Mathematical Epidemiology

Front Cover
Fred Brauer, Pauline van den Driessche, J. Wu
Springer Science & Business Media, Apr 30, 2008 - Medical - 414 pages

Based on lecture notes of two summer schools with a mixed audience from mathematical sciences, epidemiology and public health, this volume offers a comprehensive introduction to basic ideas and techniques in modeling infectious diseases, for the comparison of strategies to plan for an anticipated epidemic or pandemic, and to deal with a disease outbreak in real time. It covers detailed case studies for diseases including pandemic influenza, West Nile virus, and childhood diseases. Models for other diseases including Severe Acute Respiratory Syndrome, fox rabies, and sexually transmitted infections are included as applications. Its chapters are coherent and complementary independent units. In order to accustom students to look at the current literature and to experience different perspectives, no attempt has been made to achieve united writing style or unified notation.

Notes on some mathematical background (calculus, matrix algebra, differential equations, and probability) have been prepared and may be downloaded at the web site of the Centre for Disease Modeling (www.cdm.yorku.ca).

 

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Contents

A Light Introduction to Modelling Recurrent Epidemics
3
12 Plague
4
13 Measles
5
14 The SIR Model
6
15 Solving the Basic SIR Equations
8
16 SIR with Vital Dynamics
11
17 Demographic Stochasticity
13
19 Slow Changes in Susceptible Recruitment
14
92 Modeling Populations with Age Structure
206
921 Solutions along Characteristic Lines
208
922 Equilibria and the Characteristic Equation
209
93 AgeStructured Integral Equations Models
211
931 The Renewal Equation
212
94 AgeStructured Epidemic Models
214
95 A Simple AgeStructured AIDS Model
215
951 The Reproduction Number
216

110 Not the Whole Story
15
111 Take Home Message
16
Compartmental Models in Epidemiology
19
211 Simple Epidemic Models
22
212 The KermackMcKendrick Model
24
213 KermackMcKendrick Models with General Contact Rates
32
214 Exposed Periods
36
215 Treatment Models
38
QuarantineIsolation Model
40
217 Stochastic Models for Disease Outbreaks
45
222 The SIS Model
52
23 Some Applications 231 Herd Immunity
55
232 Age at Infection
56
233 The Interepidemic Period
57
234 Epidemic Approach to the Endemic Equilibrium
59
235 Disease as Population Control
60
24 Age of Infection Models 241 The Basic SIR Model
66
242 Equilibria
69
243 The Characteristic Equation
70
244 The Endemic Equilibrium
72
245 An SIS Model
74
246 An Age of Infection Epidemic Model
76
References
78
An Introduction to Stochastic Epidemic Models
81
32 Review of Deterministic SIS and SIR Epidemic Models
82
33 Formulation of DTMC Epidemic Models
85
332 Numerical Example
90
334 Numerical Example
93
342 Numerical Example
97
343 SIR Epidemic Model
98
35 Formulation of SDE Epidemic Models
100
352 Numerical Example
103
354 Numerical Example
105
362 Quasistationary Probability Distribution
108
363 Final Size of an Epidemic
112
364 Expected Duration of an Epidemic
115
37 Epidemic Models with Variable Population Size
117
371 Numerical Example
119
38 Other Types of DTMC Epidemic Models
121
382 Epidemic Branching Processes
124
39 MatLab Programs
125
References
128
Advanced Modeling and Heterogeneities
131
An Introduction to Networks in Epidemic Modeling
133
42 The Probability of a Disease Outbreak
134
43 Transmissibility
138
44 The Distribution of Disease Outbreak and Epidemic Sizes
140
45 Some Examples of Contact Networks
142
46 Conclusions
145
Deterministic Compartmental Models Extensions of Basic Models
147
521 KermackMcKendrick SIR Model
148
522 SEIR Model
150
53 Immigration of Infectives
152
54 General Temporary Immunity
154
References
157
Further Notes on the Basic Reproduction Number
159
62 Compartmental Disease Transmission Models
160
63 The Basic Reproduction Number
162
64 Examples
163
642 A Variation on the Basic SEIR Model
165
643 A Simple Treatment Model
166
644 A Vaccination Model
168
645 A VectorHost Model
170
646 A Model with Two Strains
171
65 Ro and the Local Stability of the DiseaseFree Equilibrium
173
66 Ro and Global Stability of the DiseaseFree Equilibrium
175
References
177
Spatial Structure Patch Models
179
72 Spatial Heterogeneity
180
73 Geographic Spread
182
74 Effect of Quarantine on Spread of 19181919 Influenza in Central Canada
185
75 Tuberculosis in Possums
188
References
189
Spatial Structure Partial Differential Equations Models
191
82 Model Derivation
192
Spatial Spread of Rabies in Continental Europe
194
Spread Rates of West Nile Virus
199
85 Remarks
202
ContinuousTime AgeStructured Models in Population Dynamics and Epidemiology
205
952 PairFormation in AgeStructured Epidemic Models
218
953 The Semigroup Method
220
96 Modeling with Discrete Age Groups
222
961 Examples
223
References
225
Distribution Theory Stochastic Processes and Infectious Disease Modelling
229
1021 Nonnegative Random Variables and Their Distributions
231
1022 Some Important Discrete Random Variables Representing Count Numbers
234
1023 Continuous Random Variables Representing TimetoEvent Durations
237
1024 Mixture of Distributions
239
1025 Stochastic Processes
241
1026 Random Graph and Random Graph Process
248
103 Formulating the Infectious Contact Process
249
1031 The Expressions for R0 and the Distribution of N such that R0 EN
251
and Homogeneity in the Transmission of Infectious Diseases
254
104 Some Models Under Stationary Increment Infectious Contact Process Kx
255
1042 Tail Properties for N
258
105 The Invasion and Growth During the Initial Phase of an Outbreak
261
1051 Invasion and the Epidemic Threshold
262
1052 The Risk of a Large Outbreak and Quantities Associated with a Small Outbreak
263
The Intrinsic Growth
273
1054 Summary for the Initial Phase of an Outbreak
280
The Final Size of Large Outbreaks
281
1061 Generality of the Mean Final Size
282
1062 Some Cautionary Remarks
283
107 When the Infectious Contact Process may not Have Stationary Increment
285
1071 The Linear Pure Birth Processes and the Yule Process
286
1072 Parallels to the Preferential Attachment Model in Random Graph Theory
288
References
291
Part III Case Studies
294
The Role of Mathematical Models in Explaining Recurrent Outbreaks of Infectious Childhood Diseases
297
112 The SIR Model with Demographics
300
113 Historical Development of Compartmental Models
302
1132 Stochasticity
306
1134 Age Structure
307
1136 Distribution of Latent and Infectious Period
308
1138 Chaos
309
1139 Transitions Between Outbreak Patterns
310
1141 Power Spectra
311
1142 Wavelet Power Spectra
313
115 Conclusions
314
References
316
Modeling Influenza Pandemics and Seasonal Epidemics
321
122 A Basic Influenza Model
322
123 Vaccination
326
124 Antiviral Treatment
330
125 A More Detailed Model
334
126 A Model with Heterogeneous Mixing
336
127 A Numerical Example
341
128 Extensions and Other Types of Models
345
References
346
Mathematical Models of Influenza The Role of CrossImmunity Quarantine and AgeStructure
349
132 Basic Model
351
133 CrossImmunity and Quarantine
354
134 AgeStructure
359
135 Discussion and Future Work
362
References
363
A Comparative Analysis of Models for West Nile Virus
365
West Nile Virus
367
143 Minimalist Model 1431 The Question
368
1433 Model Formulation
370
1434 Model Analysis
372
1435 Model Application
373
When does the DiseaseTransmission Term Matter?
374
1443 Numerical Values of R0
377
146 Model Parameterization Validation and Comparison
380
WN Control
381
Seasonal Mosquito Population
382
149 Summary
384
References
386
Suggested Exercises and Projects
391
1 Cholera
395
4 HIVAIDS
396
6 Human Papalonoma Virus
397
9 Measles
398
11 Severe Acute Respiratory Syndrome SARS
399
13 Tuberculosis
400
Index
403
Copyright

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Page 386 - Turner, J. 2002. A clarification of transmission terms in host-microparasite models: numbers, densities and areas. Epidemiology and Infection 129 147-153.

About the author (2008)

Fred Brauer is a professor emeritus at the University of Wisconsin – Madison, where he taught from 1960 to 1999, and has been an honorary professor at the University of British Columbia since 1997. He is the author or co-author of 115 papers on differential equations, mathematical population biology, and mathematical epidemiology as well as 10 books including undergraduate texts and a book on models in population biology and epidemiology jointly with Carlos Castillo – Chavez.

Pauline van den Driessche is a professor emerita in the Department of Mathematics and Statistics and an adjunct professor in the Department of Computer Science at the University of Victoria. Her research interests include mathematical biology, especially epidemiology, and matrix analysis. Recently she was awarded the 2007 Krieger – Nelson prize by the Canadian Mathematical Society, and gave the Olga Taussky Todd lecture at the International Conference in Industrial and Applied Mathematics in July 2007.

Jianhong Wu is a professor and a Senior Canada Research Chair in Applied Mathematics at York University. He is the author or coauthor of over 200 peer-reviewed publications and six monographs in the areas of nonlinear dynamical systems, delay differential equations, mathematical biology and epidemiology, neural networks, and pattern formation and recognition. He is the recipient of the Canadian Industrial and Applied Mathematics Research Prize (2003), the Alexander van Humboldt Fellowship (1996), the Paul Erdos Visiting Professorship (2000), and Cheung Kong/YangZi River Lecture Professorship (2006).

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