Modeling Infectious Diseases in Humans and Animals

Modeling Infectious Diseases in Humans and Animals

Pub. Date:
Princeton University Press


View All Available Formats & Editions
Current price is , Original price is $97.5. You
Select a Purchase Option (New Edition)
  • purchase options
    $85.31 $97.50 Save 13% Current price is $85.31, Original price is $97.5. You Save 13%.
  • purchase options


Modeling Infectious Diseases in Humans and Animals

For epidemiologists, evolutionary biologists, and health-care professionals, real-time and predictive modeling of infectious disease is of growing importance. This book provides a timely and comprehensive introduction to the modeling of infectious diseases in humans and animals, focusing on recent developments as well as more traditional approaches.

Matt Keeling and Pejman Rohani move from modeling with simple differential equations to more recent, complex models, where spatial structure, seasonal "forcing," or stochasticity influence the dynamics, and where computer simulation needs to be used to generate theory. In each of the eight chapters, they deal with a specific modeling approach or set of techniques designed to capture a particular biological factor. They illustrate the methodology used with examples from recent research literature on human and infectious disease modeling, showing how such techniques can be used in practice. Diseases considered include BSE, foot-and-mouth, HIV, measles, rubella, smallpox, and West Nile virus, among others. Particular attention is given throughout the book to the development of practical models, useful both as predictive tools and as a means to understand fundamental epidemiological processes. To emphasize this approach, the last chapter is dedicated to modeling and understanding the control of diseases through vaccination, quarantine, or culling.

  • Comprehensive, practical introduction to infectious disease modeling
  • Builds from simple to complex predictive models
  • Models and methodology fully supported by examples drawn from research literature
  • Practical models aid students' understanding of fundamental epidemiological processes
  • For many of the models presented, the authors provide accompanying programs written in Java, C, Fortran, and MATLAB
  • In-depth treatment of role of modeling in understanding disease control

Product Details

ISBN-13: 9780691116174
Publisher: Princeton University Press
Publication date: 10/08/2007
Edition description: New Edition
Pages: 408
Product dimensions: 6.90(w) x 10.00(h) x 1.10(d)

About the Author

Matt J. Keeling is professor in the Department of Biological Sciences and the Mathematics Institute at the University of Warwick. Pejman Rohani is associate professor in the Institute of Ecology and the Center for Tropical and Emerging Global Diseases at the University of Georgia.

Read an Excerpt

Modeling Infectious Diseases in Humans and Animals

By Matt J. Keeling Pejman Rohani Princeton University Press
Copyright © 2007
Princeton University Press
All right reserved.

ISBN: 978-0-691-11617-4

Chapter One Introduction

This book is designed as an introduction to the modeling of infectious diseases. Westart with the simplest of mathematical models and show how the inclusion of appropriate elements of biological complexity leads to improved understanding of disease dynamics and control. Throughout, our emphasis is on the development of models, and their use either as predictive tools or as a means of understanding fundamental epidemiological processes. Although many theoretical results can be proved analytically for very simple models, we have generally focused on results obtained by computer simulation, providing analytical results only where they lead to a more generic interpretation of model behavior. Where practical, we have illustrated the general modeling Principles with applied examples from the recent literature. We hope this book motivates readers to develop their own models for diseases of interest, expanding on the model frameworks given here.


The Oxford English Dictionary defines a disease as "a condition of the body, or of some part or organ of the body, in which its functions are disturbed or deranged; a morbid physical condition; a departure from the state of health, especially when caused by structural change." This definition encompassesa wide range of ailments from AIDS to arthritis, from the common cold to cancer. The fine-scale classification of diseases varies drastically between different scientific disciplines. Medical doctors and veterinary clinicians, for example, are primarily interested in treating human patients or animals and, as such, are most concerned about the infection's pathophysiology (affecting, for example, the central nervous system) or clinical symptoms (for example, secretory diarrhea). Microbiologists, on the other hand, focus on the natural history of the causative organism: What is the etiologicalagent (avirus, bacterium, protozoan, fungus, orprion)? and what are the ideal conditions for its growth? Finally, epidemiologists are most interested in features that determine patterns of disease and its transmission.

In general terms, we may organize diseases according to several overlapping classifications (Figure 1.1). Diseases can be either infectious or noninfectious. Infectious diseases (such as influenza) can be passed between individuals, whereas noninfectious diseases (such as arthritis) developover an individual's lifespan. The epidemiology of noninfectious diseases is primarily a study of risk factors associated with the chance of developing the disease (for example, the increased risk of lung cancer attributable to smoking). In contrast, the primary risk factor for catching an infectious disease is the presence of infectious cases in the local population-this tenet is reflected in all the mathematical models presented in this book. These two categories, infectious and noninfectious, are not necessarily mutually exclusive. Infection with the human papillomavirus (HPV), for example, is firmly associated with (although not necessary for developing) cervical cancer, thus bridging the two fields. This book focuses on infectious diseases, where models have great predictive power at the population scale and over relatively short time scales.

Infectious diseases can be further subdivided (Figure 1.1). The infecting pathogen can be either a microparasite (hatched in diagram) or a macroparasite. Microparasites, as the same suggests, are small (usually single-cell organisms) and are either viruses, bacteria, protoza, or prions; macroparasites are any larger form of pathogen and include helminths and flukes. Although the biological distinction between these two groups of organisms is clear, from a modeling perspective the boundaries are less well defined. In general, microparasitic infections develop rapidly from a small number of infecting particles so the internal dynamics of the pathogen within the host can often be safely ignored. As a result, we are not interested in the precise abundance of pathogens within the host; instead we focus on the host's infection status. In contrast, macroparasites such as helminths have a complex life cycle with in the host which often needs to be modeled explicitly. In addition, the worm burden, or the number of parasites within the host, represents an important contributing factor to pathogenicity and disease transmission. We focus in this book on microparasites, where extensive long-term data and a good mechanistic understanding of the transmission dynamics have led to a wealth of well-parameterized models.

Infectious diseases (both macro-and microparasitic) can also be subdivided into two further categories (Figure 1.1), depending on whether transmission of infection is direct (shaded gray) or indirect. Direct transmission is when infection is caught by close contact with an infectious individual. The great majority of microparasitic diseases, such as influenza, measles, and HIV, are directly transmitted, although there are exceptions such as cholera, which is waterborne. Generally, directly transmitted pathogens do not survive for long outside the host organism. In contrast, indirectly transmitted parasites are passed between hosts via the environment; most macroparasitic diseases, such as those caused by helminths and schistosomes, are indirectly transmitted, spending part of their life cycle outside of their hosts. In addition, there is a class of diseases where transmission is via a secondary host or vector, usually insects such as mosquitoes, tsetseflies, orticks. However, this transmission route can be considered as two sequential direct transmission events, from the primary host to the insect and then from the insect to another primary host.

The models and diseases of this book are focused toward the study of directly transmitted, microparasitic infectious diseases. As such, this subset represents only a fraction of the whole field of epidemiological modeling and analysis, but one in which major advances have occurred over recent decades.

Worldwide there are about 1,415 known human pathogens of which 217 (15%) are viruses or prions and 518 (38%) are bacteria or rickettsia; hence around 53% are micro-parasites (Cleaveland et al. 2001). Of these pathogens, 868 (61%) are zoonotic and can therefore be transmitted from animals to humans. Around 616 pathogens of domestic livestock are known, of which around 18% are viral and 25% bacterial. However, if we restrict our attention to the 70 pathogens listed by the Office International des Epizooties (which contain the most prominent and infectious livestock diseases), we find that 77% are microparasites (Cleaveland et al. 2001). The lower number of known livestock pathogens compared to human pathogens probably reflects to some degree our natural anthropocentric bias. Similarly, very few infectious diseases of wildlife are known or studied in any detail, and yet wildlife reservoirs may be important sources of novel emerging human infections. It is therefore clear that the study of microparasitic infectious diseases encompasses a huge variety of hosts and diseases.


The progress of an infectious microparasitic disease is defined qualitatively in terms of the level of pathogen within the host, which in turn is determined by the growth rate of the pathogen and the interaction between the pathogen and the host's immune response. Figure 1.2 shows a much simplified infection profile. Initially, the host is susceptible to infection: No pathogen is present; just a low-level nonspecific immunity within the host. At time 0, the host encounters an infectious individual and becomes infected with a microparasite; the abundance of the parasite grows over time. During this early phase the individual may exhibit no obvious signs of infection and the abundance of pathogen may be too low to allow further transmission-individuals in this phase are said to be in the exposed class. Once the level of parasite is sufficiently large within the host, the potential exists to transmit the infection to other susceptible individuals; the host is infectious. Finally, once the individual's immune system has cleared the parasite and the host is therefore no longer infectious, they are referred to as recovered.

This fundamental classification (as susceptible, exposed, infectious, or recovered) solely depends on the host's ability to transmit the pathogen. This has two implications. First, the disease status of the host is irrelevant-it is not important whether the individual is showing symptoms; an individual who feels perfectly healthy can be excreting large amounts of pathogen (Figure 1.2). Second, the boundaries between exposed and infectious (and infectious and recovered) are somewhat fuzzy because the ability to transmit does not simply switch on and off. This uncertainty is further complicated by the variability in responses between different individuals and the variability in pathogen levels over the infectious period; it is only with the recent advances in molecular techniques that these within-host individual-level details are beginning to emerge. Our classification of hosts as susceptible, exposed, infectious, or recovered can therefore be compared to the ecological concept of a meta population (Levins 1969; Hanski and Gilpin 1991), in which the within-host density of the pathogen is ignored and each host is simply classified as being in one of a limited number of categories.

Although Figure 1.2 shows an example of a disease profile that might be modeled as SEIR (susceptible-exposed-infectious-recovered), other within-host profiles are also common. Often, it is mathematically simpler and justifiable at the population scale to ignore the exposed class, reducing the number of equations by one and leading to SIR dynamics. Some infections, especially of plants, are more appropriately described by the SI (susceptible-infectious) paradigm; for such diseases, the host is infectious soon after it is infected, such that the exposed period can be safely ignored, and remains infectious until its death. Other infectious diseases, in particular sexually transmitted infections (such as gonorrhoea), are better described by an SIS (susceptible-infectious susceptible) framework, because once recovered(or following treatment) the host is once again susceptible to infection. In the majority of cases this renewed susceptibility is due to the vast antigenic variation associated with sexually transmitted diseases. Finally, many diseases have profiles that are individualistic and require specific model formulation. Smallpox has a definite short prodromal period before the symptoms emerge when the infected individual is mobile and can widely disseminate the virusbut infectiousness has not reached its peak. Hepatitis B has a carrier state such that some infected individuals do not fully recover but transmitata low level for the rest of their lives. Chlamydia (and many other sexually transmitted diseases) may be asymptomatic, such that some infected individuals do not suffer from the disease even though they are able to transmit infection. Similarly, infections such as meningitis or MRSA (methicillin resistant streptococus aurius) are widespread in the general population and usually benign, with only occasional symptomatic outbreaks. All of these more complex epidemiological behaviors require greater subdivision of the population and therefore models that deal explicitly with these extra classes.

Although such qualitative descriptions of disease dynamics allow us to understand the behavior of infection with in an individual and may even shed some light on potential transmission, if we are to extra polate from the individual-level dynamics to the population-scale epidemic, numerical values are required for many of the key parameters. Two fundamental quantities govern the population-level epidemic dynamics: the basic reproductive ratio, R0, and the timescale of infection, which is measured by the infectious period for SIS and SIR infections or by a mixture of exposed and infectious periods in diseases with SEIR dynamics (for details, see Chapter 2). The basic reproductive ratio is one of the most critical epidemiological parameters because it defines the average number of secondary cases an average primary case produces in a totally susceptible population. Among other things, this single parameter allows us to determine whether a disease can successfully invade or not, the threshold level of vaccination required for eradication, and the long-term proportion of susceptible individuals when the infection is endemic.

One of the key features of epidemiological modeling is the huge variability in infection profiles, parameter values, and timescales. Many childhood infectious diseases (such as measles, rubella, or chickenpox) follow the classic SEIR profile, have high basic reproductive ratios(R0 [approximately equal to] 17 for both measles and whooping cough in England and Wales from 1945 to 1965), and short infected periods (of less than one month). In contrast, diseases such as HIV have a much more complex infection profile with transmission rates varying as a function of time since infection, R0 is crucially dependent on sexual behavior(R0 [approximately equal to] 4 for the homosexual population in the United Kingdom, whereas R0 [approximately equal to] 11 for female prostitutes in Kenya), and infection is lifelong. Between these two extremes lies a vast array of other infectious diseases, with their own particular characteristics and parameters.


One of the primary reasons for studying infectious diseases is to improve control and ultimately to eradicate the infection from the population. Models can be a powerful tool in this approach, allowing us to optimize the use of limited resources or simply to target control measures more efficiently. Several forms of control measure exist; all operate by reducing the average amount of transmission between infectious and susceptible individuals. Which control strategy (or mixture of strategies) is used will depend on the disease, the hosts, and the scale of the epidemic.

The practice of vaccination began with Edward Jenner in 1796 who developed a vaccine against smallpox-which remains the only disease to date that has been eradicated world-wide. Vaccination acts by stimulating a host immune response, such that immunized individuals are protected against infection. Vaccination is generally applied prophylactically to a large proportion of the population, so as to greatly reduce the number of susceptible individuals. Such prophylactic vaccination campaigns have successfully reduced the incidence of many childhood infections in the developed world by vaccinating the vast majority of young children and infants. In 1988, the World Heath Organization (WHO) resolved to use similar campaigns to eradicate polio worldwide by 2005-this is still ongoing work although much progress has been made to date.

Although vaccination offers a very powerful method of disease control, there are many associated difficulties. Generally, vaccines are not 100% effective, and therefore only a proportion of vaccinated individuals are protected. Some vaccines can have adverse side effects; the vaccine against smallpox can be harmful (sometimes fatal) to those with eczema, asthma, or are immuno-suppressed, and may even cause cases of smallpox. Some vaccines provide only limited immunity, whether this is due to the natural waning of immunity in the host or to antigenic variation in the pathogen. Finally, in the face of a novel(or unexpected) epidemic, reactive vaccination may prove to be too slow to prevent a large outbreak. Therefore, in many situations, alternative control measures are necessary. Vaccination operates by reducing the number of susceptible individuals in the population.

Quarantine, or the isolation of known or suspected infectious individuals, is one of the oldest known forms of disease control. During the fifteenth and sisteenth century, Venice, Italy, practiced a policy of quarantine against all ships arriving from areas infected with plague, and in 1665 the village of Eyam in Derbyshire, UK, famously quarantined themselves in an effort to prevent the plague spreading to neighboring villages. Today quarantining is still a powerful control measure; was used to combat SARS in 2003, and it is a rapid first response against many invading pathogens. Quarantining essentially operates by preventing infectious individuals from mixing with susceptible individuals, hence stopping transmission. The primary advantage of quarantining is that it is simple and generic; quarantining is effective even when the causative agent is unknown. However, quarantining can be applied only once an infectious individual is identified, by which time the individual may have been transmitting infection for many days. In addition, unless the number of cases is small, quarantining can be a prohibitive drain on resources.


Excerpted from Modeling Infectious Diseases in Humans and Animals by Matt J. Keeling Pejman Rohani
Copyright © 2007 by Princeton University Press. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Table of Contents

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 The Threshold Phenomenon 19 Epidemic Burnout 21 Worked Example: Influenza in a Boarding School 26

2.1.2 The SIR Model With Demography 26 The Equilibrium State 28 Stability Properties 29 Oscillatory Dynamics 30 Mean Age at Infection 31

2.2 Infection-Induced Mortality and SI Models 34

2.2.1 Mortality Throughout Infection 34 Density-Dependent Transmission 35 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 High-Risk and Low-Risk Groups 57 Initial Dynamics 59 Equilibrium Prevalence 62 Targeted Control 63 Generalizing the Model 64 Parameterization 64

3.1.2 Two Applications of Risk Structure 69 Early Dynamics of HIV 71 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 Initial Dynamics 80 Equilibrium Prevalence 80 Control by Vaccination 81 Parameterization 82

3.2.2 Applications of Age Structure 84 Dynamics of Measles 84 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 Evolutionary Implications 109

4.1.2 No Cross-Immunity 112 Application: The Interaction of Measles and Whooping Cough 112 Application: Multiple Malaria Strains 115

4.1.3 Enhanced Susceptibility 116

4.1.4 Partial Cross-Immunity 118 Evolutionary Implications 120 Oscillations Driven by Cross-Immunity 122

4.1.5 A General Framework 125

4.2 Multiple Hosts 128

4.2.1 Shared Hosts 130 Application: Transmission of Foot-and-Mouth Disease 131 Application: Parapoxvirus and the Decline of the Red Squirrel 133

4.2.2 Vectored Transmission 135 Mosquito Vectors 136 Sessile Vectors 141

4.2.3 Zoonoses 143 Directly Transmitted Zoonoses 144 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 Seasonal Births 183 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 Contrasting Types of Noise 199 Advantages and Disadvantages 200

6.3 Event-Driven Approaches 200

6.3.1 Basic Methodology 201 The SIS Model 202

6.3.2 The General Approach 203 Simulation Time 203

6.3.3 Stochastic Extinctions and The Critical Community Size 205 The Importance of Imports 209 Measures of Persistence 212 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 Plants 240 Animals 241 Humans 242 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 Phocine Distemper Virus 252 Rabies in Raccoons 252

7.3 Lattice-Based Models 255

7.3.1 Coupled Lattice Models 255

7.3.2 Cellular Automata 257 The Contact Process 258 The Forest-Fire Model 259 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 Random Networks 277 Lattices 277 Small World Networks 279 Spatial Networks 279 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 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

What People are Saying About This

Andrew Dobson

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


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

Customer Reviews

Most Helpful Customer Reviews

See All Customer Reviews