SUMMARYThis work concerns the application of the optimal control theory to Dengue epidemics. The dynamics of this insect-borne disease is modelled as a set of non-linear ordinary di!erential equations including the e!ect of educational campaigns organized to motivate the population to break the reproduction cycle of the mosquitoes by avoiding the accumulation of still water in open-air recipients. The cost functional is such that it re#ects a compromise between actual "nancial spending (in insecticides and educational campaigns) and the population health (which can be objectively measured in terms of, for instance, treatment costs and loss of productivity). The optimal control problem is solved numerically using a multiple shooting method. However, the optimal control policy is di$cult to implement by the health authorities because it is not practical to adjust the investment rate continuously in time. Therefore, a suboptimal control policy is computed assuming, as the admissible set, only those controls which are piecewise constant. The performance achieved by the optimal control and the sub-optimal control policies are compared with the cases of control using only insecticides when Breteau Index is greater or equal to 5 and the case of no-control. The results show that the sub-optimal policy yields a substantial reduction in the cost, in terms of the proposed functional, and is only slightly inferior to the optimal control policy.
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 CD4 + T cells and reduction of the density of free viruses.
This work concerns a particular application of the OptimalControl Theory to a model related to HIV infection dynamics. The mathematical model adopted in this work was proposed by Nowak et al., 1996 and describes the dynamics of viral concentration in t e m of interaction with CD4 cells and the cytotoxic T lymphocytes, which are responsible for the immune defense of the organism The control variable is the drug dose, which, in tum, affects the rate of infection of CD4 cells by HIV virus. The cost function to be minimized is a weighted sum of the jinal viral load and the accumulated side effects. Simulation results show that the optimal control scheme can achieve improved quality ofthe treatment in rems of reduction in the viral load and quantity of administered drugs, but has the inconvenience related to the necessity of frequent and periodic laboratory analysis to provide feedback information to adjust the doses.
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