## Sto 0 Sto 1 I To I To

where prime denotes differentiation with respect to z.

Prove that, for all finite z, 0 < S < 1 by showing that S'(z) > 0 is monotonic for all —to < z < to. Show also that (S + I)' > 0 and hence that for all —to <

z < to, S(z) + I (z) < 1. Prove that r to r to r to

I (z' ) dz' > I (z')S(z') dz'= X I (z' ) dz' -<x J—x> J —

and hence deduce that the threshold criterion for a travelling epidemic wave solution to exist is X < 1.

2. A rabies model which includes a logistic growth for the susceptibles S and diffusive dispersal for the infectives is

— = —rIS + bS (l — , ^ = rIS — aI + D, d t V So J d t d x2

where r, b, a, D and So are positive constant parameters. Nondimensionalise the system to give ut = uxx + uv — Xu, vt = —u v + bv(1 — v), where u relates to I and v to S. Look for travelling wave solutions with u > 0 and v > 0 and hence show, by linearising far ahead of a wavefront where v ^ 1 and u ^ 0, that a wave may exist if X < 1 and if so the minimum wavespeed is 2(1 — X)1/2. What is the steady state far behind the wave?

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