## The Spatial Spread of Rabies Among Foxes I Background and Simple Model

Rabies, as mentioned in the last section, is widespread throughout the world and epidemics are quite common. During the past few hundred years, Europe has been repeatedly subjected to rabies epidemics. It is not known why rabies died out in Europe some 50 or so years before the current epidemic started. The analysis of the models here, however, provides one possible scenario.

The present European epizootic (an epidemic in animals) seems to have started about 1939 in Poland and it has moved steadily westward at a rate of 30-60 km per year. It has been slowed down, only temporarily, by such barriers as rivers, high mountains and autobahns. The red fox is the main carrier and victim of rabies in the current European epidemic. The spread of rabies is like a travelling wave as shown in Figure 13.3.

Rabies, a viral infection of the central nervous system, is transmitted by direct contact, and the dog is the principal transmitter of the disease to man. As mentioned, the incidence of rabies in man, at least in Europe and America, is now rare, with only very few deaths a year, but with considerably more in underdeveloped countries. The effect of rabies on other mammals, domestic and wild, however, is serious. In France, in 1980 alone, 314 cases of rabies in domestic animals were reported and 1280 cases in wild animals. Rabies justifiably gives cause for concern and warrants extensive study and development of control strategies, a subject we discuss later in Section 13.6.

Figure 13.3 shows the advance of the rabies epidemic in France obtained from data from the French Centre National d'Etudes sur la Rage every two years between 1969 and 1977 on the northeastern part of the country. Macdonald (1980) discusses the situation at this time in France in more detail and describes the effects of a vaccination control and what happened when it was stopped. Since this time, however, there has been a concerted effort to control the spread by vaccination through bait and it has been quite successful in several countries in Europe.

A rabies epidemic is also moving rapidly up the east coast of America: the main vector here is the racoon. In this epidemic, the progress was considerably enhanced by the importation into Virginia (by hunting clubs) of infected racoons from Georgia and Florida.

If we refer to Figure 13.1 again, we see that, just as in the spatially uniform epidemic system situation discussed in Chapter 10, Volume I, after the epidemic has passed a proportion of the susceptibles have survived. It would be useful to be able to estimate this survival fraction analytically in a spatial context. This we can do in the following very simple but still illuminating model for the spatial spread of rabies.

Red foxes account for about 70% of the recorded cases in Western Europe. Although Britain has effectively been free from rabies since about 1900, the disease could be reintroduced in the near future through the illegal importation of pets or even by infected bats from the continent. The problem would be particularly serious in Britain because of the high rural and urban density of foxes, dogs and cats. In Bristol, for example, the fox density is of the order of 12 foxes/km2 as compared with a rural population of 2-4 foxes/km2. The book on the fox and rabies by Macdonald (1980) provides many of the facts and data for Britain. General data on rabies in Europe is available from the Centre National d'Etudes sur la Rage in France. The books edited by Kaplan (1977) and Bacon (1985) are specifically concerned with the population dynamics of rabies and provide biological and ecological background together with some data on the disease.

It is important to understand how the rabies epizootic wavefront progresses into uninfected regions, what control methods might halt it and how the various parameters affect them. The remaining sections of this chapter will be concerned with these specific spatial problems. The material primarily comes from the model of Murray et al. (1986) and, in this section, from the much simpler, but less realistic, model of Kallen et al. (1985). The models and control strategies we propose in Sections 13.6 and 13.9 are specifically related to the current European fox epizootic but the type of model is applicable to many other spatially propagating epidemics.

The spatial spread of epidemics is usually a very complex process, and rabies is no exception. In modelling such a complex process we can try to incorporate as many of the facts as possible, which necessarily involves many parameters, estimations of which are difficult to obtain with extant data. An alternative approach is to start with as simple a model as possible but which captures the key elements and for which it is possible to determine estimates for the fewer parameters. There is a trade-off between comprehensiveness and thus complexity, and the difficulty of estimating many parameters and a simpler approach in which parameter values can be reasonably assessed. For the models in this chapter, we have opted for the latter strategy. In spite of their simplicity, they nevertheless pose highly relevant practical questions and give estimates for various characteristics of importance in the spatial spread of diseases. Although in this section we describe and analyse a particularly simple model, it is one for which we can obtain useful analytical results.

Although many animals are involved, a basic, and reasonable, assumption is that the ecology of foxes, the principal vectors, determines the dynamics of the spread of rabies. We further assume that the spatial spread of the epizootic is due primarily to the random erratic migration of rabid foxes. Uninfected foxes do not seem to wander far from their territory (Macdonald 1980). We divide the fox population into two groups—susceptible and rabid. Although the resulting model captures certain aspects of the spatial spread of the epizootic front, it leaves out a basic feature of rabies, namely, the long incubation period of between 12 and 150 days from the time of an infected bite to the onset of the clinical infectious stage. We include this in the more realistic model presented in Section 13.5.

To control, and ideally prevent, the spread of the disease, it is important to have some understanding of how rabies spreads so as to assess the effects of possible control strategies. It is with this in mind that we first study a particularly simple modified version of the epidemic model system (13.1), which captures some of the key elements in the spread of rabies in the fox population. We shall then use it to derive some estimates of essential facts about the epizootic wave.

We consider the foxes to be divided into two groups, infectives 1, and susceptibles S; the infectives consist of rabid foxes and those in the incubation stage. The principal assumptions are: (i) The rabies virus, contained in the saliva of the rabid fox, is transmitted from the infected fox to the susceptible fox. Foxes become infected at an average rate per head, rl, where r is the transmission coefficient which measures the rate of contact between the two groups. (ii) Rabies is invariably fatal and foxes die at a per capita rate a; that is, the life expectancy of an infected fox is 1/a. (iii) Foxes are territorial and divide the countryside into non-overlapping ranges. (iv) The rabies virus enters the central nervous system and induces behavioral changes in the fox. If the virus enters the spinal cord it induces paralysis whereas if it enters the limbic system it induces transient aggression during which it loses its sense of territory and the fox wanders about in a more or less random way. So, we assume that it is only the infectives which disperse with diffusion coefficient Dkm2/year. With these assumptions our model is then (13.1) except that the susceptible foxes do not disperse. We exclude here the migration of cubs seeking their own territory. When they do move they try to stay as close to their original territory as possible. The model system in one dimension is then dS = -rlS, dt

d I d21

dt dx2

From the analysis in the last section we expect this system to possess travelling wave solutions, whose speed of propagation depends intimately on the parameter values. The realistic estimation of these few parameters is important but still not easy.

Using the nondimensionalisation (13.2), the system (13.12) becomes (cf. (13.3))

¥ = 1S - X1 + dX2, where now S, 1, x and t are dimensionless, and, as in the last section, X = a/rS0 is a measure of the mortality rate as compared with the contact rate. As before the contact rate is crucial and is not known with any confidence. We expect the threshold value to be again X = 1 but we now verify this (see also Exercise 2).

Travelling wavefront solutions of (13.13) are of the form

S(x, t) = S(z), 1 (x, t) = 1 (z), z = x - ct, (13.14)

where c is the wavespeed and we look for solutions satisfying the boundary conditions

S(to) = 1, S'(-<x>) = 0, 1 (to) = 1 (-to) = 0. (13.15)

Refer back to Figure 13.1 for the type of wave anticipated. Note that it is the derivative of S(z) which tends to zero as z ^ — oo since we anticipate a residual number, as yet undetermined, of susceptible foxes to survive the epidemic. With (13.14) the system (13.13) becomes cS' = IS,

Linearising about I = 0 and S = 1 exactly as we did in the last section and requiring I to be always nonnegative, we find that this requires X < 1, in which case the wavespeed c > 2(1 — X)1/2, X< 1. (13.17)

With this specific model we are able to take the analysis further and find the actual fraction of susceptibles which survives the epidemic. From the first of (13.16), I = cS'/S, which on substituting into the second equation gives

Integration gives

Using the boundary conditions as z ^ œ from (13.15), where S = 1, I = 0 and with I' = 0, we determine the constant to be c. If we now let z ^ -œ, again using (13.15) with I = I' = 0, we get the following transcendental equation for the surviving susceptible population, a say, after the passage of the epizootic wavefront, a - X ln a = 1, X< 1, a = S(-œ), (13.18)

which is independent of c. Writing this in the form a - 1

ln a

From (13.19), with X = 0.4, a = 0.1 for example, whereas with X = 0.7, a = 0.5. X is a measure of the severity of the epidemic. The smaller X the fewer susceptibles survive; in other words, the worse the epidemic. Figure 13.4 illustrates the surviving susceptible fraction a as a function of X obtained from (13.18); the curve was obtained by plotting X as a function of a.

The critical bifurcation value for X is X = 1, which in dimensional terms, from (13.2), means a/(rS0) = 1. If X > 1 no epidemic wave can propagate. This is to be expected since if a > rS0 it means the mortality rate is greater than the rate of recruitment of new infectives. As before this bifurcation result says that given r and a,

there is a critical minimum fox density Sc = a/r below which rabies cannot persist in the population and any infectives introduced will not cause an epidemic.

When rabies does persist, that is, X < 1, the computed speed of propagation of the epidemic wave is the minimum of the allowable speeds, namely, c = 2(1 — X)1/2, which in dimensional terms from (13.17) and (13.2) is c = 2[ D(rSo — a)]1/2. (13.20)

Figure 13.5 shows an example of the computed travelling front solutions for S and I, from (13.13), for X = 0.5. From Figure 13.4 with X = 0.5, the surviving fraction of susceptibles a « 0.2.

Let us now compare the qualitative form of the susceptible fox population in the epidemic in Figure 13.5 with that obtained from data from continental Europe as illus-

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