Mathematical Tools for Understanding Infectious Disease Dynamics: (Princeton Series in Theoretical and Computational Biology)

Mathematical modeling is critical to our understanding of how infectious diseases spread at the individual and population levels. This book gives readers the necessary skills to correctly formulate and analyze mathematical models in infectious disease epidemiology, and is the first treatment of the subject to integrate deterministic and stochastic models and methods.

Mathematical Tools for Understanding Infectious Disease Dynamics fully explains how to translate biological assumptions into mathematics to construct useful and consistent models, and how to use the biological interpretation and mathematical reasoning to analyze these models. It shows how to relate models to data through statistical inference, and how to gain important insights into infectious disease dynamics by translating mathematical results back to biology. This comprehensive and accessible book also features numerous detailed exercises throughout; full elaborations to all exercises are provided.

• Covers the latest research in mathematical modeling of infectious disease epidemiology
• Integrates deterministic and stochastic approaches
• Teaches skills in model construction, analysis, inference, and interpretation
• Features numerous exercises and their detailed elaborations
• Motivated by real-world applications throughout

Mathematical Tools for Understanding Infectious Disease Dynamics fully explains how to translate biological assumptions into mathematics to construct useful and consistent models, and how to use the biological interpretation and mathematical reasoning to analyze these models. It shows how to relate models to data through statistical inference, and how to gain important insights into infectious disease dynamics by translating mathematical results back to biology. This comprehensive and accessible book also features numerous detailed exercises throughout; full elaborations to all exercises are provided.

• Covers the latest research in mathematical modeling of infectious disease epidemiology
• Integrates deterministic and stochastic approaches
• Teaches skills in model construction, analysis, inference, and interpretation
• Features numerous exercises and their detailed elaborations
• Motivated by real-world applications throughout

A brief outline of the book xii

I The bare bones: Basic issues in the simplest context 1

• 1 The epidemic in a closed population 3

  • 1.1 The questions (and the underlying assumptions) 3

  • 1.2 Initial growth 4

  • 1.3 The final size 14

  • 1.4 The epidemic in a closed population: summary 28

• 2 Heterogeneity: The art of averaging 33

  • 2.1 Differences in infectivity 33

  • 2.2 Differences in infectivity and susceptibility 39

  • 2.3 The pitfall of overlooking dependence 41

  • 2.4 Heterogeneity: a preliminary conclusion 43

• 3 Stochastic modeling: The impact of chance 45

  • 3.1 The prototype stochastic epidemic model 46

  • 3.2 Two special cases 48

  • 3.3 Initial phase of the stochastic epidemic 51

  • 3.4 Approximation of the main part of the epidemic 58

  • 3.5 Approximation of the final size 60

  • 3.6 The duration of the epidemic 69

  • 3.7 Stochastic modeling: summary 71

• 4 Dynamics at the demographic time scale 73

  • 4.1 Repeated outbreaks versus persistence 73

  • 4.2 Fluctuations around the endemic steady state 75

  • 4.3 Vaccination 84

  • 4.4 Regulation of host populations 87

  • 4.5 Tools for evolutionary contemplation 91

  • 4.6 Markov chains: models of infection in the ICU 101

  • 4.7 Time to extinction and critical community size 107

  • 4.8 Beyond a single outbreak: summary 124

• 5 Inference, or how to deduce conclusions from data 127

  • 5.1 Introduction 127

  • 5.2 Maximum likelihood estimation 127

  • 5.3 An example of estimation: the ICU model 130

  • 5.4 The prototype stochastic epidemic model 134

  • 5.5 ML-estimation of α and β in the ICU model 146

  • 5.6 The challenge of reality: summary 148

II Structured populations 151

• 6 The concept of state 153

  • 6.1 i-states 153

  • 6.2 p-states 157

  • 6.3 Recapitulation, problem formulation and outlook 159

• 7 The basic reproduction number 161

  • 7.1 The definition of R₀ 161

  • 7.2 NGM for compartmental systems 166

  • 7.3 General h-state 173

  • 7.4 Conditions that simplify the computation of R₀ 175

  • 7.5 Sub-models for the kernel 179

  • 7.6 Sensitivity analysis of R₀ 181

  • 7.7 Extended example: two diseases 183

  • 7.8 Pair formation models 189

  • 7.9 Invasion under periodic environmental conditions 192

  • 7.10 Targeted control 199

  • 7.11 Summary 203

• 8 Other indicators of severity 205

  • 8.1 The probability of a major outbreak 205

  • 8.2 The intrinsic growth rate 212

  • 8.3 A brief look at final size and endemic level 219

  • 8.4 Simplifications under separable mixing 221

• 9 Age structure 227

  • 9.1 Demography 227

  • 9.2 Contacts 228

  • 9.3 The next-generation operator 229

  • 9.4 Interval decomposition 232

  • 9.5 The endemic steady state 233

  • 9.6 Vaccination 234

• 10 Spatial spread 239

  • 10.1 Posing the problem 239

  • 10.2 Warming up: the linear diffusion equation 240

  • 10.3 Verbal reflections suggesting robustness 242

  • 10.4 Linear structured population models 244

  • 10.5 The nonlinear situation 246

  • 10.6 Summary: the speed of propagation 248

  • 10.7 Addendum on local finiteness 249

• 11 Macroparasites 251

  • 11.1 Introduction 251

  • 11.2 Counting parasite load 253

  • 11.3 The calculation of R₀ for life cycles 260

  • 11.4 A 'pathological' model 261

• 12 What is contact? 265

  • 12.1 Introduction 265

  • 12.2 Contact duration 265

  • 12.3 Consistency conditions 272

  • 12.4 Effects of subdivision 274

  • 12.5 Stochastic final size and multi-level mixing 278

  • 12.6 Network models (an idiosyncratic view) 286

  • 12.7 A primer on pair approximation 302

III Case studies on inference 307

• 13 Estimators of R₀ derived from mechanistic models 309

  • 13.1 Introduction 309

  • 13.2 Final size and age-structured data 311

  • 13.3 Estimating R₀ from a transmission experiment 319

  • 13.4 Estimators based on the intrinsic growth rate 320

• 14 Data-driven modeling of hospital infections 325

  • 14.1 Introduction 325

  • 14.2 The longitudinal surveillance data 326

  • 14.3 The Markov chain bookkeeping framework 327

  • 14.4 The forward process 329

  • 14.5 The backward process 333

  • 14.6 Looking both ways 334

• 15 A brief guide to computer intensive statistics 337

  • 15.1 Inference using simple epidemic models 337

  • 15.2 Inference using 'complicated' epidemic models 338

  • 15.3 Bayesian statistics 339

  • 15.4 Markov chain Monte Carlo methodology 341

  • 15.5 Large simulation studies 344

IV Elaborations 347

• 16 Elaborations for Part I 349

  • 16.1 Elaborations for Chapter 1 349

  • 16.2 Elaborations for Chapter 2 368

  • 16.3 Elaborations for Chapter 3 375

  • 16.4 Elaborations for Chapter 4 380

  • 16.5 Elaborations for Chapter 5 402

• 17 Elaborations for Part II 407

  • 17.1 Elaborations for Chapter 7 407

  • 17.2 Elaborations for Chapter 8 432

  • 17.3 Elaborations for Chapter 9 445

  • 17.4 Elaborations for Chapter 10 451

  • 17.5 Elaborations for Chapter 11 455

  • 17.6 Elaborations for Chapter 12 465

• 18 Elaborations for Part III 483

  • 18.1 Elaborations for Chapter 13 483

  • 18.2 Elaborations for Chapter 15 488

Bibliography 491

Index 497