"Mathematical models of infectious disease have proven to be a valuable component of public health planning and response, as well as an important application of population biology. Keeling and Rohani have written an accessible and much-needed introduction to this field that will be suitable for graduate students and advanced undergraduates alike."--Carl T. Bergstrom, University of Washington
"Mathematical models are now as crucial in the study of infectious diseases as are microscopes, stethoscopes, and the tools of molecular diagnosis. These models have contributed to epidemiological understanding at all levels, from projections of the magnitude of the AIDS epidemic to an understanding of the within-host interactions between pathogens and the host's immune system. This book outlines all the major developments in mathematical epidemiology that have occurred since the publication of Anderson and May's classic synthesis in Infectious Diseases of Humans. It is highly recommended to all students of infectious disease biology who require a detailed and well-organized introduction to the mathematical models needed to understand the dynamics of infectious diseases."--Andrew Dobson, Princeton University
Inhaltsverzeichnis
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 References 337 Index 361 Parameter Glossary 367
