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 $\mathcal{C}(C_{2}t)\subset B(t)\subset \mathcal{C}(C_{1}t)$ for all large t, where $\mathcal{C}(r)=[-r,r]^{d}$. In a further paper we shall use the results presented here to prove a full “shape theorem,” saying that t⁻¹B(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 D_A≠D_B, then we can only prove the upper bound $B(t)\subset \mathcal{C}(C_{1}t)$ eventually.