7.2 Spatial Heterogeneity ................................. 180
7.3 Geographic Spread ..................................... 182
7.4 Effect of Quarantine on Spread of 1918-1919
Influenza in Central Canada ........................... 185
7.5 Tuberculosis in Possums ............................... 188
7.6 Concluding Remarks .................................... 188
References ................................................. 189
8 Spatial Structure: Partial Differential Equations Models ... 191
Jianhong Wu
8.1 Introduction .......................................... 191
8.2 Model Derivation ...................................... 192
8.3 Case Study I: Spatial Spread of Rabies in
Continental Europe .................................... 194
8.4 Case Study II: Spread Rates of West Nile Virus ........ 199
8.5 Remarks ............................................... 202
References ................................................. 202
9 Continuous-Time Age-Structured Models in Population
Dynamics and Epidemiology .................................. 205
Jia Li and Fred Brauer
9.1 Why Age-Structured Models? ............................ 205
9.2 Modeling Populations with Age Structure ............... 206
9.2.1 Solutions along Characteristic Lines ........... 208
9.2.2 Equilibria and the Characteristic Equation ..... 209
9.3 Age-Structured Integral Equations Models .............. 211
9.3.1 The Renewal Equation ........................... 212
9.4 Age-Structured Epidemic Models ........................ 214
9.5 A Simple Age-Structured AIDS Model .................... 215
9.5.1 The Reproduction Number ........................ 216
9.5.2 Pair-Formation in Age-Structured Epidemic
Models ......................................... 218
9.5.3 The Semigroup Method ........................... 220
9.6 Modeling with Discrete Age Groups ..................... 222
9.6.1 Examples ....................................... 223
References ................................................. 225
10 Distribution Theory, Stochastic Processes and Infectious
Disease Modelling .......................................... 229
Ping Yan
10.1 Introduction .......................................... 230
10.2 A Review of Some Probability Theory and Stochastic
Processes ............................................. 231
10.2.1 Non-negative Random Variables a,nri Their
Distributions .................................. 231
10.2.2 Some Important Discrete Random Variables
Representing Count Numbers ..................... 234
10.2.3 Continuous Random Variables Representing
Time-to-Event Durations ........................ 237
10.2.4 Mixture of Distributions ....................... 239
10.2.5 Stochastic Processes ........................... 241
10.2.6 Random Graph and Random Graph Process .......... 248
10.3 Formulating the Infectious Contact Process ............ 249
10.3.1 The Expressions for R0 and the Distribution
of N such that R0 = E[N] ....................... 251
10.3.2 Competing Risks, Independence and Homogeneity
in the Transmission of Infectious Diseases ..... 254
10.4 Some Models Under Stationary Increment Infectious
Contact Process {K(x)} ................................ 255
10.4.1 Classification of some Epidemics Where N
Arises from the Mixed Poisson Processes ........ 255
10.4.2 Tail Properties for N .......................... 258
10.5 The Invasion and Growth During the Initial Phase of
an Outbreak ........................................... 261
10.5.1 Invasion and the Epidemic Threshold ............ 262
10.5.2 The Risk of a Large Outbreak and Quantities
Associated with a Small Outbreak ............... 263
10.5.3 Behaviour of a Large Outbreak in its Initial
Phase: The Intrinsic Growth .................... 273
10.5.4 Summary for the Initial Phase of an Outbreak ... 280
10.6 Beyond the Initial Phase: The Final Size of Large
Outbreaks ............................................. 281
10.6.1 Generality of the Mean Final Size .............. 282
10.6.2 Some Cautionary Remarks ........................ 283
10.7 When the Infectious Contact Process may not Have
Stationary Increment .................................. 285
10.7.1 The Linear Pure Birth Processes and the Yule
Process ........................................ 286
10.7.2 Parallels to the Preferential Attachment
Model in Random Graph Theory ................... 288
10.7.3 Distributions for N when {K(x)} Arises as a
Linear Pure Birth Process ...................... 288
References ................................................. 291
Part III Case Studies
11 The Role of Mathematical Models in Explaining
Recurrent Outbreaks of Infectious Childhood Diseases ....... 297
Chris T. Bauch
11.1 Introduction .......................................... 297
11.2 The SIR Model with Demographics ....................... 300
11.3 Historical Development of Compartmental Models ........ 302
11.3.1 Early Models ................................... 302
11.3.2 Stochasticity .................................. 306
11.3.3 Seasonality .................................... 306
11.3.4 Age Structure .................................. 307
11.3.5 Alternative Assumptions About Incidence
Terms .......................................... 307
11.3.6 Distribution of Latent and Infectious Period ... 308
11.3.7 Seasonality Versus Nonseasonality .............. 308
11.3.8 Chaos .......................................... 309
11.3.9 Transitions Between Outbreak Patterns .......... 310
11.4 Spectral Analysis of Incidence Time Series ............ 310
11.4.1 Power Spectra .................................. 311
11.4.2 Wavelet Power Spectra .......................... 313
11.5 Conclusions ........................................... 314
References ................................................. 316
12 Modeling Influenza: Pandemics and Seasonal Epidemics ....... 321
Fred Brauer
12.1 Introduction .......................................... 321
12.2 A Basic Influenza Model ............................... 322
12.3 Vaccination ........................................... 326
12.4 Antiviral Treatment ................................... 330
12.5 A More Detailed Model ................................. 334
12.6 A Model with Heterogeneous Mixing ..................... 336
12.7 A Numerical Example ................................... 341
12.8 Extensions and Other Types of Models .................. 345
References ................................................. 346
13 Mathematical Models of Influenza: The Role of Cross-
Immunity, Quarantine and Age-Structure ..................... 349
M. Nuño, C. Castillo-Chavez, Z. Feng and M. Martcheva
13.1 Introduction .......................................... 349
13.2 Basic Model ........................................... 351
13.3 Cross-Immunity and Quarantine ......................... 354
13.4 Age-Structure ......................................... 359
13.5 Discussion and Future Work ............................ 362
References ................................................. 363
14 A Comparative Analysis of Models for West Nile Virus ....... 365
M.J. Wonham and M.A. Lewis
14.1 Introduction: Epidemiological Modeling ................ 365
14.2 Case Study: West Nile Virus ........................... 367
14.3 Minimalist Model ...................................... 368
14.3.1 The Question ................................... 368
14.3.2 Model Scope and Scale .......................... 368
14.3.3 Model Formulation .............................. 370
14.3.4 Model Analysis ................................. 372
14.3.5 Model Application .............................. 373
14.4 Biological Assumptions 1: When does the Disease-
Transmission Term Matter? ............................. 374
14.4.1 Frequency Dependence ........................... 374
14.4.2 Mass Action .................................... 374
14.4.3 Numerical Values of 1Z0 ........................ 377
14.5 Biological Assumptions 2: When do Added Model
Classes Matter? ....................................... 377
14.6 Model Parameterization, Validation, and Comparison .... 380
14.7 Model Application #1: WN Control ...................... 381
14.8 Model Application #2: Seasonal Mosquito Population .... 382
14.9 Summary ............................................... 384
References ................................................. 386
Suggested Exercises and Projects .............................. 391
1 Cholera .................................................... 395
2 Ebola ...................................................... 395
3 Gonorrhea .................................................. 395
4 HIV/AIDS ................................................... 396
5 HIV in Cuba ................................................ 396
6 Human Papalonoma Virus ..................................... 397
7 Influenza .................................................. 397
8 Malaria .................................................... 397
9 Measles .................................................... 398
10 Poliomyelitis (Polio) ...................................... 398
11 Severe Acute Respiratory Syndrome (SARS) ................... 399
12 Smallpox ................................................... 399
13 Tuberculosis ............................................... 400
14 West Nile Virus ............................................ 400
15 Yellow Fever in Senegal 2002 ............................... 400
Index ......................................................... 403
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