Stability analysis and optimal intervention design for age-structured HIV/AIDS transmission model with protective consciousness

In keeping with the age heterogeneity observed in HIV/AIDS transmission, a novel age-structured model incorporating protective consciousness is introduced. Our research delves into the existence and stability of steady states in relation to basic reproduction number ${ℛ}_0$ (or ${\widetilde{ℛ}}_0$). Specifically, we demonstrate that when the basic reproduction number is less than 1, a globally stable disease-free steady state $E^0$ exists, this implies that the disease will disappear. In scenarios where disease-induced death is absent, the endemic steady state $E^{\ast }$ becomes the sole existing steady state and exhibits local asymptotic stability for the basic reproduction number is more than 1, thereby perpetuating the disease. To further enhance the applicability of our model, we incorporate control functions such as physical distancing, educational campaigns, and treatment and then formulate an optimal control problem and rigorously prove the existence and uniqueness of optimal control solution. This approach provides a robust theoretical foundation for designing effective intervention strategies. Numerical simulations are conducted to illustrate the core findings of our study, indicating the critical role of age variability in the dissemination dynamics of HIV/AIDS. Notably, our results suggest that targeting educational interventions to achieve self-protection awareness toward younger populations yields significantly greater effectiveness compared to implementing uniform strategies across all age groups. This insight underscores the importance of age-specific approaches in optimizing resource allocation and maximizing public health impact.