Dynamic analysis of deterministic and stochastic SEIR models incorporating the Ornstein-Uhlenbeck process

In this paper, we introduce a Susceptible-Exposed-Infected-Recovered (SEIR) epidemic model and analyze it in both deterministic and stochastic contexts, incorporating the Ornstein-Uhlenbeck process. The model incorporates a nonlinear incidence rate and a saturated treatment response. We establish the basic properties of solutions and conduct a comprehensive stability analysis of the system's equilibria to assess its epidemiological relevance. Our results demonstrate that the disease will be eradicated from the population when R0<1, while the disease will persist when R0>1. Furthermore, we explore various bifurcation phenomena, including transcritical, backward, saddle-node, and Hopf, and discuss their epidemiological implications. For the stochastic model, we demonstrate the existence of a unique global positive solution. We also identify sufficient conditions for the disease extinction and persistence. Additionally, by developing a suitable Lyapunov function, we establish the existence of a stationary distribution. Several numerical simulations are conducted to validate the theoretical findings of the deterministic and stochastic models. The results provide a comprehensive demonstration of the disease dynamics in constant as well as noisy environments, highlighting the implications of our study.