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(.) 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 t...{.} 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 .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 .(G):=inf{t:.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.1) ..d(n) and determine the asymptotic behavior of . up to a constant factor. In fact, throughout we allow the particle density . to depend on n and for d.2 we determine the asymptotic behavior of .(..d(n)) up to smaller order terms for a wide range of .=.n.