Control Strategy Based on Wave Propagation into a Nonepidemic Region Estimate of Width of a Rabies Barrier

We discuss here one possible control strategy as developed by Murray et al. (1986), namely, that of a possible protective barrier against the rabies epizootic which can be achieved by reducing the susceptible fox population below the critical density KT in areas ahead of the advancing wave. This, for example, has been successful in Denmark, specifically Jutland. It has also been carried out in some regions of Italy and Switzerland, where it has been pursued with diligence, but it has had mixed results (Macdonald 1980, Westergaard 1982). Such a barrier can be created either by killing or vaccination. Since killing releases territories, there could be a more rapid colonization by young foxes which could in fact enhance the spread of the disease. Vaccination causes less disruption in the ecology, is almost certainly more effective and is also probably more economic.

For a rabies 'break' to be effective we must have reasonable estimates of both the width and the allowable susceptible fox density within it. Here we derive estimates analytically for how wide the protective break region needs to be to keep rabies from reaching the areas beyond. We also present some of the results from numerical simulations of the full equation system (13.33). In what follows, we use the term 'infected fox' to refer to all foxes with rabies, whether infectious or not.

If we observe the passage of the rabies epizootic wave at a fixed place we note that each outbreak of the disease is followed by a long quiescent period, during which very few cases of rabies occur; refer to Figures 13.9 and 13.10. The spatial and temporal dimensions are such that the secondary epidemic wave is sufficiently far behind so that the first wave will either have moved past the break, or have effectively died out by the time the second one arrives. Each successive outbreak is weaker than the previous one. So, it seems reasonable to assume that the same population reduction schemes which eradicate the first outbreak will also be effective in stopping all subsequent outbreaks from passing through. We thus only need to consider how wide the break needs to be to stop the first outbreak. The width of the break is dependent on the size of the susceptible fox population density within it.

Since we model spatial dispersal by a deterministic diffusion mechanism it is, from a strict mathematical viewpoint, not possible for the density of infected foxes to vanish anywhere. This arises from treating the fox densities as continuous in space and time, rather than dealing with individual foxes, and from using classical diffusion to model the rabid fox dispersal. Thus we cannot simply have the epizootic wave move into a break of finite width and determine whether or not the density of infected foxes remains zero on the other side; it will always be positive, although exponentially small. Thus no matter how wide the break is, eventually enough infected foxes will in time leak through for the epizootic to start off again on the other side. Thus we must think instead of determining when the probability is acceptably small that an infected fox will reach the far side of the break.

Since the aim of any control scheme is to keep the density of foxes small, we treat the break region as one with a carrying capacity below KT, the critical threshold value (13.30) for an epidemic, and we assume that the fox density has been reduced to this value well before the epizootic front arrives. To obtain estimates for the width of the break we investigate the behaviour of the model when the region of lowered susceptible fox density starts at x = 0 and extends to infinity. We first give here the results of the numerical simulations of the full system (13.33) and later in the section obtain approximate analytic results.

Figures 13.12 and 13.13 show what happens when the epizootic wave, coming in from the left, impinges on the break region. Remember that the epizootic wave cannot propagate when the carrying capacity is below the critical value KT. Also, the point of maximum infected fox density will be at x = 0. As the infection wave moves into the region x > 0 it spreads out, decays in amplitude and the total number of infected foxes decreases. Eventually there will be less than p infected foxes/km2 remaining, where p is some small number. Let tc (p) be the time at which this occurs. We now choose p sufficiently small that the probability of a rabid fox encountering a susceptible one after this critical time is negligible. Since the wave cannot propagate in the break region it simply decays, so, for all time the density of infected foxes is greatest at the edge of the break and decays with x, exponentially as x2 in fact, as we show later. We choose the width of the break to be the point xc where the infected fox density is a given (small) fraction m of the value at the origin, that is,5

I(Xc, tc) + R(Xc, tc) = m[I(0, tc) + R(0, tc)]. (13.51)

Available evidence suggests that it has never been possible to eliminate all foxes from a region—a 70% reduction in population is about the best that can be achieved (Macdonald 1980). Figure 13.14 shows the dependence of the break width in terms of the percentage population reduction in the break, for different choices of the average duration time of clinical disease, 1/a.

In the numerical simulations for the curves in Figure 13.14, the value of pK outside the break was held at 160 yr—\ the number of infected foxes at the critical time was taken to be p = 0.5 foxes/km2, the ratio m in (13.51) was arbitrarily chosen to be 10—4 and all other parameters except a are from Table 13.1. With these assumptions, for any given choice of a, the nondimensional forms in (13.32) give d = (a +0.5 yr—1)/(160 yr—*) and (13.30) gives a carrying capacity outside the break region of K = 149/(a + 0.5 yr—x) foxes km-2 yr— l. For example, if we assume that the rabid period lasts an average of 3.8 days, then d = 0.6 and K = 1.5 foxes/km2 outside of the break. If a reduction scheme can reduce the carrying capacity to 0.4 foxes/km2 inside the break region well before the epidemic arrives, then sb = 0.26 and Figure 13.14 gives xb = 15. Assuming a diffusion coefficient of 200 km2/ yr, (13.32) gives the predicted

5Strictly the xc and tc are dimensionless here.

Figure 13.12. The behaviour of the travelling epizootic front when it encounters a break in the susceptible fox population. These plots show (a) the susceptible and (b) the rabid fox population densities for a sequence of times as the wave approaches the break region, stops and dissipates. They were obtained by solving equations (13.37)-(13.40) numerically with a carrying capacity of 2 foxes/km2 in the region outside the vertical lines and of 0.4 foxes/km2 in the region between them. Other parameter values were taken from Table 13.1. Note that the susceptible population just outside the break remains slightly higher than elsewhere, since few rabid foxes wander into this region from the right. The density of incubating foxes is proportional to the rabid population as we noted in Section 13.5: with the parameter values used, the incubating fox density is 5.6 times the rabid fox density. The times and distances are normalised values within the computer model. (From Murray et al. 1986)

Figure 13.12. The behaviour of the travelling epizootic front when it encounters a break in the susceptible fox population. These plots show (a) the susceptible and (b) the rabid fox population densities for a sequence of times as the wave approaches the break region, stops and dissipates. They were obtained by solving equations (13.37)-(13.40) numerically with a carrying capacity of 2 foxes/km2 in the region outside the vertical lines and of 0.4 foxes/km2 in the region between them. Other parameter values were taken from Table 13.1. Note that the susceptible population just outside the break remains slightly higher than elsewhere, since few rabid foxes wander into this region from the right. The density of incubating foxes is proportional to the rabid population as we noted in Section 13.5: with the parameter values used, the incubating fox density is 5.6 times the rabid fox density. The times and distances are normalised values within the computer model. (From Murray et al. 1986)

Figure 13.13. This plot shows the total infected fox density per km (the integral over x of the infected foxes I + R) as a function of time for the case shown in Figure 13.12 starting when the epidemic front first reaches the break. (From Murray et al. 1986)

Relative carrying capacity in break,

Figure 13.14. The dependence of the break width on the initial susceptible population inside the break, as predicted by the model. The break width, in nondimensional terms, is plotted against the ratio of the carrying capacity in the break to the carrying capacity outside the break for various values of the duration time of clinical disease, 1/a(d ^ a/fiK). The curves were obtained by solving (13.33) numerically until the total infected fox population in the first outbreak is 0.5 fox/km. As described in the text, we use these curves to calculate the break width, which can be put into dimensional form using relations (13.32). The dimensional break width Xc is given by (D/fiK)1/2xc, where xc is the nondimensional break width with m = 10-4. fiK was set at 160 yr-1, and all other parameters, except a, were taken from Table 13.1. For example, if we assume 1/a = 5 days then d = 0.46 and the carrying capacity outside the break is 2 foxes/km2. If the carrying capacity inside the break is assumed to be 0.4 foxes/km2 then s^ = 0.2 and this figure predicts xc = 18. Assuming D = 200 km2/yr, the predicted break width Xc is then 20 km. (From Murray et al. 1986)

break width as 17 km. Of course, the choice of p and m depends on how cautious we want to be: Murray et al. (1986) discuss the sensitivity of the model to variations in these. The maximum value of I + R at tc for all of the calculations was less than 0.15 foxes/km2. Even with m as large as m = 10—2 there are fewer than 0.0015 infected foxes per square kilometre on the protected side of the break.

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