Mathematical Modeling of HIV/AIDS and Optimal Control

We discuss compartmental models for HIV/AIDS transmission dynamics. The models are analyzed from the epidemic and mathematical point of view. Local and global stability of the equilibrium points is obtained. Fractional and stochastic versions of the SICA model are proposed. For the fractional model, uniform stability is proved using appropriate Lyapunov functions. As for the stochastic version, we prove existence of the global and positive solution, investigate the condition for the extinction of the disease and persistence in mean. The results are illustrated through numerical simulations and a concrete case study is provided using Cape Verde data from 1987 to 2014. Moreover, pre-exposure prophylaxis (PrEP) is introduced in the SICA model and we show that PrEP reduces the number of new HIV infections. However, only people who are HIV-negative and at very high risk for HIV infection should take PrEP. Therefore, the number of individuals under PrEP must be limited at each instant of time, for a fixed interval of time. In order to study this health public problem, we formulate an optimal control problem with a mixed state-control constraint. Optimal control theory allow us to derive PrEP strategies that minimize the number of HIV-infected individuals, the cost associated with PrEP and satisfy the limitations on the number of total individuals that should be under PrEP at each instant.