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 d , with jump rate 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 d , are independent. In addition, there are B-particles which perform continuous time simple random walks with jump rate 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 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 � B(t ): ={x ∈ Z d :a B-particle visits x during [0 ,t ]}.

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