Periodic solution and stationary distribution of stochastic S-DI-A epidemic models

Abstract

This paper considers two S-DI-A models in random environments. Firstly, using Has’minskii theory of periodic solution, we show that stochastic periodic S-DI-A model has a nontrivial positive periodic solution if . Then, we construct stochastic Lyapunov functions with regime switching to obtain the existence of ergodic stationary distribution of the solution to S-DI-A model perturbed by white and telephone noises. Finally, examples are introduced to illustrate the results developed.

Keywords:

• S-DI-A model
• stochastic periodic solutions
• stationary distribution
• Markov switching

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Natural Science Foundation of Shandong Province, China [grant number ZR2014AL008]; National Natural Science Foundation of P.R. China [grant number 11371085], [grant number 11401586]; the Fundamental Research Funds for the Central Universities of China [grant number 15CX02080A], [grant number 15CX08011A], [grant number 16CX02055A].