Optimal Control of HIV-Virus Dynamics

Abstract

In this paper we consider a mathematical model of HIV-virus dynamics and propose an efficient control strategy to keep the number of HIV virons under a pre-specified level and to reduce the total amount of medications that patients receive. The model considered is a nonlinear third-order model. The third-order model describes dynamics of three most dominant variables: number of healthy white blood cells (T-cells), number of infected T-cells, and number of virus particles. There are two control variables in this model corresponding to two categories of antiviral drugs: reverse transcriptase inhibitors (RTI) and protease inhibitors (PI). The proposed strategy is based on linearization of the nonlinear model at the equilibrium point (steady state). The corresponding controller has two components: the first one that keeps the system state variables at the desired equilibrium (set-point controller) and the second-one that reduces in an optimal way deviations of the system state variables from their desired equilibrium values. The second controller is based on minimization of the square of the error between the actual and desired (equilibrium) values for the linearized system (linear-quadratic optimal controller). The obtained control strategy recommends to HIV researchers and experimentalists that the constant dosages of drugs have to be administrated at all times (set point controller, open-loop controller) and that the variable dosages of drugs have to be administrated on a daily basis (closed-loop controller, feedback controller).