Stationary distribution and extinction of the DS-I-A model disease with periodic parameter function and Markovian switching

This paper introduces the DS-I-A model with periodic parameter function and Markovian switching. First, we will prove that the solution of the system is positive and global. Furthermore, we draw a conclusion that there exists nontrivial positive periodic solution for the stochastic system and we establish sufficient conditions for extinction of system. Moreover, we construct stochastic Lyapunov functions with regime switching to obtain the existence of ergodic stationary distribution of the solution to DS-I-A model perturbed by white and telephone noises and we also establish sufficient conditions for extinction of system with regime switching. Finally, we test our theory conclusion by simulations.