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