The Institute for Mathematical Sciences at the National University of Singapore hosted a research program on Mathematical Modeling of Infectious Diseases: Dynamics and Control from 15 August to 9 October 2005. As part of the program, tutorials for graduate students and junior researchers were given by leading experts in the field.
This invaluable volume is a collection of three expanded lecture notes of those tutorials which cover a wide range of topics including basic mathematical details for various epidemic models, statistical distribution theory, and applications with real-life examples with implications for health policy makers.
Contents: The Basic Epidemiology Models: Models, Expressions for R 0, Parameter Estimation, and Applications (H W Hethcote); Epidemiology Models with Variable Population Size (H W Hethcote); Age-Structured Epidemiology Models and Expressions for R 0 (H W Hethcote); Clinical and Public Health Applications of Mathematical Models (J W Glasser); Non-Identifiables and Invariant Quantities in Infectious Disease Models (P Yan).
Author(s): Stefan Ma, Stefan Ma, Yingcun Xia
Series: Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore
Publisher: World Scientific Publishing Company
Year: 2008
Language: English
Pages: 240
CONTENTS......Page 6
Foreword......Page 8
Preface......Page 10
Contents......Page 12
1.1 Introduction......Page 13
1.2 Why do epidemiology modeling?......Page 17
1.3 Definitions, assumptions and model formulations......Page 27
1.3.1 Formulating epidemiology models......Page 30
1.3.2 Three threshold quantities: R0, σ, and R......Page 32
1.4 The basic SIS endemic model......Page 33
1.5 The basic SIR epidemic model......Page 35
1.6 The basic SIR endemic model......Page 39
1.7 Similar models with M and E epidemiological states......Page 43
1.7.1 The SEIR epidemic model......Page 44
1.7.2 The MSEIRS endemic model......Page 46
1.8 Threshold estimates using the two SIR models......Page 51
1.9 Comparisons of some directly transmitted diseases......Page 52
1.10.1 Smallpox......Page 54
1.10.2 Poliomyelitis......Page 55
1.10.3 Measles......Page 56
1.10.4 Rubella......Page 57
1.10.5 Chickenpox (varicella)......Page 58
1.10.6 Influenza......Page 59
1.11 Other epidemiology models with thresholds......Page 60
References......Page 63
Contents......Page 74
2.1. Introduction......Page 75
2.2. Epidemiological compartment structures......Page 77
2.3. Horizontal incidences......Page 78
2.4. Waiting times in the E, I and R compartments......Page 79
2.5. Demographic structures......Page 80
2.6. Epidemiological-demographic interactions......Page 81
2.7. SIRS model with recruitment-death and standard incidence......Page 82
2.8. SIRS model with logistic demographics and standard incidence......Page 84
2.9. SIRS model with exponential demographics and standard incidence......Page 88
2.10. Discussion of the 3 previous SIRS models......Page 89
2.11. Periodicity in SEI endemic models......Page 91
2.11.1. The SEI model for fox rabies with mass action incidence......Page 92
2.11.2. An SEI model with standard incidence......Page 93
2.11.2.1. The equilibria in T......Page 94
2.11.2.2. Asymptotic behavior......Page 95
2.11.3. Discussion of the cause of periodicity in SEI models......Page 96
References......Page 97
Contents......Page 102
3.1. Introduction......Page 103
3.2. Three threshold quantities: R0, σ, and R......Page 104
3.3.1. The demographic model with continuous age......Page 105
3.3.2. The demographic model with age groups......Page 107
3.4. The MSEIR model with continuous age structure......Page 109
3.4.1. Formulation of the MSEIR model......Page 110
3.4.2. The basic reproduction number R0 and stability......Page 113
3.4.3. Expressions for the average age of infection A......Page 116
3.4.4. Expressions for R0 and A with negative exponential survival......Page 117
3.4.5. The MSEIR model with vaccination at age Av......Page 119
3.4.6. Expressions for R0 and A for a step survival function......Page 121
3.5.1. Formulation of the SEIR model with age groups......Page 123
3.5.2. The basic reproduction number R0 and stability......Page 124
3.5.3. Expressions for the average age of infection A......Page 127
3.6. Application to measles in Niger......Page 128
3.7. Application to pertussis in the United States......Page 130
3.8. Discussion......Page 134
References......Page 136
1. Introduction......Page 140
1.1. Modeling......Page 142
2. Measles......Page 144
2.1. Outbreak in S ao Paulo, 1997......Page 145
3. Congenital Rubella Syndrome......Page 148
3.1. Assessing the burden......Page 149
3.1.1. Mitigating the burden — Costa Rica......Page 151
3.1.2. Mitigating the burden — Romania......Page 152
4. Pertussis......Page 157
5. Varicella and Herpes Zoster......Page 159
6. Smallpox......Page 167
7. Emerging Infectious Diseases......Page 171
8. Conclusions and Outlook......Page 174
References......Page 175
Non-identifiables and Invariant Quantities in Infectious Disease Models Ping Yan......Page 178
1. Introduction......Page 179
1.1. Observed data versus stochastic mechanisms that manifest data......Page 180
1.2. Stochastic mechanisms that lead to transmission of an infectious disease......Page 182
2. Some statistical models and methods for identifying the non-identifiables......Page 183
2.1. Retrospectively ascertained data......Page 184
2.2. Partially identifiable information in retrospectively ascertained data......Page 187
2.2.1. The identi.able part of F(x)......Page 188
2.2.2. Discrete time model and non-parametric approach......Page 189
2.2.3. Continuous time model and parametric approach......Page 191
2.3. Data without retrospective ascertainment......Page 196
2.3.1. The back-calculation philosophy......Page 198
2.3.2. Back-calculation methods in a nutshell......Page 201
2.3.3. Some modeling paradigms for i1(t; θ)......Page 202
3. Stochastic aspects of disease transmission mechanisms......Page 206
3.1. Formulating the stochastic mechanisms in infectious disease transmission......Page 207
3.1.1. The distributions for the latent and infectious periods......Page 209
3.1.2. The infectious contact process {K(x), x ∈ [0,∞)}......Page 211
4. The role of stationary increment infectious contact processes in modeling disease transmission......Page 213
4.1. With respect to the intrinsic growth rate ρ......Page 214
4.2. With respect to the basic reproduction number R0 and uncertainties......Page 215
4.3. Relations between R0 and ρ when {K(x)} has
stationary increment......Page 216
4.3.1. A remark on serial intervals and a formula used in [12]......Page 218
4.3.2. Generalized formula for R0 as a function of ρ with gamma
distributed latent and infectious periods......Page 220
4.4. Stationary increment infectious contact process {K(x)} and the final size of a large outbreak......Page 222
5. Robustness, invariance and identifiability......Page 223
5.1.1. Robustness......Page 225
5.2. Identifiability......Page 226
5.3.1. Based on .nal size equations......Page 227
5.3.2. The case with limited supply......Page 231
5.3.3. On the reduction of intrinsic growth rate ρ......Page 233
6. Some conclusion remarks......Page 237
Acknowledgements......Page 238
References......Page 239
