A novel time-delayed stochastic epidemic modeling approach incorporating Crowley–Martin incidence and nonlinear holling type II treatment rate

Mathematical modeling of infectious disease is essential for understanding the impact of various epidemiological factors and stochastic influences on disease spread. In this study, we investigate a stochastic compartmental epidemic model with time delays, featuring a Crowley–Martin (C-M) incidence rate alongside a holling type II (HT-II) treatment rate. Initially, we demonstrate the existence and uniqueness of a positive global solution to the model. Subsequently, we establish sufficient conditions that lead to the extinction of the disease. A suitability constructed Lyapunov function is used to confirm the presence of a stationary distribution (SD). In epidemiology, the presence of a stationary distribution indicates that the disease will persist over the long term. Additionally, the Fokker–Planck equation is solved to obtain the exact analytical form of the probability density function (PDF) that describes the behavior of the stochastic model near its unique endemic quasi-equilibrium. In statistical analysis, the explicit density function can capture and represent all the dynamical features of an epidemic model. Finally, a comprehensive simulation is provided to support and illustrate our theoretical results, offering practical insights into the model’s behavior. This work contributes to the development of more accurate predictive models that can assist public health policymakers in designing effective disease control strategies and intervention plans to mitigate the spread of infectious diseases.