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Short and long period optimization of drug doses in the treatment of AIDS. (English) Zbl 1006.92019
Summary: Numerical optimization techniques are useful in solving problems of computing the best inputs for systems described by mathematical models and when the objectives can be stated in a quantitative form. This work concerns the problem of optimizing the drug doses in the treatment of AIDS in terms of achieving a balance between the therapeutic response and the side effects. A mathematical model describing the dynamics of HIV viruses and CD4 cells is used to compute the short term optimal drug doses in the treatments of patients with AIDS by a direct method of optimization using a cost function of Bolza type. The model parameters were fitted to actual published clinical data. In order to simplify the numerical procedures, the control law is expressed as a series and the sub-optimal control is obtained by truncating the higher terms. When the patient reaches a clinically satisfactory state, the LQR – Linear Quadratic Regulator – technique is used to determine the long period maintenance doses for the drugs. The doses computed using the LQR technique tend to be smaller than equivalent constant-dose therapy in terms of increase in the counts of ${\text{CD4}}^{+}T$ cells and reduction of the density of free viruses.

##### MSC:
 92C50 Medical applications of mathematical biology 65K10 Optimization techniques (numerical methods) 65C20 Models (numerical methods)
##### Keywords:
simulation; drugs; treatment of AIDS; HIV viruses; CD4 cells