A phase transition in a model for the spread of an infection
Abstract
We show that a certain model for the spread of an infection has a phase transition in the recuperation rate. The model is as follows: There are particles or individuals of type A and type B, interpreted as healthy and infected, respectively. All particles perform independent, continuous time, simple random walks on Zd with the same jump rate D. The only interaction between the particles is that at the moment when a B-particle jumps to a site which contains an A-particle, or vice versa, the A-particle turns into a B-particle. All B-particles recuperate (that is, turn back into A-particles) independently of each other at a rate lamda. We assume that we start the system with NA(x,0-) A-particles at x, and that the NA(x,0-), x in Zd, are i.i.d., mean muA Poisson random variables. In addition we start with one additional B-particle at the origin. We show that there is a critical recuperation rate lambdac > 0 such that the B-particles survive (globally) with positive probability if lambda < lamdac and die out with probability 1 if lambda > c.
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