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 Zd, 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∈ Zd, are i.i.d., mean-μAPoisson 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$\tilde{B}(t)\coloneq \{x\in {\Bbb Z}^{d}\colon {\rm a}\ B\text{-}\text{particle visits}\ x\ \text{during}\ [0,t]\}$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 DA=DB(so that the A- and B-particles perform the same random walk), then there exist constants $0