Stability and optimal control of a delayed HIV/AIDS-PrEP model

In this paper, we propose a time-delayed HIV/AIDS-PrEP model which takes into account the delay on pre-exposure prophylaxis (PrEP) distribution and adherence by uninfected persons that are in high risk of HIV infection, and analyze the impact of this delay on the number of individuals with HIV infection. We prove the existence and stability of two equilibrium points, for any positive time delay. After, an optimal control problem with state and control delays is proposed and analyzed, where the aim is to find the optimal strategy for PrEP implementation that minimizes the number of individuals with HIV infection, with minimal costs. Different scenarios are studied, for which the solutions derived from the Minimum Principle for Multiple Delayed Optimal Control Problems change depending on the values of the time delays and the weights constants associated with the number of HIV infected individuals and PrEP. We observe that changes on the weights constants can lead to a passage from bang-singular-bang to bang-bang extremal controls.