Understanding cascading failures or epidemics in networks is crucial for developing effective defensive mechanisms for many critical systems and infrastructures (e.g. biological, social and cyber networks). Most of the existing works treat the network topology as being exogenously given and study under what conditions an epidemic breaks out and/or extinguishes. However, if agents are able to strategically decide their connections according to their own self-interest, the network will instead be endogenously formed and evolving. In such systems, the epidemic, agents' strategic decisions and the network structure become complexly coupled and co-evolve. As a result, existing knowledge may no longer be applicable. Built on a continuous time Susceptible-Infected-Susceptible epidemic model with strong mixing, this paper studies stochastic epidemic networks consisting of strategic agents, who decide the number of links to form based on a careful evaluation of its current obtainable benefit and the potential future cost due to infection by forming links. A game theoretical framework is developed to analyze such networks and a number of important insights are obtained. One key result is that whereas an epidemic eventually dies out if the effective spreading rate is sufficiently low in exogenously given networks, it never dies out when agents are strategic regardless of the effective spreading rate. This property leads to reduced achievable system efficiency and considerably different optimal protection mechanisms. Without understanding the strategic behavior of agents, significant security cost may incur.