## Annals of Probability

### 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 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-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 (t):={x∈ℤd: a B-particle visits x during [0,t]} and B(t)=(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<Ci<∞ 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 DADB, then we can only prove the upper bound $B(t)\subset \mathcal{C}(C_{1}t)$ eventually.

#### Article information

Source
Ann. Probab., Volume 33, Number 6 (2005), 2402-2462.

Dates
First available in Project Euclid: 7 December 2005

https://projecteuclid.org/euclid.aop/1133965862

Digital Object Identifier
doi:10.1214/009117905000000413

Mathematical Reviews number (MathSciNet)
MR2184100

Zentralblatt MATH identifier
1111.60074

#### Citation

Kesten, Harry; Sidoravicius, Vladas. The spread of a rumor or infection in a moving population. Ann. Probab. 33 (2005), no. 6, 2402--2462. doi:10.1214/009117905000000413. https://projecteuclid.org/euclid.aop/1133965862

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