The other dimensional parameter values in (13.70) are given in Table 13.1 together with the diffusion coefficient DR = 200 km2 year-1; recall the difficulty in estimating the diffusion coefficient.
We first consider the case in which immune foxes have susceptible offspring. The four-class model was solved numerically by Murray and Seward (1992) for five values of the percentage of immune foxes: p = 2%, 5%, 10%, 15%, 20%. Although we believe the larger values are not appropriate in this model for the European situation, we include them to see the effect on the model predictions. The shapes of both the initial wave and the recurrent outbreaks were found to vary only slightly between the three-class and the four-class models: they look like the shapes in Figures 13.9 and 13.10. However, they found that the effects of introducing an immune population are:
(i) the speed of the initial wave decreases;
(ii) the levels of the infected and rabid populations are not as high in the initial outbreak;
(iii) the susceptible population is not reduced as severely when the rabies outbreak occurs;
(iv) the time between recurrent outbreaks is reduced.
The first three of these are as we would expect intuitively while the fourth point follows from them. These effects become more marked as the immune percentage increases. From the asymptotic analysis and numerical results for the three-class system in Section 13.5 the speed of the initial wave is about 51 year—1 when K = 2 fox km—2 and 103 km year—1 when K = 4-6 fox km—2. The computed wavespeeds in the four-class cases are given in Table 13.3.
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