Abstract
In this paper, we discuss a stochastic SIS epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction number $R_{0}$ . When the perturbation is large, the number of infected decays exponentially to zero and the solution converges to the disease-free equilibrium regardless of the magnitude of $R_{0}$ . Moreover, we get the same exponential stability and the convergence if $R_{0}<1$ . When the perturbation and the disease-related death rate are small, we derive that the disease will persist, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model on average in time if $R_{0}>1$ . Furthermore, we prove that the system is persistent in the mean. Finally, the results are illustrated by computer simulations.
Highlights
1 Introduction Epidemiology is the study of the spread of diseases with the objective to trace factors that are responsible for or contribute to their occurrence
If R ≤, the disease-free equilibrium is a unique equilibrium in this type of epidemic model and it is globally asymptotically stable; if R >, this type of model has a unique endemic equilibrium, which is globally asymptotically stable
On the other hand we investigate the asymptotic behavior of the solution of the system ( . ) according to R > the solution of system ( . ) does not converge to P∗
Summary
Epidemiology is the study of the spread of diseases with the objective to trace factors that are responsible for or contribute to their occurrence. In a simple epidemic model, there is generally a threshold, R. If R ≤ , the disease-free equilibrium is a unique equilibrium in this type of epidemic model and it is globally asymptotically stable; if R > , this type of model has a unique endemic equilibrium, which is globally asymptotically stable. The threshold R determines the extinction and persistence of the epidemic. Controlling infectious diseases has been an increasingly complex issue in recent years. When the immunity is temporary, the immunity can be lost after a period of time. It is used in many references [ – ] where one assumes the process of losing immunity is in the exponential form
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