The spread of a rumor or infection in a moving population

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 B(t):= {x ∈ 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 < ∞ such that almost surely C(C 2 t) C B(t) C 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 ¬= D B , then we can only prove the upper bound B(t) C C(C 1 t) eventually.