title: Epidemics on networks with heterogeneous population and stochastic infection rates.
abstract: 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.
authors.affiliation: Department of Mathematics, University of Trento, Via Sommarive 14, Povo 38123, Trento, Italy. Electronic address: stefano.bonaccorsi@unitn.it.
authors.forename: Stefano
authors.initials: S
authors.lastname: Bonaccorsi
authors.affiliation: Department of Mathematics, University of Trento, Via Sommarive 14, Povo 38123, Trento, Italy; CREATE-NET, Via alla Cascata 56/d, Povo 38123, Trento, Italy. Electronic address: sottaviano@create-net.org.
authors.forename: Stefania
authors.initials: S
authors.lastname: Ottaviano
mesh_terms: D003141:Communicable Diseases / Q000635:transmission*; D058872:Epidemics / Q000706:statistics & numerical data*; D006801:Humans; D008962:Models, Theoretical; D013269:Stochastic Processes
keywords: Heterogeneous population; SIS stochastic models; Stability of the null solution; Stochastic persistence