On an epidemic model on finite graphs

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 $\lambda=\lambda_n$.