## Mathematical EpidemiologyFred Brauer, Pauline van den Driessche, J. Wu 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 |

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 |

403 | |