Threshold of a stochastic SIQRS epidemic model with nonlinear incidence rate and lognormal Ornstein–Uhlenbeck process

In this paper, we first develop a deterministic susceptible–infected–quarantined–recovered–susceptible epidemic model with nonlinear incidence rate and investigate the global stability of the equilibria of the model. Then, we extend the deterministic model to a stochastic framework by introducing the lognormal Ornstein–Uhlenbeck process to model the inherent randomness of the disease transmission. Following that, we analyze the stochastic dynamics of the model in detail. More precisely, by adopting the Markov semigroup theory and Lyapunov function techniques, we first establish sufficient criteria for the existence and uniqueness of an invariant probability measure of the model when the parameter R0S1, indicating the strong persistence of the disease. Afterward, under the same conditions as the global stability of the endemic equilibrium, we achieve the concrete form of the local probability density function near the quasi-endemic equilibrium of the stochastic model. Simultaneously, we also show that the invariant global probability density function can be approximated by the local probability density function when the noise intensity approaches zero. Subsequently, sufficient criteria for disease extinction are presented when the parameter R0S1. Crucially, we obtain the threshold that determines the outbreak or extinction of the disease. Finally, several examples together with comprehensive numerical simulations are performed to confirm our analytical findings.