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

In this paper we are concerned with the Susceptible-Infective-Removed model with random transition rates on complete graphs $C_n$ with $n$ vertices. We assign 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. When $\lambda<\lambda_c$, the proportion of vertices which have ever been infective converges to $0$ weakly as $n\rightarrow+\infty$ while when $\lambda>\lambda_c$, there exist $c(\lambda)>0$ and $b(\lambda)>0$ such that for each $n\geq 1$ with probability at least $b(\lambda)$ the proportion of vertices which have ever been infective is at least $c(\lambda)$. Furthermore, we prove that $\lambda_c$ is the inverse of the production of the mean of $\rho$ and the mean of the inverse of $\xi$.