{
  "id": "1610.04301",
  "version": "v1",
  "published": "2016-10-14T00:32:09.000Z",
  "updated": "2016-10-14T00:32:09.000Z",
  "title": "On an epidemic model on finite graphs",
  "authors": [
    "Itai Benjamini",
    "Luiz Renato Fontes",
    "Jonathan Hermon",
    "Fabio Prates Machado"
  ],
  "comment": "25 pages. arXiv admin note: text overlap with arXiv:1609.08738",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a system of random walks, known as the frog model, starting from an independent Poisson($\\lambda$) particle's profile with one additional active particle planted at some vertex $\\mathbf{o}$ of a finite connected simple graph $\\mathcal{G}=(\\mathcal{V},\\mathcal{E})$. Initially, only the particles occupying $\\mathbf{o}$ are active. Active particles perform $t \\in \\mathbb{N} \\cup \\{\\infty \\}$ steps of the walk they picked before vanishing and activate all inactive particles they hit. This system is often taken as a model for the spread of an epidemic over a population. Let $\\mathcal{R}_t$ be the set of vertices which are visited by the process, when active particles vanish after $t$ steps. We study the susceptibility of the process on the underlying graph, defined as the random quantity $\\mathcal{S}(\\mathcal{G}):=\\inf \\{t:\\mathcal{R}_t=\\mathcal{V} \\}$, the time it takes for the entire population get infected. We consider the cases that the underlying graph is either a regular expander or a $d$-dimensional torus of side length $n$ (for all $d \\ge 1$) and determine asymptotic bounds for $\\mathcal{S} $ up to a constant factor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-14T00:32:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82C41",
      "60K35",
      "82B43",
      "60J10"
    ],
    "keywords": [
      "finite graphs",
      "epidemic model",
      "finite connected simple graph",
      "determine asymptotic bounds",
      "additional active particle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}