  Pre-exposure prophylaxis (PrEP) consists in the use of an antiretroviral
medication to prevent the acquisition of HIV infection by uninfected
individuals and has recently demonstrated to be highly efficacious for HIV
prevention. We propose a new epidemiological model for HIV/AIDS transmission
including PrEP. Existence, uniqueness and global stability of the disease free
and endemic equilibriums are proved. The model with no PrEP is calibrated with
the cumulative cases of infection by HIV and AIDS reported in Cape Verde from
1987 to 2014, showing that it predicts well such reality. An optimal control
problem with a mixed state control constraint is then proposed and analyzed,
where the control function represents the PrEP strategy and the mixed
constraint models the fact that, due to PrEP costs, epidemic context and
program coverage, the number of individuals under PrEP is limited at each
instant of time. The objective is to determine the PrEP strategy that satisfies
the mixed state control constraint and minimizes the number of individuals with
pre-AIDS HIV-infection as well as the costs associated with PrEP. The optimal
control problem is studied analytically. Through numerical simulations, we
demonstrate that PrEP reduces HIV transmission significantly.
