A PHASE TRANSITION IN A MODEL FOR THE SPREAD OF AN INFECTION

摘要

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 Z~d 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 λ. 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 we start with one additional B-particle at the origin. We show that there is a critical recuperation rate λ_c > 0 such that the B-particles survive (globally) with positive probability if λ < λ_c and the out with probability 1 if λ > λ_c.