Open Access
February 2020 On an epidemic model on finite graphs
Itai Benjamini, Luiz Renato Fontes, Jonathan Hermon, Fábio Prates Machado
Ann. Appl. Probab. 30(1): 208-258 (February 2020). DOI: 10.1214/19-AAP1500

Abstract

We study a system of random walks, known as the frog model, starting from a profile of independent Poisson(λ) 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 tN{} 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 Rt 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:Rt=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 d1) Td(n) and determine the asymptotic behavior of S up to a constant factor. In fact, throughout we allow the particle density λ to depend on n and for d2 we determine the asymptotic behavior of S(Td(n)) up to smaller order terms for a wide range of λ=λn.

Citation

Download Citation

Itai Benjamini. Luiz Renato Fontes. Jonathan Hermon. Fábio Prates Machado. "On an epidemic model on finite graphs." Ann. Appl. Probab. 30 (1) 208 - 258, February 2020. https://doi.org/10.1214/19-AAP1500

Information

Received: 1 January 2018; Revised: 1 February 2019; Published: February 2020
First available in Project Euclid: 25 February 2020

zbMATH: 07200527
MathSciNet: MR4068310
Digital Object Identifier: 10.1214/19-AAP1500

Subjects:
Primary: 60J10 , 60K35 , 82B43 , 82C41

Keywords: Cover time , epidemic spread , frog model , infection spread , multiple random walks , rumor spread , Simple random walk , susceptibility

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.30 • No. 1 • February 2020
Back to Top