"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

"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

Mathematical Tools for Understanding Infectious Disease Dynamics

PRINCETON UNIVERSITY PRESS

Contents

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 models one has an explicit expression for R₀ in terms of the parameters. Indeed, from the assumptions above, we find

R₀ = pc(T₂ - T₁) (1.1)

where c is the contact rate introduced in Section 1.1.

We conclude that whether or not the introduction of an infectious agent leads to an epidemic explosion is determined by the value of the generation multiplication factor R₀ relative to the threshold value one. At least for simple sub-models for the contact process and infectivity, one can determine R₀ explicitly in terms of parameters of these sub-models.

Exercise 1.1 Female mosquitoes strive for a _xed number of blood meals per unit of time, in order to be able to lay eggs. Show that consequently the mean number of bites that one human receives per unit of time is proportional to D_mosquito=D_human, i.e., to the ratio of the two densities D.

Exercise 1.2 Consider one infected mosquito. Assume it stays infected for an expected period of time T_m during which it bites (different) people at a rate c. Assume that each bite results in successful transmission with probability p_m. How many people is this mosquito expected to infect?

Exercise 1.3 Consider one infected human. Assume it stays infected for an expected period of time T_h during which it is bitten by (different) mosquitoes at a rate k. Let each bite result in successful transmission with probability p_h. How many mosquitoes is this human expected to infect?

Exercise 1.4 Argue that for the above crude description of malaria transmission the quantity

c²T_mT_hp_mp_h D_mosquito/D_human

is a threshold parameter with threshold value 1. Spell out the meaning of 'threshold parameter' in some detail.

1.2.2 The influence of demographic stochasticity

Within the idealized deterministic description of infection transmission, we found in the preceding subsection that a newly introduced infectious agent starts to spread when R₀ > 1, while going extinct more or less immediately when R₀ <1. However, for the deterministic description to be warranted, we need not only a large number of susceptibles but also a large number of infectives. Yet the very essence of the introduction of the agent is that it is present in only a few hosts. So we need to take account of demographic stochasticity, i.e., the chance fluctuations associated with the fact that individual hosts are discrete units, counted with integers and either infected or not (rather than fractionally). Only when the infectives form a small fraction of the large population, and not just a small number, does the deterministic description apply.

To refine the analysis, we need branching processes, as introduced below, but we keep counting on a generation basis. We postpone more precise formulations, and the development in real time, to Section 3.3. As infecting another host is very similar to producing offspring, we shall freely use the standard terminology of true reproduction, even though it does not apply literally to the context of disease spread that we consider here. We repeat that the number of susceptibles is assumed to be very large so that we can, in the initial phase of an epidemic, neglect the depletion of susceptibles by their conversion into infectives. So the finite population we are going to consider in the following is the sub-population of infected individuals.

Consider this finite population from a generation perspective and assume that individuals reproduce independently from each other, the number of offspring for each being taken from the same probability distribution {f_q}^∞_k=0. This means that any individual begets k offspring with probability q_k and that [summation]^∞_k=0 q_k = 1. We note right away that the expected number of offspring R₀ can be found from {q_k} as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Exercise 1.5 To warm up, we consider the situation in which an individual has either two descendants, one descendant, or no offspring at all. In other words: q_k = 0 for k ≥ 3. We exclude the uninteresting case q₁ = 1 (the results are not valid in that case).

i) Show that R₀ > 1 if and only if q₂ > q₀;

ii) Consider one individual. Let z be the probability that its line of descent will stop (sooner or later). Next consider two individuals. What is the probability that the line of descent of both of them will stop?

iii) Explain why z should satisfy the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

iv) Rewrite this equation such that it becomes easy to see that either z = 1 or z = q₀/q₂:

v) Show that z = 1 if R₀ ≤ 1. Does this surprise you?

vi) What do you expect z to be if R₀ > 1?

To deal with the general situation we introduce, as an auxiliary tool, the (probability) generating function g defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

(Recall the convention z⁰ = 1.)

Exercise 1.6 Check that

i) g(0) = q₀,

ii) g(1) = 1,

iii) g'(1) = R₀,

iv) g'(z) > 0,

v) g''(z) > 0:

Hint (for 1.6-iv and v): Realize that q_k ≥ 0, with strict inequality for at least one value of k, which is implicitly contained in the interpretation. Which additional assumptions are needed to make the inequalities strict?

Now assume that q₀ > 0, which means that there is a positive probability that an individual will beget no offspring at all. Let us start the process with just one individual. Then clearly q₀ is also the probability that the population will be extinct after one step along the generation ladder.

Let z_n denote the probability that the population will be extinct after n steps along the generation ladder (so z_n equals the probability that the population will go extinct in at most n steps). Then our last observation translates into z₁ = q₀. We claim that the z_n can be computed recursively from the difference equation

z_n = g(z_n-1). (1.4)

To substantiate this claim, we argue as follows. If in the first generation there are k offspring, then the lines descending from each of these should go extinct in n - 1 generation-steps in order for the population to go extinct in or before the nth generation. For each separate line the probability is, by definition, z_n-1. By independence, the probability that all k-lines go extinct in n-1 steps is then simply (z_n-1)^k. It remains to sum over all possible values of k with the appropriate weight q_k. Thus we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as claimed.

Because g is increasing, the sequence z_n must be increasing and so, because it is also bounded by 1, it has a limit z_∞ = lim_{n[right arrow]∞} z_n. By definition z_∞ is the probability that the population started by the first individual will go extinct. We refer to this as the probability of a minor outbreak. When z_∞ = 1, the population goes extinct with probability 1. When 0 <z_∞ <1, there exists a complementary probability 1 - z_∞ that, as further arguments show (see books on branching processes, such as Haccou et al. 2005, Jagers 1975, Mode 1971 and Harris 1963), exponential growth sets in and the deterministic description applies. So we expect that R₀ <1 implies z_∞ = 1, while for R₀ > 1 the inequality 0 <z_∞ <1 holds. That our expectation is correct follows most easily from a graphical consideration, see Figures 1.1 and 1.2.

Exercise 1.7 Argue that z_∞ is the smallest root in [0, 1] of the equation

z = g(z) (1.5)

(and interpret this equation as a consistency condition that the probability to go extinct should satisfy. Hint: Recall the derivation of the difference equation (1.4)).

Exercise 1.8 Use Exercise 1.6-iii to show analytically that z_∞ = 1 for R₀ <1 while 0 <z_∞ <1 for R₀ > 1. Hint: Also use Exercise 1.6-v.

Exercise 1.9 Determine z_∞ for the critical case R₀ = 1.

We conclude that even in the situation where the infectious agent has the potential of exponential growth, i.e., R₀ > 1, it still may go extinct due to an unlucky (for the agent) combination of events while numbers are low. The probability that such an extinction happens, when we start out with exactly one primary case, can be computed as a specific root of the equation z = g(z).

But how do we derive the function g (or, equivalently, the probabilities q_k) from the kind of specification of a transmission model that we employed so far? (Recall Sections 1.1 and 1.2.1.)

When during a time interval of length [increment of T] contacts are made according to a Poisson process with rate c, the probability that k contacts are made equals

(cΔT)^k/k! e^-cΔT.

In other words, the number of contacts follows the Poisson distribution with mean parameter cΔT. When contacts lead to successful transmission with probability p, the number of 'successful' contacts is again Poisson-distributed with the parameter modified to pcΔT.

Exercise 1.10 Prove the last statement.

The generating function for the Poisson distribution with parameter λ equals

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

where the first equality is the definition and the second equality follows from the Taylor expansion of the exponential function. Together with R₀ = pcΔT = pc(T₂ - T₁) we arrive at the conclusion that for this particular sub-model z_∞ is, for R₀ > 1, the unique root in the interval (0, 1) of the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

(recall Figure 1.1). In order to avoid the wrong impression that this is a general result, we add that the minor outbreak probability z_∞ satisfies

z_∞ = 1/R₀ (1.8)

for another sub-model for infectivity, viz., the one where [increment of T] above is not a fixed quantity but a random variable following the exponential distribution. The following exercises provide the details underlying this assertion (see also Exercises 2.10 and 2.11).

We conclude that the probability z_∞ that the introduction of an infected host from outside does not lead to an epidemic can, for some simple sub-models, be either determined by a graphical construction or be expressed explicitly in terms of the parameters. Different sub-models that yield the same value for R₀ may lead to different values of z_∞.

(Continues...)

Excerpted from Mathematical Tools for Understanding Infectious Disease Dynamicsby Odo Diekmann Hans Heesterbeek Tom Britton Copyright © 2013 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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