VIRAL PROCESSES BY RANDOM WALKS ON RANDOM REGULAR GRAPHS

• 成果类型: Article
• 署名作者: Abdullah, Mohammed; Cooper, Colin; Draief, Moez
• 署名单位: University of Birmingham; University of London; King's College London; Imperial College London
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP1000
• 发表日期: 2015
• 页码: 477-522
• 关键词: cover time; infection

摘要

We study the SIR epidemic model with infections carried by k particles making independent random walks on a random regular graph. Here we assume k <= n(epsilon), where n is the number of vertices in the random graph, and epsilon is some sufficiently small constant. We give an edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erdos-Renyi random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcritical regime, O (ln k) particles are infected. In the supercritical regime, for a constant beta is an element of (0, 1) determined by the parameters of the model, beta k get infected with probability beta, and O(ln k) get infected with probability (1 - beta). Finally, there is a regime in which all k particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We exploit this to give a completion time of the process for the SI case.