The spread of a rumor or infection in a moving population

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 ∈ 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 B̃(t) := {x ∈ ℤ d : a B-particle visits x during [0, t]} and B(t) = B̃(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 < ∈ such that almost surely script C sign(C 2t) ⊂ B(t) ⊂ script C sign(C 1t) for all large t, where script C sign(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 ≠ D B, then we can only prove the upper bound B(t) ⊂ script C sign(C 1t) eventually. © Institute of Mathematical Statistics, 2005.