Modeling and Control of HIV/AIDS Epidemics: A Stochastic and Optimal Control Perspective

Human Immunodeficiency Virus (HIV) impairs the immune system by targeting CD4+ lymphocytes, and its progression leads to AIDS, presenting a persistent global health challenge. Although various HIV/AIDS models exist, most do not fully capture the inherent randomness in disease transmission and the dynamic impact of interventions. This study develops a comprehensive and realistic framework for modeling and controlling HIV/AIDS spread by combining stochastic differential equations (SDEs) with optimal control theory. The proposed stochastic model incorporates environmental and demographic uncertainties through multiplicative white noise processes, thereby reflecting the unpredictability of real-world epidemics. We analyze the model’s mathematical properties and establish the existence, uniqueness, and global positivity of solutions. Furthermore, we derive sufficient conditions for the existence of a unique ergodic stationary distribution using Lyapunov functional techniques. Optimal intervention strategies—such as preventive measures (e.g., condom use, PrEP, PEP), awareness programs, and accessible treatment—are identified through Pontryagin’s Maximum Principle (PMP), minimizing both the infection burden and total intervention costs. In addition, a detailed sensitivity analysis of the basic reproduction number $$R_0$$ is conducted to identify key parameters driving the epidemic threshold. Results reveal that transmission rates, treatment efficacy, and awareness-related parameters have the most significant influence on $$R_0$$ , highlighting them as primary targets for control. The sensitivity indices are visualized via bar charts and 3D contour plots, offering insight into parameter interactions and intervention priorities. Numerical simulations are carried out using the Milstein method. The results emphasize the crucial role of stochastic effects on HIV dynamics and demonstrate the superior cost-effectiveness of optimal controls under uncertainty compared to deterministic settings. A comparison with the deterministic model highlights the importance of randomness and the effectiveness of control strategies under uncertainty. This work provides a robust, data-driven decision-making framework that helps policymakers design more effective and adaptive HIV/AIDS intervention programs.