Estimates for the spreading velocity of an epidemic model

Abstract

We present numerical estimates for the spreading velocity of an epidemic model. The model consists of a growing set of simple random walks (SRWs) on Z d ( d=1, 2), also known as frog model. The dynamics is described as follows. It is a discrete time process in which at any time there are active particles, which perform independent SRWs on Z d , and inactive particles, which initially do not move. When an inactive particle is hit by an active particle, the former becomes active too. We consider the case where initially there is one inactive particle per site except for the active particle which is placed at the origin. Alves et al. [Ann. Appl. Probab. 12 (2) (2002) 533] have recently proved that the set of the original positions of all active particles, re-scaled by the elapsed time, converges to a compact convex set A without being able to identify rigorously the actual limit shape. Numerical estimates coming from simulations show that for d=2 the limit shape A is not an Euclidean ball.