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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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