  In this paper we study the diffusion of an SIS-type epidemics on a network
under the presence of a random environment, that enters in the definition of
the infection rates of the nodes. Accordingly, we model the infection rates in
the form of independent stochastic processes. To analyze the problem, we apply
a mean field approximation, which allows to get a stochastic differential
equations for the probability of infection in each node, and classical tools
about stability, which require to find suitable Lyapunov's functions. Here, we
find conditions which guarantee, respectively, extinction and stochastic
persistence of the epidemics. We show that there exists two regions, given in
terms of the coefficients of the model, one where the system goes to extinction
almost surely, and the other where it is stochastic permanent. These two
regions are, unfortunately, not adjacent, as there is a gap between them, whose
extension depends on the specific level of noise. In this last region, we
perform numerical analysis to suggest the true behavior of the solution.
