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Thresholds and travelling waves in an epidemic model for rabies. (English) Zbl 0547.92016
This paper discusses a reaction diffusion system, $(1)\quad u_ t=\Delta u+uv-ru,\quad v_ t=-uv,\quad u(x,0)\geq 0,\quad v(x,0)=1,$ that models the present rabies epidemic in foxes in Europe, where u is the density of infectives and v is the density of susceptibles. Existence and uniqueness results for traveling wave solutions of (1) are given under various conditions on r. The asymptotic decay of the solution toward zero as $$t\to\infty$$, is proven using the parabolic maximum principle. This asymptotic results is an analogue of the well known D. G. Kendall pandemic threshold theorem [Mathematical models of the spread of infection, in: Mathematics and Computer Science in Biology and Medicine, 213-225 H.M.S.O., London (1965)].
Reviewer: S.Lenhart

##### MSC:
 92D25 Population dynamics (general) 49J40 Variational inequalities 35B40 Asymptotic behavior of solutions to PDEs 35K15 Initial value problems for second-order parabolic equations
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##### References:
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