Mathematical Tools for Understanding Infectious Disease Dynamics

Preface xi 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 alpha and ss 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