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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($\lambda $) particles per site, with one additional active particle planted at some vertex $\mathbf{o}$ of a finite connected simple graph $G=(V,E)$. 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 $\mathcal{S}(G):=\inf \{t:\mathcal{R}_{t}=V\}$ (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 $d\ge 1$) $\mathbb{T}_{d}(n)$ and determine the asymptotic behavior of $\mathcal{S}$ up to a constant factor. In fact, throughout we allow the particle density ${\lambda }$ to depend on $n$ and for $d\ge 2$ we determine the asymptotic behavior of $\mathcal{S}(\mathbb{T}_{d}(n))$ up to smaller order terms for a wide range of ${\lambda }={\lambda }_{n}$.

Citation

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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

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 1 • February 2020
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