Dynamical behavior of a stochastic SIQR epidemic model with Ornstein–Uhlenbeck process and standard incidence rate after dimensionality reduction

Due to the continuous interference of environmental white noise, the dynamical behaviors of a stochastic SIQR epidemic model with mean-reverting Ornstein–Uhlenbeck process and standard incidence are considered. Several conclusions can be verified after dimensionality reduction. We study the existence and uniqueness of positive solution by construct a nonnegative Lyapunov function at first. Then, a sufficient condition for extinction of the diseases is derived by constructing a suitable Lyapunov function. In addition, we also obtain the stationary distribution of the model by constructing a complex Lyapunov function. Particularly, we propose the exact local expression of the density function of the random model near the unique endemic equilibrium. Finally, the numerical simulations illustrate our above theoretical results. • System of a SIQR model with Ornstein–Uhlenbeck process and standard incidence rate is studied. • In order to get more meaningful conclusions and reduce the dimension of system SIQR, it has also been shown. • Existence of stationary distribution is proved. • New method in constructing Lyapunov function on high-dimensional system is given. • Density function of the stochastic model around the unique endemic equilibrium is derived.