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 Z, with jumprate DA. We assume that “just before the start” the number of A-particles at x, NA(x, 0−), has a mean μA Poisson distribution and that the NA(x, 0−), x ∈ Z , are independent. In addition, there are B-particles which perform continuous time simple random walks with jumprate DB . 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 non-random. 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. [KSb] gave some basic estimates for the growth of the set B̃(t) := {x ∈ Z : a B-particle visits x during [0, t]}. In this article we show that if DA = DB , then B(t) = B̃(t) + [− 1 2 , 1 2 ] grows linearly in time with an asymptotic shape, i.e., there exists a non-random set B0 such that (1/t)B(t) → B0, in a sense which will be made precise.