Hybrid deterministic–stochastic modeling of epidemic spread: Analytical insights and numerical evidence from an Ornstein–Uhlenbeck SEIR framework

Understanding how environmental variability influences infectious disease dynamics is fundamental to realistic epidemic modeling and control. In this work, we develop and analyze a hybrid deterministic-stochastic SEIR (susceptible-exposed-infectious-recovered) model in which the transmission and treatment rates evolve as mean-reverting Ornstein-Uhlenbeck (OU) stochastic processes. This formulation captures correlated environmental fluctuations that cause key epidemiological parameters to vary above and below their mean levels, bridging the gap between classical deterministic frameworks and realistic stochastic dynamics. We first establish the basic qualitative properties of the deterministic subsystem, including positivity, boundedness, and the existence of biologically feasible equilibria. A closed-form expression for the basic reproduction number R0 is derived, and local as well as global stability of the disease-free and endemic equilibria are rigorously characterized. Bifurcation analysis reveals rich nonlinear behavior, including transcritical, backward, saddle-node, and Hopf bifurcations, indicating the potential for bistability and self-sustained epidemic oscillations. The stochastic extension preserves positivity almost surely and admits analytical conditions for extinction, persistence, and stationary distributions. Numerical simulations demonstrate that environmental noise can induce or suppress outbreaks, alter epidemic amplitude and frequency, and even generate noise-driven periodic cycles. Global sensitivity analysis using partial rank correlation coefficients identifies the transmission rate β, treatment rate a, and noise intensity σβ as dominant factors shaping epidemic outcomes. The results collectively show that stochastic fluctuations are not minor disturbances but can fundamentally reshape epidemic thresholds and long-term behavior. This hybrid OU-driven framework provides a unified and biologically consistent approach for exploring epidemic dynamics under uncertainty, with direct relevance for designing robust and adaptive disease control strategies.