Unique stationary distribution of a stochastic SEIR model with general incidence and logarithmic Ornstein–Uhlenbeck process

This study examines a stochastic SEIR model that incorporates a general incidence rate alongside a logarithmic Ornstein–Uhlenbeck process. At the outset, we present the foundational concepts pertinent to this research. Then, we prove the existence and uniqueness of a global positive solution through the construction of an appropriate Lyapunov function. Subsequently, by applying the theory of Markov semigroup and the Fokker–Planck equation, we establish the existence of a unique invariant probability measure. This finding subsequently implies the existence of a unique stationary distribution for the stochastic model. Following this, we explicitly determine the probability density function in the vicinity of the unique endemic equilibrium of the stochastic model and prove that the global probability density function can be approximated by the local probability density function. Moreover, we derive a sufficient condition for disease extinction. Finally, numerical simulations are conducted to corroborate the theoretical findings, and a concise summary of the research is presented.