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 epsilon Z(d), are i.i.d., mean-mu(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) over tilde (t) := {x epsilon Z(d) : a B-particle visits x during [0, t]) and B(t) = (B) over tilde (t) + [- 1/2, 1/2](d), then we investigate the asymptotic behavior of B(t). Our principal result states that if DA = DB (so that the A- and B-particles perform the same random walk), then there exist constants 0 < C-i < infinity such that almost surely C(C(2)t) subset of B(t) subset of 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 not equal D-B, then we can only prove the upper bound B(t) subset of C(C(1)t) eventually.
