Run for your life; a note on the asymptotic speed of propagation of an epidemic : (preprint)

We study the large-time behaviour of the solution of a nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic. For this model it is known that there exists a minimal wave speed c0 (i.e., travelling wave solutions with speed c exist if ¦c¦ > c0 and do not exist if ¦c¦ < c0). In this paper we show that c0 is the asymptotic speed of propagation (i.e., for any c1, c2with 0 < c1 < c0 < c2 the solution tends to zero uniformly in the region ¦x¦ ⩾ c2t, whereas it is bounded away from zero uniformly in the region ¦x¦ ⩽ c1t for t sufficiently large).