Mathematical Epidemiology

by Brauer, Fred; Van Den Driessche, Pauline; Wu, Jianhong; Allen, L. J. S. (CON); Bauch, C. T. (CON)

• ISBN13: 9783540789109
• ISBN10: 3540789103
• Format: Paperback
• Copyright: 2008-06-03
• Publisher: Springer Verlag

Summary

Based on lecture notes of two summer schools with a mixed audience from mathematical sciences, epidemiology and public health, this volume offers a comprehensive introduction to basic ideas and techniques in modeling infectious diseases, for the comparison of strategies to plan for an anticipated epidemic or pandemic, and to deal with a disease outbreak in real time. It covers detailed case studies for diseases including pandemic influenza, West Nile virus, and childhood diseases. Models for other diseases including Severe Acute Respiratory Syndrome, fox rabies, and sexually transmitted infections are included as applications. Its chapters are coherent and complementary independent units. In order to accustom students to look at the current literature and to experience different perspectives, no attempt has been made to achieve united writing style or unified notation.Notes on some mathematical background (calculus, matrix algebra, differential equations, and probability) have been prepared and may be downloaded at the web site of the Centre for Disease Modeling (www.cdm.yorku.ca).

Author Biography

Fred Brauer is a professor emeritus at the University of Wisconsin - Madison, where he taught from 1960 to 1999, and has been an honorary professor at the University of British Columbia since 1997. He is the author or co-author of 115 papers on differential equations, mathematical population biology, and mathematical epidemiology as well as 10 books including undergraduate texts and a book on models in population biology and epidemiology jointly with Carlos Castillo - Chavez.Pauline van den Driessche is a professor emerita in the Department of Mathematics and Statistics and an adjunct professor in the Department of Computer Science at the University of Victoria. Her research interests include mathematical biology, especially epidemiology, and matrix analysis. Recently she was awarded the 2007 Krieger - Nelson prize by the Canadian Mathematical Society, and gave the Olga Taussky Todd lecture at the International Conference in Industrial and Applied Mathematics in July 2007.Jianhong Wu is a professor and a Senior Canada Research Chair in Applied Mathematics at York University. He is the author or coauthor of over 200 peer-reviewed publications and six monographs in the areas of nonlinear dynamical systems, delay differential equations, mathematical biology and epidemiology, neural networks, and pattern formation and recognition. He is the recipient of the Canadian Industrial and Applied Mathematics Research Prize (2003), the Alexander van Humboldt Fellowship (1996), the Paul Erdos Visiting Professorship (2000), and Cheung Kong/YangZi River Lecture Professorship (2006).

Table of Contents

Mathematical Epidemiology	p. 1
Introduction and General Framework
A Light Introduction to Modelling Recurrent Epidemics	p. 3
Introduction	p. 3
Plague	p. 4
Measles	p. 5
The SIR Model	p. 6
Solving the Basic SIR Equations	p. 8
SIR with Vital Dynamics	p. 11
Demographic Stochasticity	p. 13
Seasonal Forcing	p. 13
Slow Changes in Susceptible Recruitment	p. 14
Not the Whole Story	p. 15
Take Home Message	p. 16
References	p. 16
Compartmental Models in Epidemiology	p. 19
Introduction	p. 19
Simple Epidemic Models	p. 22
The Kermack-McKendrick Model	p. 24
Kermack-McKendrick Models with General Contact Rates	p. 32
Exposed Periods	p. 36
Treatment Models	p. 38
An Epidemic Management (Quarantine-Isolation) Model	p. 40
Stochastic Models for Disease Outbreaks	p. 45
Models with Demographic Effects	p. 45
The SIR Model	p. 45
The SIS Model	p. 52
Some Applications	p. 55
Herd Immunity	p. 55
Age at Infection	p. 56
The Interepidemic Period	p. 57
"Epidemic" Approach to the Endemic Equilibrium	p. 59
Disease as Population Control	p. 60
Age of Infection Models	p. 66
The Basic SI* R Model	p. 66
Equilibria	p. 69
The Characteristic Equation	p. 70
The Endemic Equilibrium	p. 72
An SI* S Model	p. 74
An Age of Infection Epidemic Model	p. 76
References	p. 78
An Introduction to Stochastic Epidemic Models	p. 81
Introduction	p. 81
Review of Deterministic SIS and SIR Epidemic Models	p. 82
Formulation of DTMC Epidemic Models	p. 85
SIS Epidemic Model	p. 85
Numerical Example	p. 90
SIR Epidemic Model	p. 90
Numerical Example	p. 93
Formulation of CTMC Epidemic Models	p. 93
SIS Epidemic Model	p. 93
Numerical Example	p. 97
SIR Epidemic Model	p. 98
Formulation of SDE Epidemic Models	p. 100
SIS Epidemic Model	p. 100
Numerical Example	p. 103
SIR Epidemic Model	p. 103
Numerical Example	p. 105
Properties of Stochastic SIS and SIR Epidemic Models	p. 105
Probability of an Outbreak	p. 105
Quasistationary Probability Distribution	p. 108
Final Size of an Epidemic	p. 112
Expected Duration of an Epidemic	p. 115
Epidemic Models with Variable Population Size	p. 117
Numerical Example	p. 119
Other Types of DTMC Epidemic Models	p. 121
Chain Binomial Epidemic Models	p. 121
Epidemic Branching Processes	p. 124
MatLab Programs	p. 125
References	p. 128
Advanced Modeling and Heterogeneities
An Introduction to Networks in Epidemic Modeling	p. 133
Introduction	p. 133
The Probability of a Disease Outbreak	p. 134
Transmissibility	p. 138
The Distribution of Disease Outbreak and Epidemic Sizes	p. 140
Some Examples of Contact Networks	p. 142
Conclusions	p. 145
References	p. 145
Deterministic Compartmental Models: Extensions of Basic Models	p. 147
Introduction	p. 147
Vertical Transmission	p. 148
Kermack-McKendrick SIR Model	p. 148
SEIR Model	p. 150
Immigration of Infectives	p. 152
General Temporary Immunity	p. 154
References	p. 157
Further Notes on the Basic Reproduction Number	p. 159
Introduction	p. 159
Compartmental Disease Transmission Models	p. 160
The Basic Reproduction Number	p. 162
Examples	p. 163
The SEIR Model	p. 163
A Variation on the Basic SEIR Model	p. 165
A Simple Treatment Model	p. 166
A Vaccination Model	p. 168
A Vector-Host Model	p. 170
A Model with Two Strains	p. 171
R[subscript o] and the Local Stability of the Disease-Free Equilibrium	p. 173
R[subscript o] and Global Stability of the Disease-Free Equilibrium	p. 175
References	p. 177
Spatial Structure: Patch Models	p. 179
Introduction	p. 179
Spatial Heterogeneity	p. 180
Geographic Spread	p. 182
Effect of Quarantine on Spread of 1918-1919 Influenza in Central Canada	p. 185
Tuberculosis in Possums	p. 188
Concluding Remarks	p. 188
References	p. 189
Spatial Structure: Partial Differential Equations Models	p. 191
Introduction	p. 191
Model Derivation	p. 192
Case Study I: Spatial Spread of Rabies in Continental Europe	p. 194
Case Study II: Spread Rates of West Nile Virus	p. 199
Remarks	p. 202
References	p. 202
Continuous-Time Age-Structured Models in Population Dynamics and Epidemiology	p. 205
Why Age-Structured Models?	p. 205
Modeling Populations with Age Structure	p. 206
Solutions along Characteristic Lines	p. 208
Equilibria and the Characteristic Equation	p. 209
Age-Structured Integral Equations Models	p. 211
The Renewal Equation	p. 212
Age-Structured Epidemic Models	p. 214
A Simple Age-Structured AIDS Model	p. 215
The Reproduction Number	p. 216
Pair-Formation in Age-Structured Epidemic Models	p. 218
The Semigroup Method	p. 220
Modeling with Discrete Age Groups	p. 222
Examples	p. 223
References	p. 225
Distribution Theory, Stochastic Processes and Infectious Disease Modelling	p. 229
Introduction	p. 230
A Review of Some Probability Theory and Stochastic Processes	p. 231
Non-negative Random Variables and Their Distributions	p. 231
Some Important Discrete Random Variables Representing Count Numbers	p. 234
Continuous Random Variables Representing Time-to-Event Durations	p. 237
Mixture of Distributions	p. 239
Stochastic Processes	p. 241
Random Graph and Random Graph Process	p. 248
Formulating the Infectious Contact Process	p. 249
The Expressions for R[subscript 0] and the Distribution of N such that R[subscript 0] = E[N]	p. 251
Competing Risks, Independence and Homogeneity in the Transmission of Infectious Diseases	p. 254
Some Models Under Stationary Increment Infectious Contact Process {K(x)}	p. 255
Classification of some Epidemics Where N Arises from the Mixed Poisson Processes	p. 255
Tail Properties for N	p. 258
The Invasion and Growth During the Initial Phase of an Outbreak	p. 261
Invasion and the Epidemic Threshold	p. 262
The Risk of a Large Outbreak and Quantities Associated with a Small Outbreak	p. 263
Behaviour of a Large Outbreak in its Initial Phase: The Intrinsic Growth	p. 273
Summary for the Initial Phase of an Outbreak	p. 280
Beyond the Initial Phase: The Final Size of Large Outbreaks	p. 281
Generality of the Mean Final Size	p. 282
Some Cautionary Remarks	p. 283
When the Infectious Contact Process may not Have Stationary Increment	p. 285
The Linear Pure Birth Processes and the Yule Process	p. 286
Parallels to the Preferential Attachment Model in Random Graph Theory	p. 288
Distributions for N when {K(x)} Arises as a Linear Pure Birth Process	p. 288
References	p. 291
Case Studies
The Role of Mathematical Models in Explaining Recurrent Outbreaks of Infectious Childhood Diseases	p. 297
Introduction	p. 297
The SIR Model with Demographics	p. 300
Historical Development of Compartmental Models	p. 302
Early Models	p. 302
Stochasticity	p. 306
Seasonality	p. 306
Age Structure	p. 307
Alternative Assumptions About Incidence Terms	p. 307
Distribution of Latent and Infectious Period	p. 308
Seasonality Versus Nonseasonality	p. 308
Chaos	p. 309
Transitions Between Outbreak Patterns	p. 310
Spectral Analysis of Incidence Time Series	p. 310
Power Spectra	p. 311
Wavelet Power Spectra	p. 313
Conclusions	p. 314
References	p. 316
Modeling Influenza: Pandemics and Seasonal Epidemics	p. 321
Introduction	p. 321
A Basic Influenza Model	p. 322
Vaccination	p. 326
Antiviral Treatment	p. 330
A More Detailed Model	p. 334
A Model with Heterogeneous Mixing	p. 336
A Numerical Example	p. 341
Extensions and Other Types of Models	p. 345
References	p. 346
Mathematical Models of Influenza: The Role of Cross-Immunity, Quarantine and Age-Structure	p. 349
Introduction	p. 349
Basic Model	p. 351
Cross-Immunity and Quarantine	p. 354
Age-Structure	p. 359
Discussion and Future Work	p. 362
References	p. 363
A Comparative Analysis of Models for West Nile Virus	p. 365
Introduction: Epidemiological Modeling	p. 365
Case Study: West Nile Virus	p. 367
Minimalist Model	p. 368
The Question	p. 368
Model Scope and Scale	p. 368
Model Formulation	p. 370
Model Analysis	p. 372
Model Application	p. 373
Biological Assumptions 1: When does the Disease-Transmission Term Matter?	p. 374
Frequency Dependence	p. 374
Mass Action	p. 374
Numerical Values of R[subscript 0]	p. 377
Biological Assumptions 2: When do Added Model Classes Matter?	p. 377
Model Parameterization, Validation, and Comparison	p. 380
Model Application #1: WN Control	p. 381
Model Application #2: Seasonal Mosquito Population	p. 382
Summary	p. 384
References	p. 386
Suggested Exercises and Projects	p. 391
Cholera	p. 395
Ebola	p. 395
Gonorrhea	p. 395
HIV/AIDS	p. 396
HIV in Cuba	p. 396
Human Papalonoma Virus	p. 397
Influenza	p. 397
Malaria	p. 397
Measles	p. 398
Poliomyelitis (Polio)	p. 398
Severe Acute Respiratory Syndrome (SARS)	p. 399
Smallpox	p. 399
Tuberculosis	p. 400
West Nile Virus	p. 400
Yellow Fever in Senegal 2002	p. 400
Index	p. 403
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