A shape theorem for the spread of an infection

Abstract

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 $\mu_A$ 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 $\widetilde{B}(t):= \{x \in \mathbb{Z}^d: \hbox{ a } B\hbox{-particle visits } x\hbox{ during }[0,t]\}$. In this article we show that if $D_A=D_B$, then $B(t) := \widetilde{B}(t) + [-\frac 12, \frac 12]^d$ grows linearly in time with an asymptotic shape, i.e., there exists a nonrandom set $B_0$ such that $(1/t)B(t) \to B_0$, in a sense which will be made precise.