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Preface xi
1 Some History 1
1.1 An empirical approach 1
1.2 A deterministic model 2
1.2.1 The Law of Mass Action 6
1.3 Prom curve-fitting to homogeneous mixing models 7
1.4 Stochastic modelling 11
1.5 Model fitting and prediction 13
1.6 Some general observations and summary 15
1.6.1 Methods and models 16
1.6.2 Some terminology 16
1.7 Exercises and Complements to Chapter 1 17
2 Deterministic Models 20
2.1 The simple epidemic in continuous time 20
2.2 The simple epidemic in interacting groups 23
2.3 The general epidemic in a homogeneous population 27
2.4 The general epidemic in a stratified population 35
2.5 Generation-wise evolution of epidemics 38
2.6 Carrier models 45
2.7 Endemicity of a vector-borne disease 46
2.8 Discrete time deterministic models 48
2.9 Exercises and Complements to Chapter 2 53
3 Stochastic Models in Continuous Time 56
3.1 The simple stochastic epidemic in continuous time 57
3.1.1 Analysis of the Markov chain 58
3.1.2 A simplifying device 61
3.1.3 Distribution of the duration time 62
3.2 Probability generating function methods for Markov chains 63
3.3 The general stochastic epidemic 66
3.3.1 Solution by the p.g.f. method 68
3.3.2 Whittle's threshold theorem for the general stochastic
epidemic 73
3.4 The ultimate size of the general stochastic epidemic 77
3.4.1 The total size distribution using the p.g.f. method 77
3.4.2 Embedded jump processes 79
3.4.3 The total size distribution using the embedded jump chain 81
3.4.4 Behaviour of the general stochastic epidemic model: a
composite picture 83
3.5 The general stochastic epidemic in a stratified population 85
3.6 The carrier-borne epidemic 94
3.7 Exercises and Complements to Chapter 3 100
4 Stochastic Models in Discrete Time 105
4.1 The Greenwood and Reed-Frost Models 105
4.1.1 P.g.f. methods for the Greenwood model 108
4.2 Further properties of the Reed-Frost model 111
4.3 Chains with infection probability varying between households 115
4.4 Chain binomial models with replacement 118
4.5 Final size of epidemic with arbitrary infectious period 123
4.6 A pairs-at-parties model: exchangeable but not
homogeneous mixing 126
4.7 Exercises and Complements to Chapter 4 131
5 Rumours: Modelling Spread and its Cessation 133
5.1 Rumour models 133
5.2 Deterministic analysis of rumour models 138
5.3 Embedded random walks for rumour models 141
5.4 A diffusion approximation 145
5.5 P.g.f. solutions of rumour models 149
5.6 Exercises and Complements to Chapter 5 151
6 Fitting Epidemic Data 154
6.1 Influenza epidemics: a discrete time deterministic model 155
6.2 Extrapolation forecasting for AIDS: a continuous time model 158
6.3 Measles epidemics in households: chain binomial models 162
6.3.1 Final number of cases infected 162
6.3.2 Cases infected for different types of chain 163
6.4 Variable infectivity in chain binomial models 164
6.5 Incubation period of AIDS and the back-calculation method 168
6.5.1 The distribution of the AIDS incubation period 172
6.6 Exercises and Complements to Chapter 6 174
7 The Control of Epidemics 175
7.1 Control by education 176
7.2 Control by immunization 179
7.3 Control by screening and quarantine 184
7.3.1 The single prison model 186
7.3.2 Interaction of a prison with the outside world 187
7.3.3 A quarantine policy in prison 189
7.4 Exercises and Complements to Chapter 7 192
References and Author Index 194
Subject Index 205
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