Abstract
A regime-switching SIRS model with Beddington–DeAngelis incidence rate is studied in this paper. First of all, the property that the model we discuss has a unique positive solution is proved and the invariant set is presented. Secondly, by constructing appropriate Lyapunov functionals, global stochastic asymptotic stability of the model under certain conditions is proved. Then, we leave for studying the asymptotic behavior of the model by presenting threshold values and some other conditions for determining disease extinction and persistence. The results show that stochastic noise can inhibit the disease and the behavior will have different phenomena owing to the role of regime-switching. Finally, some examples are given and numerical simulations are presented to confirm our conclusions.
Highlights
Infectious diseases are one of the greatest enemies of human beings
Some authors used Lyapunov functions to study the stability of the model [1,2,3]. e authors in [2] proposed a new technique to study stability of an SIR model with a nonlinear incidence rate by establishing a transformation of variable
Some scholars have studied the dynamic behavior of epidemic models and gave the threshold values of disease extinction and persistence so as to give control strategies for disease [6,7,8,9,10]. e authors in [6] have proved that the number RS0 can govern the dynamics of the model under intervention strategies by using the Markov semigroup theory
Summary
Infectious diseases are one of the greatest enemies of human beings. Whenever they happen, they will bring great disasters to human beings. erefore, it is of great significance to model and study the infectious mechanism of infectious diseases for disease control. In order to describe this perturbation, stochastic noise driven by continuous Brownian motion is widely studied in epidemic models and other systems with various incidence functions [7,8,9, 12, 13, 16, 17]. Beddington–DeAngelis function is an important incidence rate with the form f(S, I) (SI/m1 + m2S + m3I), Mathematical Problems in Engineering which has been studied by some scholars [17,18,19]. As far as we know, there are a great many number of research studies on the epidemic models with Markovian switching, there is little work on the properties of the regime-switching SIRS model the with Beddington–DeAngelis incidence rate.
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