Stability and Asymptotic Behavior of a Regime-Switching SIRS Model with Beddington–DeAngelis Incidence Rate

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.

On the Stochastic Stability and Boundedness of Solutions for Stochastic Delay Differential Equation of the Second Order

We present two qualitative results concerning the solutions of the following equation: x¨(t)+g(x˙(t))+bx(t-h)+σx(t)ω˙(t)=p(t,x(t),x˙(t),x(t-h)); the first result covers the stochastic asymptotic stability of the zero solution for the above equation in case p≡0, while the second one discusses the uniform stochastic boundedness of all solutions in case p≢0. Sufficient conditions for the stability and boundedness of solutions for the considered equation are obtained by constructing a Lyapunov functional. Two examples are also discussed to illustrate the efficiency of the obtained results.

Stability of invariant sets of Itô stochastic differential equations with Markovian switching

Consider the nonlinear Itô stochastic differential equations with Markovian switching, some sufficient conditions for the invariance, stochastic stability, stochastic asymptotic stability, and instability of invariant sets of the equations are derived.