Stationary distribution and probability density for a stochastic SISP respiratory disease model with Ornstein–Uhlenbeck process

In this paper, we establish and analyze a stochastic SISP respiratory disease system with Ornstein–Uhlenbeck process, which is used to describe the transmission dynamics of respiratory disease in the population. Firstly, we verify that there is a unique global solution to the proposed model for any initial value. Then we adopt a stochastic Lyapunov function method to obtain sufficient criteria for the existence of a stationary distribution of positive solutions to the proposed model, which reflects the strong persistence of the disease and the PM2.5. In particular, under some mild conditions which are used to ensure the existence and local stability of the positive equilibrium of the deterministic system, we derive the specific form of probability density near the quasi-endemic equilibrium of the stochastic system. Finally, numerical simulations are conducted to confirm our analytical results. •A stochastic SISP respiratory model with Ornstein-Uhlenbeck process is studied.•We prove the existence and uniqueness of the global solution to the stochastic model.•We establish sufficient criteria for the existence of a stationary distribution.•We obtain the specific form of probability density of the stochastic system.•Numerical simulations are conducted to confirm our analytical results.