Stochastic Modeling and Optimal Control of HIV-1 Infection Dynamics Under Combination Antiretroviral Therapy

HIV-1 remains a formidable global health challenge, as complete viral eradication is still unattainable despite considerable advances in combination antiretroviral therapy (cART). To address this, we develop a stochastic differential equation (SDE) model that incorporates environmental noise into a classical HIV-1 infection dynamics framework, establishing two key advances. Mathematically, we derive the stochastic basic reproduction number $R^s$ and establish the corresponding threshold dynamics: when $R^s<1$ (under mild conditions), the infection is almost surely cleared, whereas for $R^s>1$ , the virus persists stochastically, following an ergodic stationary distribution. Epidemiologically, we demonstrate that environmental noise profoundly influences HIV-1 dynamics and reaffirm the central role of cART. Using optimal control theory, we evaluate three intervention strategies: Strategy 1 (cART enhancement), Strategy 2 (immune modulation), and Strategy 3 (a combined cART-immune approach). Both statistical indicators and dynamical outcomes confirm that Strategy 3 provides a clear advantage in promoting rapid viral suppression by integrating cART enhancement with immune modulation. Moreover, we observe that this combined intervention remains highly effective even under stringent cost constraints, and further reductions in intervention cost could improve its cost-efficiency. These results provide not only a novel theoretical framework for understanding HIV-1 infection dynamics, but also actionable clinical insights for optimizing treatment protocols, underscoring the critical importance of cost considerations in HIV-1 management.