Ferrita

In reality, a specic event,e.g. the capture of a particular prey by a particular predator, occurs at

a time governed by many factors, not the least of which is chance. In a model, however, we must be much more specic about quantifying this. Only the current state of the system, and chance, can possibly eect the time (or rate) at which future events occur. (The extent to which chance is dierent from just more environmental interactions (bad weather, being seen, attracting a mate...) is debatable right down to the question of whether or not we live in a deterministic universe. However, most people would agree that, even when all reasonable factors are taken into account, luck still has a part to play. The rabbit slipping on wet ground or choosing to dart left instead of right may be all that stands between life and death in the jaws of a fox.)

Sometimes, time is an important factor and must be included in specifying the state of the system. It may manifest itself as a seasonal variation in some phenomenon (e.g. annual reproduction, or

excess winter mortality), or through the duration of a lengthy process (e.g. pregnancy). But it is

often convenient to ignore time and leave it out of the equations. This implies that nothing changes between events and the state of the system must therefore remain constant. This has important consequences for the allowable event rates, which must be independent of the length of time since the last event. We are lead to a uniform rate of occurrence in the sense that in any small interval of timet, the probability of an event happening is

t+O(t

2) (1.16)

Here, is the event rate which in general will depend on the current state of the system and the

specic event in question, but not on time itself. The probabilityP

(

t) of waiting a timetuntil the event rst occurs in the nexttcan be calculated

by writing t = nt where n2 N. It corresponds to n failures of the event to happen in intervals

0t]t2t]:::(n;1)tt], and then success in the interval tt+t]: P ( t)(1;t) n t+O(t 2)

On substituting fort and using the fact (1 +x=n) n !e xas n!1, we nd P ( t) =e ;t t+O(t 2)

The waiting time for such an event is therefore exponentially distributed and has a probability density function given by

(t) = 8 < : e ;t if t0 0 ift<0 (1.17)

A random variable with this distribution is often called a Sojourn time. It has mean 1=and vari-

ance 1=

2. A more detailed discussion and derivation is given in Rozanov (1977) where the topic is

approached from the viewpoint of a continuous Markov process.

It is important to note that the probability of such an exponentially distributed event happening in an interval is independent of any past length of time during which the event has not happened. The event remembers no history. A useful fact is the following: IfT

1and T

2are two independent random

variables, exponentially distributed with rates (or parameter) 1 and 2 then T = min(T 1 T 2) is

exponentially distributed with parameter 1+

2. This is easily veried from rst principles by

noting that the probability of either one of, or both, events in an intervalt is 1 t(1; 2 t) + 2 t(1; 1 t) + 1 t 2 t+O(t 2) = ( 1+ 2) t+O(t 2)

In general, the waiting time until the rst event for any nite number of such events is similarly exponentially distributed with parameter the sum of the individual rates.

The mathematics is appealing, and for this reason, the case of exponential waiting times is by far the most useful for constructing model systems and in many cases it can be easily justied. Disease transmission through the contact of an infected and susceptible individual, for example, is often a rather opportunistic and random aair, so the assumption of a constant rate for an eective contact taking place seems realistic, or at least as good as any other. But other events are not so naturally exponentially distributed and this is a drawback of the approach. Consider the time for recovery of an infected individual (to an immune class, or perhaps back to being susceptible). For many infections, this time is observed to be much more normally distributed about a certain mean than exponentially distributed, and even when the mean is the same, the dierence can be substantial. However, the eect this assumption will have on the system's behaviour, even when clearly unjusti- able biologically, will generally depend on the system itself. Whilst assuming a human pregnancy lasts an exponentially distributed time may seem bizarre at the individual level, it may not have a signicant eect of the dynamics of a whole population compared to the more usual assumption of a nine month average and standard deviation measured in days, especially if reproduction takes place at all times of the year. (But see Keeling and Grenfell (1997) for an example from epidemiology where the exponential distribution is abandoned to important eect).

Models that do explicitly depend upon time are called autonomous. The epidemiology of measles provides an example, where it is thought that school age children are crucial to the continued transmission. Many models incorporate a seasonal eect to model the passage of school terms and holidays, between which contact rates dier. More will be said on this subject in Chapter 6.

Whilst explicitly including time helps to overcome some problems, like those where an interaction rate is seasonally modulated, the question of replacing an unnatural exponential distribution for processes like disease recovery is still not fully answered short of modelling every individual's state (or at least the distribution of their states). It is typically events which do not rely on interactions directly, but embody the result of some undescribed internal process, that are the diculty. Nevertheless, there is a partial solution available within the framework of exponentially distributed event times. Instead of modelling a group of infected individuals, for example, by just one class,I,

with exponential decay to a recovered class R, a sequence of infected classes I 1

I 2

:::I

n can be

used. Individuals then move (exponentially) between the classes in order from classI 1 to I 2, then I 2 to I 3

etc. until nally fromI n to

R. If the n transition rates between the classes are identical,

then it can be shown that the expected time for an individual to move through allninfected classes

into the recovered class is the same as for just one infected class at an n times slower rate. The

distribution, though, is otherwise dierent. Figure (1.6) shows this dierence as a graph of the proportion of infected individuals (i.e. the sum of all those in the infected classes) remaining against

time for the case of one, two and seven infection classes. As the number of classes is increased, the initially exponential decay becomes closer to the (backwards) S shaped curve associated with a normal distribution. If members of all the infected classes are treated identically from a dynamical point of view, this technique of increasing the number of classes can alter the eective decay rate assumptions. An unnatural exponential distribution of waiting times can be replaced by a more natural, more normal one with a modest increase in the complexity of the equations.

In view of this possibility, and despite other shortcomings, we should not lose sight of the advantages of not having to model every individual that make uniform rates and the corresponding exponential distributions an obvious starting point in most circumstances.