Stochastic optimal control of pre-exposure prophylaxis for HIV infection.

The aim of the paper is to apply the stochastic optimal control problem in order to optimize the number of individual which will have the pre-exposure prophylaxis (PReP) treatment in the stochastic model for HIV/AIDS with PReP. By using the stochastic maximum principle, we derive the stochastic optimal control of PReP for the unconstrained control problem. Furthermore, by combining the stochastic maximum principle with a version of the Lagrange multiplier method, we solve the PReP problem for two different types of budget constrains with a given constrain for the costs (possible of different kind, transportation, price of the treatment, etc.). Obtained results for the different percentage of the individuals who got the vaccine, as well as results for unconstrained and constrained problems, are illustrated by a numerical example.