Optimal Controls of the Highly Active Antiretroviral Therapy
In this paper, we study generic properties of the optimal, in a certain sense, highly active antiretroviral therapy (or HAART). To address this problem, we consider a control model based on the 3-dimensional Nowak–May within-host HIV dynamics model. Taking into consideration that precise forms of fu...
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| Language: | English |
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Wiley
2020-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2020/8107106 |
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| author | Ellina V. Grigorieva Evgenii N. Khailov Andrei Korobeinikov |
| author_facet | Ellina V. Grigorieva Evgenii N. Khailov Andrei Korobeinikov |
| author_sort | Ellina V. Grigorieva |
| collection | DOAJ |
| description | In this paper, we study generic properties of the optimal, in a certain sense, highly active antiretroviral therapy (or HAART). To address this problem, we consider a control model based on the 3-dimensional Nowak–May within-host HIV dynamics model. Taking into consideration that precise forms of functional responses are usually unknown, we introduce into this model a nonlinear incidence rate of a rather general form given by an unspecified function of the susceptible cells and free virus particles. We also add a term responsible to the loss of free virions due to infection of the target cells. To mirror the idea of highly active anti-HIV therapy, in this model we assume six controls that can act simultaneously. These six controls affecting different stage of virus life cycle comprise all controls possible for this model and account for all feasible actions of the existing anti-HIV drugs. With this control model, we consider an optimal control problem of minimizing the infection level at the end of a given time interval. Using an analytical mathematical technique, we prove that the optimal controls are bang-bang, find accurate estimates for the maximal possible number of switchings of these controls and establish qualitative types of the optimal controls as well as mutual relationships between them. Having the estimate for the number of switchings found, we can reduce the two-point boundary value problem for Pontryagin Maximum Principle to a considerably simpler problem of the finite-dimensional optimization, which can be solved numerically. Despite this possibility, the obtained theoretical results are illustrated by numerical calculations using BOCOP–2.0.5 software package, and the corresponding conclusions are made. |
| format | Article |
| id | doaj-art-08ada90a02714b5fbd6a80f976d79654 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
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| series | Abstract and Applied Analysis |
| spelling | doaj-art-08ada90a02714b5fbd6a80f976d796542025-08-20T03:55:32ZengWileyAbstract and Applied Analysis1085-33751687-04092020-01-01202010.1155/2020/81071068107106Optimal Controls of the Highly Active Antiretroviral TherapyEllina V. Grigorieva0Evgenii N. Khailov1Andrei Korobeinikov2Department of Mathematics and Computer Sciences, Texas Woman’s University, Denton, TX 76204, USADepartment of Computational Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow 119992, RussiaCentre de Recerca Matemática, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona, SpainIn this paper, we study generic properties of the optimal, in a certain sense, highly active antiretroviral therapy (or HAART). To address this problem, we consider a control model based on the 3-dimensional Nowak–May within-host HIV dynamics model. Taking into consideration that precise forms of functional responses are usually unknown, we introduce into this model a nonlinear incidence rate of a rather general form given by an unspecified function of the susceptible cells and free virus particles. We also add a term responsible to the loss of free virions due to infection of the target cells. To mirror the idea of highly active anti-HIV therapy, in this model we assume six controls that can act simultaneously. These six controls affecting different stage of virus life cycle comprise all controls possible for this model and account for all feasible actions of the existing anti-HIV drugs. With this control model, we consider an optimal control problem of minimizing the infection level at the end of a given time interval. Using an analytical mathematical technique, we prove that the optimal controls are bang-bang, find accurate estimates for the maximal possible number of switchings of these controls and establish qualitative types of the optimal controls as well as mutual relationships between them. Having the estimate for the number of switchings found, we can reduce the two-point boundary value problem for Pontryagin Maximum Principle to a considerably simpler problem of the finite-dimensional optimization, which can be solved numerically. Despite this possibility, the obtained theoretical results are illustrated by numerical calculations using BOCOP–2.0.5 software package, and the corresponding conclusions are made.http://dx.doi.org/10.1155/2020/8107106 |
| spellingShingle | Ellina V. Grigorieva Evgenii N. Khailov Andrei Korobeinikov Optimal Controls of the Highly Active Antiretroviral Therapy Abstract and Applied Analysis |
| title | Optimal Controls of the Highly Active Antiretroviral Therapy |
| title_full | Optimal Controls of the Highly Active Antiretroviral Therapy |
| title_fullStr | Optimal Controls of the Highly Active Antiretroviral Therapy |
| title_full_unstemmed | Optimal Controls of the Highly Active Antiretroviral Therapy |
| title_short | Optimal Controls of the Highly Active Antiretroviral Therapy |
| title_sort | optimal controls of the highly active antiretroviral therapy |
| url | http://dx.doi.org/10.1155/2020/8107106 |
| work_keys_str_mv | AT ellinavgrigorieva optimalcontrolsofthehighlyactiveantiretroviraltherapy AT evgeniinkhailov optimalcontrolsofthehighlyactiveantiretroviraltherapy AT andreikorobeinikov optimalcontrolsofthehighlyactiveantiretroviraltherapy |