On an epidemic model on finite graphs

TL;DR

This paper analyzes the spread of an epidemic modeled by the frog model on finite graphs, determining how long it takes for the entire population to become infected based on graph structure and particle density.

Contribution

It provides asymptotic estimates for the susceptibility of the frog model on regular expanders and tori, considering variable particle densities and different dimensions.

Findings

Susceptibility scales with graph properties and particle density.

Results apply to regular expanders and tori of any dimension.

Asymptotic behavior characterized for a wide range of densities.

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' lifetime 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 $\lambda=\lambda_n$.