Phase transition for SIR model with random transition rates on complete graphs

We are concerned with the susceptible-infective-removed (SIR) model with random transition rates on complete graphs C (n) with n vertices. We assign independent and identically distributed (i.i.d.) copies of a positive random variable xi on each vertex as the recovery rates and i.i.d. copies of a positive random variable rho on each edge as the edge infection weights. We assume that a susceptible vertex is infected by an infective one at rate proportional to the edge weight on the edge connecting these two vertices while an infective vertex becomes removed with rate equals the recovery rate on it, then we show that the model performs the following phase transition when at t = 0 one vertex is infective and others are susceptible. There exists lambda (c) > 0 such that when lambda < lambda (c) ; the proportion ra of vertices which have ever been infective converges to 0 weakly as n -> +a while when lambda > lambda (c) ; there exist c(lambda) > 0 and b(lambda) > 0 such that for each n ae1 with probability p aeb(lambda); the proportion r (a) aec(lambda): Furthermore, we prove that lambda (c) is the inverse of the production of the mean of rho and the mean of the inverse of xi.