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, with jump rate DA. 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 NA(x,0−) A-particles at x, and that the NA(x,0−), x ∈ Z , are i.i.d., mean-μA Poisson random variables. In addition, there are B-particles which perform continuous-time simple random walks with jump rate DB . 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 : a B-particle visits x during [0, t]} and B(t) = B̃(t)+[−1/2,1/2], then we investigate the asymptotic behavior of B(t). Our principal result states that if DA =DB (so that the Aand B-particles perform the same random walk), then there exist constants 0<Ci <∞ such that almost surely C(C2t)⊂B(t)⊂ C(C1t) for all large t, where C(r) = [−r, r]. In a further paper we shall use the results presented here to prove a full “shape theorem,” saying that tB(t) converges almost surely to a nonrandom set B0, with the origin as an interior point, so that the true growth rate for B(t) is linear in t. If DA 6= DB , then we can only prove the upper bound B(t) ⊂ C(C1t) eventually.