## Analytic Approximation for the Width of the Rabies Control Break

We can determine analytically an approximate functional dependence of the break width on the parameters. The behaviour of the various fox population densities in the break region after the epizootic wave has reached it should be similar to the situation in which a concentrated localised density of infected and rabid foxes at time t = 0 (with the same total number of I and R as for the epizootic wave) is introduced at x = 0 in a domain where the carrying capacity is everywhere equal to the initial fox density in the break. We can then obtain an estimate of the break width by looking at the following idealised problem. Suppose that the carrying capacity is zero for all x, which implies that the susceptible fox density s = 0. At time t = 0, take r = r05(x) and q = q05(x), where 5(x) is a Dirac delta function (that is, we consider all of the r0 rabid foxes are initially concentrated at x = 0).

We start by assuming that for x > 0, all of the susceptible foxes have been eliminated, for example, by immunization or killing. In our analysis here we make the added approximation that the nonlinear terms in the equations for the incubating and rabid foxes can be neglected. Since e and 5 are small parameters, this should be a reasonable approximation. A further justification for these approximations comes from the numerical computations of the break width, where it was found that the computed break width did not change if these terms were neglected. With these assumptions, equations (13.33) reduce to the linear form d q (x, t ) dt

dr(x, t) , „ , , „ ^ d2r(x, t) —-— = xq (x, t ) - dr(x, t ) +--^—.

By symmetry, instead of considering the problem of a 5-function source of infected foxes at x = 0 and t = 0 which then move into the region x > 0, the initial conditions can be replaced by q(x, 0) = 2q05(x), r(x, 0) = 2r05(x) (13.53)

and we then consider instead the region -œ < x < œ. The propagation of infected foxes into the break is described by equations (13.52) with initial conditions (13.53). The specific quantities of interest are the time tc at which the population in the break has decayed to a given level, p, defined implicitly by the formula