ON AN EPIDEMIC MODEL ON FINITE GRAPHS

• 成果类型: Article
• 署名作者: Benjamini, Itai; Fontes, Luiz Renato; Hermon, Jonathan; Machado, Fabio Prates
• 署名单位: Weizmann Institute of Science; Universidade de Sao Paulo; University of Cambridge
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1500
• 发表日期: 2020
• 页码: 208-258
• 关键词: random-walks; large deviations; shape theorem; cover times; frog model; transience; recurrence; SPREAD; infection; discrete

摘要

We study a system of random walks, known as the frog model, starting from a profile of independent Poisson(lambda) particles per site, with one additional active particle planted at some vertex o of a finite connected simple graph G = (V, E). Initially, only the particles occupying o are active. Active particles perform t is an element of N boolean OR {infinity} 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 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 S(G) := inf{t : R-t = V} (essentially, the shortest particles' lifespan required for the entire population to 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 >= 1) T-d(n) and determine the asymptotic behavior of S up to a constant factor. In fact, throughout we allow the particle density lambda to depend on n and for d >= 2 we determine the asymptotic behavior of S(T-d(n)) up to smaller order terms for a wide range of lambda = lambda(n).