Hybrid stochastic SIS epidemic models with vaccination: Stability of the disease-free state and applications

In this paper, we consider a stochastic SIS epidemic model with vaccination in random switching environment. The system is formulated as a hybrid stochastic differential equation. We provide a threshold number that characterizes completely its longtime behavior. It turns out that if the threshold is negative, the number of the infected class converges to zero or the extinction happens. The rate of convergence is also obtained. In contrast, if the threshold is positive, the infection is endemic. We are able to obtain an algebraic formula for the threshold, which helps us to study some strategies for controlling the disease such as: (i) determining the minimum vaccination rate needed to keep the population from the disease and (ii) determining the strategy with minimum cost of vaccination and treatment. To illustrate the results, a number of mathematical simulations and numerical examples are also presented.