The spread of a rumor or infection in a moving population

Harry Kesten; Vladas Sidoravicius

doi:10.1214/009117905000000413

paper

The spread of a rumor or infection in a moving population

arXiv:math/0312496 · doi:10.1214/009117905000000413

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 $Z^{d}$ , with jump rate $D_{A}$ . These particles are called $A$ -particles and move independently of each other. They are regarded as individuals who are ignorant of a rumor or are healthy. We assume that we start the system with $N_{A} (x, 0 -)$ $A$ -particles at $x$ , and that the $N_{A} (x, 0 -), x \in Z^{d}$ , are i.i.d., mean- $μ_{A}$ Poisson random variables. 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. $B$ -particles are interpreted as individuals who have heard a certain rumor or who are infected. 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. We investigate how fast the rumor, or infection, spreads. Specifically, if $B (t) := {x \in Z^{d} :$ a $B$ -particle visits $x$ during $[0, t]}$ and $B (t) = B (t) + [- 1/2, 1/2]^{d}$ , then we investigate the asymptotic behavior of $B (t)$ . Our principal result states that if $D_{A} = D_{B}$ (so that the $A$ - and $B$ -particles perform the same random walk), then there exist constants $0 < C_{i} < \infty$ such that almost surely $C (C_{2} t) \subset B (t) \subset C (C_{1} t)$ for all large $t$ , where $C (r) = [- r, r]^{d}$ . In a further paper we shall use the results presented here to prove a full ``shape theorem,'' saying that $t^{- 1} B (t)$ converges almost surely to a nonrandom set $B_{0}$ , with the origin as an interior point, so that the true growth rate for $B (t)$ is linear in $t$ . If $D_{A} \neq = D_{B}$ , then we can only prove the upper bound $B (t) \subset C (C_{1} t)$ eventually.

Published at http://dx.doi.org/10.1214/009117905000000413 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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