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A phase transition in a model for the spread of an infection

Harry Kesten·Vladas Sidoravicius
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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 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 lamda. 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 Z^d, are i.i.d., mean mu_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 lambda_c > 0 such that the B-particles survive (globally) with positive probability if lambda < lamda_c and die out with probability 1 if lambda > \lamda_c.

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Harry Kesten, Vladas Sidoravicius. 2004-10-17. A phase transition in a model for the spread of an infection. https://arxiv.org/abs/math/0410371

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