## Annals of Probability

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

### The spread of a rumor or infection in a moving population

Harry Kesten and Vladas Sidoravicius

#### 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*^{−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}≠*D*_{B}, 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/009117905000000413

**Mathematical Reviews number (MathSciNet)**

MR2184100

**Zentralblatt MATH identifier**

1111.60074

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60J15

**Keywords**

Spread of infection random walks interacting particle system large deviations for density of Poisson system of random walks

#### 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