Deterministic and stochastic mean-field sirs models on heterogeneous networks

In this paper we study a model for the spread of an SIRS-type epidemics on a network, both in a deterministic setting and 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 that is known in the literature as NIMFA model, which allows to get a differential equation for the probability of infection in each node. We discover a sufficient condition which guarantees the extinction of the epidemics both in the deterministic and in the stochastic setting.