Simulation and optimal control of stochastic delay differential models for hepatitis C virus epidemics.

The hepatitis C virus (HCV) is recognized as a significant global public health concern due to its complex transmission dynamics and long-term health consequences. In this study, a stochastic delay differential model was examined to enhance the understanding of HCV transmission. Time delays were incorporated into the mathematical model to represent incubation periods, while stochastic perturbations were introduced to reflect random environmental and demographic variations. The model was formulated to account for key compartments, including susceptible individuals, acute and chronic infections, and recovered individuals, along with disease-induced mortality and progression rates. By representing delays associated with incubation periods and asymptomatic stages, the model was used to explore HCV transmission mechanisms under stochastic influences. Additionally, strategies for disease control through vaccination and treatment were investigated. Stochastic fluctuations were included to capture uncertainties arising from environmental and demographic factors. Using stochastic Lyapunov functional techniques, the existence of a unique global solution was established, and conditions for disease extinction, persistence, and the existence of stationary distributions were derived. Optimal control strategies were developed with the goal of minimizing infection prevalence and associated intervention costs, focusing on measures such as treatment, public health education, and vaccination. The optimal control trajectories under the influence of delays and stochastic effects were determined by applying Pontryagin's Maximum Principle (PMP), thereby ensuring practical relevance. Numerical simulations were conducted to demonstrate the effects of time delays and stochastic variables on HCV dynamics and to highlight the effectiveness of the proposed control strategies. Overall, this work provides important insights into the interplay between stochastic processes, time delays, and optimal interventions, offering a comprehensive framework for the effective management and potential eradication of HCV epidemics.