Optimal therapy strategies for a free boundary parabolic–hyperbolic HIV infection model with antiretroviral drugs application

Abstract

In this paper, we delve into the study of two optimal control theory problems, which are formulated for a free boundary problem that simulates the development of HIV infection under the influence of Highly Active Antiretroviral Therapy (HAART). The adopted free boundary model is a multicellular HIV spheroid model, involving eight time-varying partial differential equations. The paper provides a detailed description of five first-order hyperbolic equations that elucidate the dynamics of the HIV infection, as well as three second-order parabolic equations that describe the diffusion of antiretroviral therapy drug concentration. We present two objective functionals(first order objective functional and second order objective functional) and derive the necessary conditions of the optimal controls by using the tangent-normal cone method. Furthermore, by applying the Ekeland variational principle, we demonstrate that there is a unique optimal control solution corresponding to each optimal control problem.