Modeling Infectious Diseases in Humans and Animals

Matt J. Keeling and Pejman Rohani

Publisher:

Princeton University Press

Publication Date:

2008

Number of Pages:

366

Format:

Hardcover

Price:

65.00

ISBN:

978-0-691-11617-4

Category:

Textbook

MAA Review
Table of Contents

[Reviewed by

William J. Satzer

, on

03/26/2008

]

Many textbooks in mathematical biology are “large M” “small b” books that tend to treat the biology as an interesting and valuable application of mathematics. This is not necessarily a bad thing — particularly when it’s used to show mathematics students part of the wide universe of applications. The approach is less well-suited for an audience primarily interested in the biology; there it looks too much like a fixation on the tools instead of what you do with them.

Modeling Infectious Disease in Humans and Animals presents more of a balance. It is an introduction to real-time and predictive modeling of infectious disease intended primarily for health care professionals, epidemiologists and evolutionary biologists. It focuses on directly transmitted infection by microparasites (viruses, bacteria, protozoa and prions) where there is extensive long-term data and a good understanding of the transmission dynamics. This is only a small portion of the whole field of epidemiological modeling and analysis, but it is an area where substantial progress has been made over the last two decades.

The book begins with a very nice introductory discussion of the types and characteristics of diseases. Here it becomes clear that modeling infectious disease is not a “one size fits all” operation. This also establishes the context and scope of the authors’ presentation. The second chapter introduces simple epidemic models, and it does so in a way that is considerably more nuanced than comparable treatments. Most often authors will discuss the basic susceptible-infected-recovered (SIR) model, perhaps examine a few examples, and then move on. In such treatments, it’s difficult to see what implicit assumptions have been made, so it’s unclear when the model is indeed applicable. In this book, the authors look carefully at the model with several different sets of assumptions. These include inclusion or exclusion of births, deaths and migrations, density- and frequency-dependent transmission of infection, as well as models without immunity, with waning immunity, or with a latent period.

Succeeding chapters treat more complex environments. A chapter on host heterogeneities describes models appropriate to populations where distinct groups have different susceptibilities to catch and transmit an infection. A chapter on multi-pathogen, multi-host models treats diseases that can be caught and transmitted by numerous host species. Such models may be critical for the prediction of worldwide influenza strains and the possibility of pandemics.

Other topics treated here are temporally forced models (when diseases are subject to periodic forcing by some environmental variable), stochastic dynamics and spatial models (dealing with spatial variations of infected populations and the propagation of infections across geographic regions). A final chapter on controlling infectious diseases describes how the models from the previous chapters can be used to optimize control measures in order to minimize the spread of infection.

Somewhat surprisingly, the authors do not address data analysis or any related statistical issues. This is disappointing because – especially in this area of epidemiology that is rich in good data — the study of modeling is much enhanced when challenged with real data.

Many of the epidemiological problems addressed in this book are analytically intractable. The authors have provided a website with software (in Java, C, Fortran and Matlab) to allow students to explore at length the models presented in the text or to develop new models. This book includes no exercises, and, if used in as a course text, would need supplementing with problem sets and project suggestions. Probably its best use in a mathematics class would be as background or supplementary reading.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Acknowledgments xiii

Chapter 1: Introduction 1
1.1 Types of Disease 1
1.2 Characterization of Diseases 3
1.3 Control of Infectious Diseases 5
1.4 What Are Mathematical Models? 7
1.5 What Models Can Do 8
1.6 What Models Cannot Do 10
1.7 What Is a Good Model? 10
1.8 Layout of This Book 11
1.9 What Else Should You Know? 13

Chapter 2: Introduction to Simple Epidemic Models 15
2.1 Formulating the Deterministic SIR Model 16
2.1.1 The SIR Model Without Demography 19
2.1.1.1 The Threshold Phenomenon 19
2.1.1.2 Epidemic Burnout 21
2.1.1.3 Worked Example: Influenza in a Boarding School 26
2.1.2 The SIR Model With Demography 26
2.1.2.1 The Equilibrium State 28
2.1.2.2 Stability Properties 29
2.1.2.3 Oscillatory Dynamics 30
2.1.2.4 Mean Age at Infection 31
2.2 Infection-Induced Mortality and SI Models 34
2.2.1 Mortality Throughout Infection 34
2.2.1.1 Density-Dependent Transmission 35
2.2.1.2 Frequency Dependent Transmission 36
2.2.2 Mortality Late in Infection 37
2.2.3 Fatal Infections 38
2.3 Without Immunity: The SIS Model 39
2.4 Waning Immunity: The SIRS Model 40
2.5 Adding a Latent Period: The SEIR Model 41
2.6 Infections with a Carrier State 44
2.7 Discrete-Time Models 46
2.8 Parameterization 48
2.8.1 Estimating R0 from Reported Cases 50
2.8.2 Estimating R0 from Seroprevalence Data 51
2.8.3 Estimating Parameters in General 52
2.9 Summary 52

Chapter 3: Host Heterogeneities 54
3.1 Risk-Structure: Sexually Transmitted Infections 55
3.1.1 Modeling Risk Structure 57
3.1.1.1 High-Risk and Low-Risk Groups 57
3.1.1.2 Initial Dynamics 59
3.1.1.3 Equilibrium Prevalence 62
3.1.1.4 Targeted Control 63
3.1.1.5 Generalizing the Model 64
3.1.1.6 Parameterization 64
3.1.2 Two Applications of Risk Structure 69
3.1.2.1 Early Dynamics of HIV 71
3.1.2.2 Chlamydia Infections in Koalas 74
3.1.3 Other Types of Risk Structure 76
3.2 Age-Structure: Childhood Infections 77
3.2.1 Basic Methodology 78
3.2.1.1 Initial Dynamics 80
3.2.1.2 Equilibrium Prevalence 80
3.2.1.3 Control by Vaccination 81
3.2.1.3 Parameterization 82
3.2.2 Applications of Age Structure 84
3.2.2.1 Dynamics of Measles 84
3.2.2.2 Spread and Control of BSE 89
3.3 Dependence on Time Since Infection 93
3.3.1 SEIR and Multi-Compartment Models 94
3.3.2 Models with Memory 98
3.3.3 Application: SARS 100
3.4 Future Directions 102
3.5 Summary 103

Chapter 4: Multi-Pathogen/Multi-Host Models 105
4.1 Multiple Pathogens 106
4.1.1 Complete Cross-Immunity 107
4.1.1.1 Evolutionary Implications 109
4.1.2 No Cross-Immunity 112
4.1.2.1 Application: The Interaction of Measles and Whooping Cough 112
4.1.2.2 Application: Multiple Malaria Strains 115
4.1.3 Enhanced Susceptibility 116
4.1.4 Partial Cross-Immunity 118
4.1.4.1 Evolutionary Implications 120
4.1.4.2 Oscillations Driven by Cross-Immunity 122
4.1.5 A General Framework 125
4.2 Multiple Hosts 128
4.2.1 Shared Hosts 130
4.2.1.1 Application: Transmission of Foot-and-Mouth Disease 131
4.2.1.2 Application: Parapoxvirus and the Decline of the Red Squirrel 133
4.2.2 Vectored Transmission 135
4.2.2.1 Mosquito Vectors 136
4.2.2.2 Sessile Vectors 141
4.2.3 Zoonoses 143
4.2.3.1 Directly Transmitted Zoonoses 144
4.2.3.2 Vector-Borne Zoonoses: West Nile Virus 148
4.3 Future Directions 151
4.4 Summary 153

Chapter 5: Temporally Forced Models 155
5.1 Historical Background 155
5.1.1 Seasonality in Other Systems 158
5.2 Modeling Forcing in Childhood Infectious Diseases: Measles 159
5.2.1 Dynamical Consequences of Seasonality: Harmonic and Subharmonic Resonance 160
5.2.2 Mechanisms of Multi-Annual Cycles 163
5.2.3 Bifurcation Diagrams 164
5.2.4 Multiple Attractors and Their Basins 167
5.2.5 Which Forcing Function? 171
5.2.6 Dynamical Trasitions in Seasonally Forced Systems 178
5.3 Seasonality in Other Diseases 181
5.3.1 Other Childhood Infections 181
5.3.2 Seasonality in Wildlife Populations 183
5.3.2.1 Seasonal Births 183
5.3.2.2 Application: Rabbit Hemorrhagic Disease 185
5.4 Summary 187

Chapter 6: Stochastic Dynamics 190
6.1 Observational Noise 193
6.2 Process Noise 193
6.2.1 Constant Noise 195
6.2.2 Scaled Noise 197
6.2.3 Random Parameters 198
6.2.4 Summary 199
6.2.4.1 Contrasting Types of Noise 199
6.2.4.2 Advantages and Disadvantages 200
6.3 Event-Driven Approaches 200
6.3.1 Basic Methodology 201
6.3.1.1 The SIS Model 202
6.3.2 The General Approach 203
6.3.2.1 Simulation Time 203
6.3.3 Stochastic Extinctions and The Critical Community Size 205
6.3.3.1 The Importance of Imports 209
6.3.3.2 Measures of Persistence 212
6.3.3.3 Vaccination in a Stochastic Environment 213
6.3.4 Application: Porcine Reproductive and Respiratory Syndrome 214
6.3.5 Individual-Based Models 217
6.4 Parameterization of Stochastic Models 219
6.5 Interaction of Noise with Heterogeneities 219
6.5.1 Temporal Forcing 219
6.5.2 Risk Structure 220
6.5.3 Spatial Structure 221
6.6 Analytical Methods 222
6.6.1 Fokker-Plank Equations 222
6.6.2 Master Equations 223
6.6.3 Moment Equations 227
6.7 Future Directions 230
6.8 Summary 230

Chapter 7: Spatial Models 232
7.1 Concepts 233
7.1.1 Heterogeneity 233
7.1.2 Interaction 235
7.1.3 Isolation 236
7.1.4 Localized Extinction 236
7.1.5 Scale 236
7.2 Metapopulations 237
7.2.1 Types of Interaction 240
7.2.1.1 Plants 240
7.2.1.2 Animals 241
7.2.1.3 Humans 242
7.2.1.4 Commuter Approximations 243
7.2.2 Coupling and Synchrony 245
7.2.3 Extinction and Rescue Effects 246
7.2.4 Levins-Type Metapopulations 250
7.2.5 Application to the Spread of Wildlife Infections 251
7.2.5.1 Phocine Distemper Virus 252
7.2.5.2 Rabies in Raccoons 252
7.3 Lattice-Based Models 255
7.3.1 Coupled Lattice Models 255
7.3.2 Cellular Automata 257
7.3.2.1 The Contact Process 258
7.3.2.2 The Forest-Fire Model 259
7.3.2.3 Application: Power laws in Childhood Epidemic Data 260
7.4 Continuous-Space Continuous-Population Models 262
7.4.1 Reaction-Diffusion Equations 262
7.4.2 Integro-Differential Equations 265
7.5 Individual-Based Models 268
7.5.1 Application: Spatial Spread of Citrus Tristeza Virus 269
7.5.2 Applilcation: Spread of Foot-and-mouth Disease in the
United Kingdom 274
7.6 Networks 276
7.6.1 Network Types 277
7.6.1.1 Random Networks 277
7.6.1.2 Lattices 277
7.6.1.3 Small World Networks 279
7.6.1.4 Spatial Networks 279
7.6.1.5 Scale-Free Networks 279
7.6.2 Simulation of Epidemics on Networks 280
7.7 Which Model to Use? 282
7.8 Approximations 283
7.8.1 Pair-Wise Models for Networks 283
7.8.2 Pair-Wise Models for Spatial Processes 286
7.9 Future Directions 287
7.10 Summary 288

Chapter 8: Controlling Infectious Diseases 291
8.1 Vaccination 292
8.1.1 Pediatric Vaccination 292
8.1.2 Wildlife Vaccination 296
8.1.3 Random Mass Vaccination 297
8.1.4 Imperfect Vaccines and Boosting 298
8.1.5 Pulse Vaccination 301
8.1.6 Age-Structured Vaccination 303
8.1.6.1 Application: Rubella Vaccination 304
8.1.7 Targeted Vaccination 306
8.2 Contact Tracing and Isolation 308
8.2.1 Simple Isolation 309
8.2.2 Contact Tracing to Find Infection 312
8.3 Case Study: Smallpox, Contact Tracing, and Isolation 313
8.4 Case Study: Foot-and-Mouth Disease, Spatial Spread, and Local Control 321
8.5 Case Study: Swine Fever Virus, Seasonal Dynamics, and Pulsed Control 327
8.5.1 Equilibrium Properties 329
8.5.2 Dynamical Properties 331
8.6 Future Directions 333
8.7 Summary 334