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 o of a finite connected simple graph G = (V, E). Initially, only the particles occupying o are active. Active particles perform t is an element of N boolean OR {infinity} 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 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 S(G) := inf{t : 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 >= 1) T-d(n) and determine the asymptotic behavior of S up to a constant factor. In fact, throughout we allow the particle density lambda to depend on n and for d >= 2 we determine the asymptotic behavior of S(T-d(n)) up to smaller order terms for a wide range of lambda = lambda(n).