Dynamical Behavior of a Stochastic SIRS Epidemic Model

¹ Ecole doctorale Pierre Louis de santé publique, Université Pierre et Marie Curie, France
² Faculty of Mathematics, Informatics and Mechanics, Vietnam National University 334 Nguyen Trai road, Hanoi, Vietnam.
³ UMI 209 IRD UMMISCO, Centre IRD France Nord 32 avenue Henri Varagnat, 93143 Bondy cedex, France.
⁴ Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

^⋆ Corresponding author. E-mail: dunh@vnu.edu.vn

Abstract

In this paper we study the Kernack - MacKendrick model under telegraph noise. The telegraph noise switches at random between two SIRS models. We give out conditions for the persistence of the disease and the stability of a disease free equilibrium. We show that the asymptotic behavior highly depends on the value of a threshold λ which is calculated from the intensities of switching between environmental states, the total size of the population as well as the parameters of both SIRS systems. According to the value of λ, the system can globally tend towards an endemic state or a disease free state. The aim of this work is also to describe completely the ω-limit set of all positive solutions to the model. Moreover, the attraction of the ω-limit set and the stationary distribution of solutions will be shown.