Marco Antonio Leonel Caetano

Short and long period optimization of drug doses in the treatment of AIDS

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.

State estimation and optimal long period clinical treatment of HIV seropositive patients

Optimal control theory provides a very interesting quantitative method that can be used to assist the decision making process in several areas of application, such as engineering, biology, economics and sociology. The main idea is to determine the values of the manipulated variables, such as drug doses, so that some cost function is minimized, subject to physical constraints. In this work, the cost function reflects the number of CD4+T cells, viral particles and the drug doses. It is worth noticing that high drug doses are related to more intense side-effects, apart from the impact on the actual cost of the treatment. In a previous paper by the authors, the LQR - Linear Quadratic Regulator approach was proposed for the computation of long period maintenance doses for the drugs, which turns out to be of state feedback form. However, it is not practical to determine all the components of the state vector, due to the fact that infected and uninfected CD4+T cells are not microscopically distinguishable. In order to overcome this difficulty, this work proposes the use of Extended Kalman Filter to estimate the state, even though, because of the nonlinear nature of the involved state equations, the separation principle may not be valid. Extensive simulations were then carried out to investigate numerically if the control strategy consisting of the feedback of estimated states yielded satisfactory clinical results.

Estimating total claim size in the auto insurance industry: a comparison between tweedie and zero-adjusted inverse gaussian distribution

The objective of this article is to estimate insurance claims from an auto dataset using the Tweedie and zeroadjusted inverse Gaussian (ZAIG) methods. We identify factors that influence claim size and probability, and compare the results of these methods which both forecast outcomes accurately. Vehicle characteristics like territory, age, origin and type distinctly influence claim size and probability. This distinct impact is not always present in the Tweedie estimated model. Auto insurers should consider estimating total claim size using both the Tweedie and ZAIG methods. This allows for an estimation of confidence interval based on empirical quantiles using bootstrap simulation. Furthermore, the fitted models may be useful in developing a strategy to obtain premium pricing.

Optimal medication in HIV seropositive patient treatment using fuzzy cost function

This work concerns the optimization of medication doses using fuzzy relations to represent the compromise between the side-effects and therapeutic effects in the clinical treatment of HIV (human immunodeficiency virus) seropositive patients. The optimization process is carried out using a mathematical model which relates immunological variables such as the number of CD4+T helper cells (immune cells) and the viral load. The control variables used in the optimization process are the doses of reverse transcriptase inhibitor and protease inhibitor. The cost function uses fuzzy logic rules because clinical states are well expressed by linguistic variables by the medical staff. The results are illustrated with a model fitted to the actual data clinical data of an HIV seropositive patient.

New Iterative Method to Solve Optimal Control Problems with Terminal Constraints

Introduction T HE objective of this work is to present a new iterative method for solving optimal control problems with terminal constraints. The main idea in this work is to use a linear approximation to the state equations and a quadratic approximation to the cost functional, so that at each iteration efficient algorithms to solve linear quadratic (LQ) problems can be used to compute incremental corrections to the control signal, to make the terminal state approach the prescribed value. Many works on iterative techniques using second variations to solve optimal control problems can be found in the literature, including Refs. 1-3. The proposed scheme, however, differs from the neighboring extremal method because here the exact solution to an approximation is used in an iterative way, whereas Refs. 1-3 concern approximate solutions to the original nonlinear nonquadratic problem. The performance of the proposed method was studied in a number of examples, including a problem of atmospheric re-entry of Shuttle vehicles.

A NEW MATHEMATICAL INDEX FOR THE OPTIMAL CONTROL OF DRUG DOSES

Abstract A new quantitative index is proposed to assign a numerical value that reflects the compromise between the therapeutic and side effects during treatment of AIDS using drugs that are active against HIV. The model considers that the clinical conditions are related to the values of CD4 + T cell counts and viral load. The therapeutic effect is associated with increase in the CD4 + T cell counts, while the side effects are assumed related to the drug doses. The proposed index might be useful in computer programs for numerical evaluation and optimization of different types of drug administration schemes.