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In this paper we are concerned with the contact process with random recoveryrates and edge weights on complete graph with $n$ vertices. We show that themodel has a critical value which is inversely proportional to the product ofthe mean of the edge weight and the mean of the inverse of the recovery rate.In the subcritical case, the process dies out before a moment with order$O(\log n)$ with high probability as $n\rightarrow+\infty$. In thesupercritical case, the process survives at a moment with order $\exp\{O(n)\}$with high probability as $n\rightarrow+\infty$. Our proof for the subcriticalcase is inspired by the graphical method introduced in \cite{Har1978}. Ourproof for the supercritical case is inspired by approach introduced in\cite{Pet2011}, which deal with the case where the contact process is withrandom vertex weights.

Contact processes with random recovery rates and edge weights on complete graphs