The spread of a rumor or infection in a moving population

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

2003-12-30

arxiv.org/abs/math/0312496v2

Annals of Probability 2005, Vol. 33, No. 6, 2402-2462

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000413 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000413

We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous-time simple random walk on $\mathbb{Z}^d$, 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 $N_A(x,0-),x\in\mathbb{Z}^d$, are i.i.d., mean-$\mu_A$ 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 $\widetilde{B}(t):=\{x\in\mathbb{Z}^d:$ a $B$-particle visits $x$ during $[0,t]\}$ and $B(t)=\widetilde{B}(t)+[-1/2,1/2]^d$, 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

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