Linear incidence rate: Its influence on the asymptotic behavior of a stochastic epidemic model

Diseases are an important fact in the real world and concentrate the attention of a great number of researchers. Many of them are caused by nematodes, fungi, bacteria, or viruses. Nevertheless, there exists another, which are transmitted from the mothers to offspring (vertical transmission). In this paper, the dynamics of an suceptible-infectious (SI) epidemic model are analyzed considering a linear (bilinear or standard) incidence in the deterministic and stochastic regimes, assuming that the newborns are infected from their own mothers. A long-term behavior of the proportion of infected individuals depending on the system parameters and initial conditions is established. Then, we consider the case where this linear transmission rate, not previously used for this model, has a stochastic component described by a white noise which leads to a stochastic differential equation (SDE). The existence and uniqueness of the solution of the SDE is proved. The extinction of the disease is characterized, and an exponential decay to extinction is obtained under certain restrictions of the parameters. By assuming time-independent solutions of the Fokker-Plank equation, we determine a stationary measure of the probability density, and some of its properties are provided. Numerical simulations are performed to show the dynamics of the system in different regimes and to illustrate some differences between deterministic and stochastic effects.