Threshold of a regime-switching SIRS epidemic model with a ratio-dependent incidence rate.

Abstract In this paper, we study the dynamical behavior of a regime-switching SIRS epidemic model with a ratio-dependent incidence rate. We propose a stochastic reproduction number R 0 S which can be regarded as a threshold to use in identifying the stochastic extinction and persistence: if R 0 S < 1 , the disease dies out exponentially with probability one; while if R 0 S > 1 , there exists a unique ergodic stationary distribution of the positive solutions to the system which implies the stochastic persistence of the infectious disease. Highlights • A regime-switching SIRS epidemic model with a ratio-dependent incidence rate is studied. • A stochastic reproduction number R 0 S is proposed to be regarded as a threshold. • If R 0 S < 1 , the disease dies out exponentially with probability one. • If R 0 S < 1 , there is a unique ergodic stationary distribution of the system. [ABSTRACT FROM AUTHOR]