{
  "id": "math/0410371",
  "version": "v1",
  "published": "2004-10-17T01:39:10.000Z",
  "updated": "2004-10-17T01:39:10.000Z",
  "title": "A phase transition in a model for the spread of an infection",
  "authors": [
    "Harry Kesten",
    "Vladas Sidoravicius"
  ],
  "comment": "1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-17T01:39:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J15"
    ],
    "keywords": [
      "phase transition",
      "a-particle",
      "simple random walks",
      "poisson random variables",
      "particles perform independent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10371K"
    }
  }
}