# Mathematical Tools for Understanding Infectious Disease Dynamics:

Odo Diekmann
Hans Heesterbeek
Tom Britton
Pages: 568
https://www.jstor.org/stable/j.cttq9530

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xiv)
4. I The bare bones:: Basic issues in the simplest context
• Chapter One The epidemic in a closed population
(pp. 3-32)

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...

• Chapter Two Heterogeneity: The art of averaging
(pp. 33-44)

We start with an example. Assume that the latency period and the infectious period of all individuals are the same, but that their infectivities during the infectious period may differ. With reference to the situation and notation introduced in Section 1.2.1, we say that${T_1}$and${T_2}$are fixed whilepmay differ from one individual to another. To describe the whole population, rather than each individual separately, we can specify thedistributionofp-values (so we imagine that all individuals carry a label specifying thep-value they will have, should they happen to become infected).

During the very first period...

• Chapter Three Stochastic modeling: The impact of chance
(pp. 45-72)

In this chapter we introduce epidemic models that incorporate randomness at the population level, so-called demographic stochasticity. Even in the context of rather simple stochastic epidemic models it is quite complicated to characterize the distribution of the final size (in particular there is no closed-form solution). We therefore postpone this topic to Section 12.5.3 where we will treat a special situation concerning models for populations structured into households. Now we first focus on the derivation of approximations for large (but finite) population size. These approximations concern the initial phase of an epidemic, the main phase when the epidemic is full-blown...

• Chapter Four Dynamics at the demographic time scale
(pp. 73-126)

If population turnover is slow relative to the transmission of infection, we reach almost the final size of the epidemic in a closed population before the gradual inflow of new susceptibles has any effect. When an agent has struck a virgin (or naive) population, the susceptible fraction of the population is then of the order of${e^{ - {R_0}}}$(for${R_0}$large, see Exercise 1.20), and it will therefore take a long time before susceptibles will constitute a substantial fraction of the population again. During this period there are so few infectives that demographic stochasticity will lead to extinction of...

• Chapter Five Inference, or how to deduce conclusions from data
(pp. 127-150)

A general scientific aim is to draw conclusions from data. Models may be useful along the way, as a tool. Or it may be that the conclusion concerns the choice of the most appropriate model from a family of candidates that differ, qualitatively or quantitatively or both, in the way a phenomenon is related to underlying mechanisms (so that, in the end, the conclusion amounts to pointing out the mechanism that is likely to underlie the phenomenon that is captured by the data).

Rarely, if ever, do we have absolute confidence in our conclusions. In the spirit of Popper we...

5. II Structured populations
• Chapter Six The concept of state
(pp. 153-160)

The basic idea of dynamic structured population models is to distinguish individuals from one another according to characteristics that determine the birth, death and resource consumption rates — more generally, the interaction with the environment — and to describe the rates with which an individual’s characteristics themselves change. Since we are mainly interested in infectious diseases, we limit ourselves to those characteristics that influence the force of infection of a given infectious agent, i.e., those traits that influence the rate with which susceptible individuals become infected (encompassing both infectivity, susceptibility and contact pattern).

The first step in building a structured population model...

• Chapter Seven The basic reproduction number
(pp. 161-204)

The basic reproduction number (or ratio)${R_0}$is arguably the most important quantity in infectious disease epidemiology. It is among the quantities most urgently estimated for infectious diseases in outbreak situations, and its value provides insight when designing control interventions for established infections. From a theoretical point of view${R_0}$plays a vital role in the analysis of, and consequent insight from, infectious disease models. There is hardly a paper on dynamic epidemiological models in the literature where${R_0}$does not play a role.${R_0}$is defined as the average number of new cases of an infection caused by one...

• Chapter Eight Other indicators of severity
(pp. 205-226)

This chapter is devoted to the initial real-time growth rater, the probability of a major outbreak, the final size, and the endemic level, in structured populations, with special attention for computational simplifications in the case of separable mixing.

In the previous chapter we studied the basic reproduction number${R_0}$for epidemic models in populations manifesting various forms of heterogeneity. According to the definition,${R_0}$is the per-generation growth factor of the number infected during the early stages of an outbreak in a previously unaffected community. It was illustrated that${R_0}$depends on the transmission parameters, contact rates, the infectious...

• Chapter Nine Age structure
(pp. 227-238)

Especially in the context of infectious diseases among humans, ‘age’ is often used to characterize individuals. Partly this reflects our system of public health administration (and, perhaps, our preoccupation with age). Indeed, we can exploit that data on the distribution of the random variable ‘age at (first) infection’ contain information about the prevailing force of infection in an endemic situation.

There is, however, also a more ‘mechanistic’ reason to incorporate age structure: patterns of human social behavior and sexual activity correlate with age. In addition, the effect that the infective agent has on the host sometimes depends heavily on the...

(pp. 239-250)

As an example of a situation where spatially structured models are relevant, think of a fungal pathogen affecting an agricultural crop. A farmer having ascertained that his field is affected wants to know: How fast is the infection spreading? What fraction of the yield do I stand to lose if I do not spray with fungicides? The tradeoff here could be that spraying is expensive and bad for the environment. Suppose that harvest is three months away. Do I take the loss of plants or do I invest in fungicide and accept the concomitant pollution?

So, the key question is:...

• Chapter Eleven Macroparasites
(pp. 251-264)

As we have seen in Chapter 6, the defining mathematical distinction between microparasites and macroparasites is that for macroparasites, as a rule, re-infection through the environment is essential to get an increase in individual infectious load and consequent infectious output. In this chapter, we give a brief introduction to the consequences that this distinction has for formulating epidemic models for macroparasites. For the largest part, we concentrate on the definition and calculation of${R_0}$.

Typically, macroparasites are multicellular organisms (e.g., helminths and other worm-like parasites) where definite stages in a life cycle can be distinguished. Several of these stages live...

• Chapter Twelve What is contact?
(pp. 265-306)

In this section we reflect on the various aspects of contacts and the contact process. The interpretation of the word contact varies for different infectious agents, and only refers tohappeningswhere infection transmission could occur — to resurrect a term invented by Ross (1911), who referred to his work on epidemic phenomena as his ‘theory of happenings.’

We first note that a contact can refer to two separate types of happenings, i.e., both the actual event of a transmission opportunity and the pairing of two individuals during which several such opportunities can arise. In general, pairings last longer than transmission...

6. III Case studies on inference
• Chapter Thirteen Estimators of ${R_0}$ derived from mechanistic models
(pp. 309-324)

Up to now we derived, occasionally and non-systematically, formulas expressing reproduction numbers in terms of observable/measurable quantities, on the basis of various assumptions concerning demography, contact and transmission. In the present chapter we want to provide a modest selection of tools for the estimation of the basic reproduction number from data. We shall do so by providing (slightly systematically) a, by no means exhaustive, review of the literature. The estimation of reproduction numbers is important, as these play a major role, for example, in gauging outbreak potential, and in public health decisions on prevention and control effort.

There are, at...

• Chapter Fourteen Data-driven modeling of hospital infections
(pp. 325-336)

As already explained in Section 4.6, the treatment of hospital patients suffering from bacterial infections is increasingly hampered by antibiotic resistance. When the means of curing infections diminish, the prevention of infection gains importance. Thus, for example, Scandinavian countries and the Netherlands have implemented already in the 1980s a ‘search and destroy’ policy in order to prevent the rise of MRSA prevalence.¹

Before we proceed, we first have to clarify the terminology. We shall say that a patient is ‘colonized,’ when this person carries the antibiotic resistant bacteria at a detectable level, meaning that when a swap/sample is taken and...

• Chapter Fifteen A brief guide to computer intensive statistics
(pp. 337-346)

In Chapters 5, 13 and 14, we have presented methods for making inference about infectious diseases from available data. This is of course one of the main motivations for modeling: learning about important features, such as${R_0}$, the initial growth rate, potential outbreak sizes and what effect different control measures might have in the context of specific infections. As we have seen, quite a lot can be learned from such modeling and inference. The models considered in these chapters have all been simple enough to obtain more or less explicit estimates of just a few relevant parameters. In more complicated...

7. IV Elaborations
• Chapter Sixteen Elaborations for Part I
(pp. 349-406)

Exercise 1.1 Letcdenote the number of blood meals a mosquito takes per unit of time. Suppose a human receiveskbites per unit of time. Consistency requires that$k{D_{{\text{human}}}} = c{D_{{\text{mosquito}}}}.$Our assumption is thatcis a given constant. Hence necessarily

$k = c\frac{{{D_{{\text{mosquito}}}}}} {{{D_{{\text{human}}}}}}.$

Exercise 1.2$c{T_m}{p_m}.$

Exercise 1.3$k{T_h}{p_h}.$

Exercise 1.4 Consider one infected mosquito. It is expected to infect$c{T_m}{p_m}$humans, each of which is expected to infect$k{T_h}{p_h}$mosquitoes. So, going full circle, we have a multiplication factor

$c{T_m}{p_m}k{T_h}{p_h} = {c^2}{T_m}{T_h}{p_m}{p_h}\frac{{{D_{{\text{mosquito}}}}}}{{{D_{{\text{human}}}}}}.$

When this multiplication factor is below one, an initial infection will die out in a small number of ‘generations.’...

• Chapter Seventeen Elaborations for Part II
(pp. 407-482)

Exercise 7.2 i) We have$Kx = {c_1}K{\psi ^{(1)}} + {c_2}K{\psi ^{(2)}} = {c_1}{\lambda _1}{\psi ^{(1)}} + \;{c_2}{\lambda _2}{\psi ^{(2)}}$and, by induction,${K^m}x\; = \;{c_1}\lambda _{\;1}^m{\psi ^{(1)}} + \;{c_2}\lambda _{\;2}^m{\psi ^{(2)}}.$

ii) We can rewrite${K^m}x$as

${K^m}x\; = \;{c_1}\lambda _{\;1}^m\;\left[ {{\psi ^{(1)}} + \frac{{{c_2}}} {{{c_1}}}\;{{\left( {\frac{{{\lambda _2}}} {{{\lambda _1}}}} \right)}^n}{\psi ^{(2)}}} \right]$

(provided${c_1} \ne 0$) and since$\,{\lambda _2}\,/\,{\lambda _1}\,|\; < \;1,$the second term within the square brackets approaches zero for$m\; \to \;\infty .$The influence of the initial condition (the zeroth generation) is restricted to the value of${c_1}$. When${\lambda _1} > \;1$we observe exponential growth, when$0\; < \;{\lambda _1} < \;1$exponential decline.

Exercise 7.5 i) Thekth element of the vector${K^m}{e_j},$where${e_j}$denotes thejth unit vector (which has a one at positionjand zeros everywhere else), is precisely${({K^m})_{kj}}.$So ifKis irreducible, there exists anmfor...

• Chapter Eighteen Elaborations for Part III
(pp. 483-490)

Exercise 13.1 By definition,${R_0}$is a generation property: it gives, after a sufficiently large number of generations, the per-generation growth factor of the infected population, when circumstances remain fixed as they were at the moment of introduction. One immediately sees three issues: generation data are not observed, it depends on the situation studied what ‘sufficiently large’ is (even when such a ‘quantity’ could be well-defined), and circumstances as a rule start to change immediately after an outbreak has been discovered, for example because of control actions. With regard to the generation data: what are observed are as a rule...

8. Bibliography
(pp. 491-496)
9. Index
(pp. 497-502)