Optimal control of an HIV model

Abstract

In this article we propose an optimal control problem, a drug regimen  that inhibits the rate at which the uninfected cells become infected,  inhibits  the influx of the virus from the external  lymphoid compartment  and at the same time minimizing the drug cost. The model considered here has a different interpretation of the effects of treatment, since the production of virus from the external lymphoid compartment is not immediately blocked by treatment and thus influences the viral decay rate.  The model utilizes a system of ordinary differential equations which describes the interaction of the immune system with the human immunodeficiency virus (HIV). The optimal control problem is transferred into a modified problem in measure space, one in which existence of the solution is guaranteed by compactness of space. By an approximation, we obtain a finite dimensional  linear programming problem  which gives an approximate solution to the original problem. Our numerical solutions obtained with the help of  MATHCAD shows treatment protocols which could maximize the survival time of patients in the short and long term process. Simulations are given with the treatment parameter corresponding to suppression of virus influx from the lymphoid compartment.