Infection Spread In Random Geometric Graphs
In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G (n, r(n), f) of n nodes independently distributed in S = [-1/2, 1/2](2) according to a certain density f(.). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D(1)nr(n)(2). In the second part of the paper we consider the contact process xi(t) on G where infection spreads at rate lambda > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every lambda > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c(1) and c(2) such that, with probability at least 1 - c(1)/n(4), the contact process xi(1)(t) starting with all nodes infected survives up to time t(n) = exp(c(2)n/log n) for all n.