A Shape Theorem for the Spread of an Infection

In [KSb] we studied the following model for the spread of a rumor or infection: There is a "gas" of so-called A-particles, each of which performs a continuous time simple random walk on $\mathbb{Z}^d $ , with jump rate $D_A $ . We assume that "just before the start" the number of A-particles at $x,N_A (x,0 - )$ , has a mean µA Poisson distribution and that the $N_A (x,0 - ),x \in \mathbb{Z}^d $ , are independent. In addition, there are . B-particles which perform continuous time simple random walks with jump rate $D_B $ . We start with a finite number of B-particles in the system at time 0. The positions of these initial B-particles are arbitrary, but they are nonrandom. The B-particles move independently of each other. The only interaction occurs when a. B-particle and an A-particle coincide; the latter instantaneously turns into a B-particle. [KSb] gave some basic estimates for the growth of the set $\tilde B(t): = \{ x \in \mathbb{Z}^d $ : a B-particle visits X during [0, t]}. In this article we show that if $D_A = D_B $ , then $B(t): = \tilde B(t) + \left[ { - \frac{1} {{2'}}\frac{1} {2}} \right]^d $ grows linearly in time with an asymptotic shape, i.e., there exists a nonrandom set B₀ such that (1/t) B(t) — B₀, in a sense which will be made precise.