A shape theorem for the spread of an infection

Abstract

We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous time simple random walk on the d-dimensional lattice. These particles are called A-particles and move independently of each other. We assume that we start the system with a Poisson number of particles at each lattice site x, with the number of particles at different x's i.i.d. In addition, there are a finite number of B-particles which perform the same continuous time simple random walks as the A-particles. A- and B-particles are interpreted as individuals who are healthy or infected, respectively. The B-particles move independently of each other. The only interaction is that when a B-particle and an A-particle coincide, the latter instantaneously turns into a B-particle. Let B(t) be the set of sites visited by a B-particle during [0,t]. We show that B(t) grows linearly in time and has an asymptotic shape; more precisely, there exists a non-random convex, compact set B_0 such that almost surely, for all 0 < a <1, (1-a)tB_0 is contained in B(t) and B(t) is contained in (1+a)tB_0 eventually.