Abstract: Considering that environmental noise has a significant impact on COVID-19, In this paper, a stochastic SIRC model of nonlinear perturbation with cross immunity term is proposed and its stochastic properties are studied. First, the stochastic stochastically ultimately boundedness and stochastic persistence of the system solution are proved, and then sufficient conditions for the unique ergodic stationary distribution of the system and the extinction of the disease are obtained by establishing an appropriate Lyapunov function. Finally, the above conclusions were verified by numerical simulation, and the time of disease extinction under different intensities of noise was analyzed. The results showed that the greater the intensity of noise interference, the more conducive to disease prevention and control.

Key words:  Nonlinear random variables , Stochastically ultimately boundedness , Ergodic stationary distribution , Disease persistence and extinction