The optimal control of an HIV/AIDS reaction-diffusion epidemic model

In this paper, we consider a mathematical model describing the dynamics of the HIV/AIDS epidemic. We study the local and global asymptotic stability of the model's equilibria in the presence of a diffusion term. An optimal controller is presented that considers the use of three different measures to combat the spread of HIV/AIDS, namely: the use of condoms and the screening and treatment of unaware infective individuals. The objective of the optimal controller is to minimize the size of the susceptible and infected populations. The study starts with an investigation of the existence and uniqueness of solutions. Then, we establish estimates of the controlled system's positive strong solution by means of the semigroup theory of operators. We also make use of minimal sequence techniques to show the existence of an optimal control. In doing so, we establish the necessary optimality conditions of the developed scheme.