Mathematical tools for understanding infectious diseases

517 Pages·2013·6.376 MB·English

by  Diekmann, O., Heesterbeek, Hans, Britton, Tom

Master-NE-PUP July 26, 2012 7x10
Mathematical Tools for Understanding
Infectious Disease Dynamics
Master-NE-PUP July 26, 2012 7x10
PRINCETON SERIES
IN THEORETICAL AND COMPUTATIONAL BIOLOGY
Series Editor, Simon A. Levin
Mathematical Tools for Understanding Infectious Disease Dynamics,
by Odo Diekmann, Hans Heesterbeek, and Tom Britton
The Calculus of Selfishness,
by Karl Sigmund
The Geographic Spread of Infectious Diseases: Models and Applications,
by Lisa Sattenspiel with contributions from Alun Lloyd
Theories of Population Variation in Genes and Genomes,
by Freddy Bugge Christiansen
Analysis of Evolutionary Processes,
by Fabio Dercole and Sergio Rinaldi
Mathematics in Population Biology,
by Horst R. Thieme
Individual-based Modeling and Ecology,
by Volker Grimm and Steven F. Railsback
Master-NE-PUP July 26, 2012 7x10
Mathematical Tools for Understanding
Infectious Disease Dynamics
Odo Diekmann, Hans Heesterbeek, and
Tom Britton
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Master-NE-PUP July 26, 2012 7x10
Copyright (cid:13)c 2013 by Princeton University Press
Published by Princeton University Press
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press
6 Oxford Street, Woodstock, Oxfordshire, OX20 1TW
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Diekmann, O.
Mathematical tools for understanding infectious disease dynamics / Odo Diekmann,
Hans Heesterbeek, and Tom Britton.
p. cm. – (Princeton series in theoretical and computational biology)
Includes bibliographical references and index.
ISBN 978-0-691-15539-5 (hardback)
1. Epidemiology–Mathematicalmodels–Congresses. 2. Epidemiology–Mathematical
models. 3. Communicable diseases–Mathematical models. I. Heesterbeek, Hans, 1960-
II. Britton, Tom. III. Title.
RA652.2.M3D54 2013
614.4–dc23
2012012058
British Library Cataloging-in-Publication Data is available
This book has been composed in LATEX
The publisher would like to acknowledge the authors of this volume for providing the
camera-ready copy from which this book was printed.
Printed on acid-free paper. ∞
press.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
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I simply wish that, in a matter which so closely concerns the well-being of
the human race, no decision shall be made without all knowledge which a
little analysis and calculation can provide.
Daniel Bernoulli, 1760, on smallpox inoculation
As a matter of fact all epidemiology, concerned as it is with variation of
disease from time to time or from place to place, must be considered
mathematically (...) and the mathematical method of treatment is really
nothing but the application of careful reasoning to the problems at hand.
Sir Ronald Ross, 1911, The Prevention of Malaria
Weshallendbyestablishinganewscience. Butfirstletyouandmeunlock
the door and then anybody can go in who likes.
Sir Ronald Ross in a letter to A.G. McKendrick, 1911
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Contents
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
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viii CONTENTS
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 R 161
0
7.2 NGM for compartmental systems 166
7.3 General h-state 173
7.4 Conditions that simplify the computation of R 175
0
7.5 Sub-models for the kernel 179
7.6 Sensitivity analysis of R 181
0
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
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CONTENTS ix
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 R for life cycles 260
0
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 R derived from mechanistic models 309
0
13.1 Introduction 309
13.2 Final size and age-structured data 311
13.3 Estimating R from a transmission experiment 319
0
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