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Odo Diekmann, Hans Heesterbeek & Tom Britton

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"This landmark volume describes for readers how one should view the theoretical side of mathematical epidemiology as a whole. A particularly important need is for a book that integrates deterministic and stochastic epidemiological models, and this is the first one that does this. I know of no better overview of the subject. It belongs on the shelf of everyone working in mathematical epidemiology."--Fred Brauer, University of British Columbia

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"This landmark volume describes for readers how one should view the theoretical side of mathematical epidemiology as a whole. A particularly important need is for a book that integrates deterministic and stochastic epidemiological models, and this is the first one that does this. I know of no better overview of the subject. It belongs on the shelf of everyone working in mathematical epidemiology."--Fred Brauer, University of British Columbia

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Mathematical Tools for Understanding Infectious Disease Dynamics

PRINCETON UNIVERSITY PRESS

Contents

Preface............................................................................xiI The bare bones: Basic issues in the simplest context.............................11 The epidemic in a closed population..............................................32 Heterogeneity: The art of averaging..............................................333 Stochastic modeling: The impact of chance........................................454 Dynamics at the demographic time scale...........................................735 Inference, or how to deduce conclusions from data................................127II Structured populations..........................................................1516 The concept of state.............................................................1537 The basic reproduction number....................................................1618 Other indicators of severity.....................................................2059 Age structure....................................................................22710 Spatial spread..................................................................23911 Macroparasites..................................................................25112 What is contact?................................................................265III Case studies on inference......................................................30713 Estimators of R0 derived from mechanistic models.....................30914 Data-driven modeling of hospital infections.....................................32515 A brief guide to computer intensive statistics..................................337IV Elaborations....................................................................34716 Elaborations for Part I.........................................................34917 Elaborations for Part II........................................................40718 Elaborations for Part III.......................................................483Bibliography.......................................................................491Index..............................................................................497

Chapter One

1.1 THE QUESTIONS (AND THE UNDERLYING ASSUMPTIONS)

In general, populations of hosts show demographic turnover: old individuals disappear by death and new individuals appear by birth. Such a demographic process has its characteristic time scale (for humans on the order of 10 years). The time scale at which an infectious disease sweeps through a population is often much shorter (e.g., for influenza it is on the order of weeks). In such a case we choose to ignore the demographic turnover and consider the population as 'closed' (which also means that we do not pay any attention to emigration and immigration).

Consider such a closed population and assume that it is 'virgin' or 'naive,' in the sense that it is completely free from a certain infectious agent in which we are interested. Assume that, in one way or another, the infectious agent is introduced in at least one host. We may ask the following questions:

• Does this cause an epidemic?

• If so, at what rate does the number of infected hosts increase during the rise of the epidemic?

• What proportion of the population will ultimately have experienced infection?

Here we assume that we deal with microparasites, which are characterized by the fact that a single infection triggers an autonomous process in the host. We assume in addition that this process finally results in either death or lifelong immunity, so that no individual can be infected twice (this assumption is somewhat implicitly contained in the formulation of the third question).

In order to answer these questions, we first have to formulate assumptions about transmission. For many diseases transmission can take place when two hosts 'contact' each other, where the meaning of 'contact' depends on the context (think of 'mosquito biting man' for malaria, sexual contact for gonorrhea, traveling in the same bus for influenza, SARS, ...) and may, in fact, sometimes be a little bit vague (for fungal plant diseases transmitted through air transport of spores it is even far-fetched to think in terms of 'contact'). It is then helpful to follow a three-step procedure:

• Model the contact process.

• Model the mixing of susceptible and infective (i.e., infectious) individuals (which we shall refer to as 'susceptibles' and 'infectives,' respectively); that is, specify what fraction of the contacts of an infective are with a susceptible, given the population composition in terms of susceptibles and infectives.

• Specify the probability that a contact between an infective and a susceptible actually leads to transmission.

As an easy phenomenological approach to the first step we assume for the time being that individuals have a certain expected number c of contacts per unit of time with other individuals. So we postpone more mechanistic reasoning, and in particular a discussion of how c may relate to population size and/or density.

1.2 INITIAL GROWTH

1.2.1 Initial growth on a generation basis

During the initial phase of a potential epidemic, there are only a few infected individuals amidst a sea of susceptibles. So if we focus on an infected individual we may simply assume that all its contacts are with susceptibles. This settles the second step in the procedure sketched in Section 1.1.

For many diseases the probability that a contact between a susceptible and an infective actually leads to transmission depends on the time elapsed since the infective was itself infected. To be specific, let us assume that this probability equals

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where τ denotes the infection age (i.e., the time since infection took place), 0 < p ≤ 1, and where we have assumed that there is a latency period (i.e., the period of time between becoming infected and becoming infectious) of length T₁ followed by an infectious period of length T₂ - T₁. (What happens at the end of the infectious period is unspecified at this point; it may be that the host dies or it may be that its immune system managed to 'defeat' the agent, with a then-immune host surviving the infection; we shall come back to this point later on.)

In order to distinguish between avalanche-like growth and almost-immediate extinction, we introduce the basic reproduction number (or basic reproduction ratio):

R₀ := expected number of secondary cases per primary case in a 'virgin' population.

In other words, R₀ is the initial growth rate (more accurately: multiplication factor; note that R₀ is dimensionless) when we consider the population on a generation basis (with 'infecting another host' likened to 'begetting a child'). Consequently, R₀ has threshold value 1, in the sense that an epidemic will result from the introduction of the infectious agent when R₀ > 1, while the number of infecteds is expected to decline (on a generation basis) right after the introduction when R₀ < 1. The advantage of measuring growth on a generation basis is that for many...