  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 $\xi$.
