Contact Processes with Random Recovery Rates and Edge Weights on Complete Graphs

In this paper we are concerned with the contact process with random recovery rates and edge weights on the complete graph with n vertices. We show that the model has a critical value which is inversely proportional to the product of the mean of the edge weight and the mean of the inverse of the recovery rate. In the subcritical case, the process dies out before an amount of time with order $$O(\log n)$$ with high probability as $$n\rightarrow +\infty $$ . In the supercritical case, the process survives for an amount of time with order $$\exp \{O(n)\}$$ with high probability as $$n\rightarrow +\infty $$ .