Optimal Controls of the Highly Active Antiretroviral Therapy

Abstract

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.

Optimal CAR T-cell Immunotherapy Strategies for a Leukemia Treatment Model

“…imply the required continuation of the nonnegative solutions h 0 (t) and h 1 (t We note that such an approach for analyzing the continuation of solutions to quadratic systems was previously presented in [27].…”

“…The definition of the function G(t) implies its continuity on the interval [0, T]. Then, the neighborhood of t = T is defined, in which due to (27) and the second initial condition from (26), this function is negative if α 2 y * (T) − α 3 µz * (T) > 0, and positive if α 2 y * (T) − α 3 µz * (T) < 0. Then, integrating in this neighborhood the first equation of system (25) with the appropriate initial condition from (26), we find the formula…”

“…Previously, such an approach to facilitate the analysis of solutions to quadratic systems was applied in [25][26][27].…”

Optimal Control and Bifurcation Analysis of HIV Model

“…However, in safe practice, the risk of HIV transmission can be reduced by using principle of abstinence–be faithful–use condom (ABC) [ 36 – 38 ]. Optimal control intervention through public health education, using of condom and treatment benefits a lot for continuing human life in safe [ 1 – 3 , 7 , 11 , 15 , 19 , 25 – 27 , 33 , 39 – 44 ].…”

Impact of a cost functional on the optimal control and the cost-effectiveness: control of a spreading infection as a case study

“…Di Liddo [4] remarked that, if u(t) is the fraction of a population N(t) treated by some drug and p is the unitary cost of the drugs, then a natural way to define the cost C of the treatment is C = pu(t)N(t). Moreover, as an ultimate way to exclude the controversy of the intervention cost, Grigorieva et al [11,9,7,8,6] suggested to completely disregard the cost of the intervention under the assumption that it is "tolerable" and small compared to the losses that can be inflicted otherwise. However, one has to take into consideration that the optimal control with the L 1 -type functionals (and in particular with the cost depending on the state variables) and the functionals independent of the controls are generally less amenable to the mathematical analysis, and usually involves singular and bang-bang controls.…”