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This paper introduces a mathematical model which describes the dynamics of the spread of HIV in the human body. This model is comprised of a system of ordinary differential equations that involve susceptible cells, infected cells, HIV, immune cells and immune active cells. The distinguishing feature in the proposed model with respect to other models in the literature is that it takes into account cells that represent two distinct mechanisms of the immune system in the defense against HIV: the non-HIV-activated cells and the HIV-activated cells. With a view at minimizing the side effects of a treatment that employs a drug combination designed to attack the HIV at various stages of its life cycle, we introduce control variables that represent the infected patient’s medication. The optimal control rule that prescribes the medication for a given time period is obtained by means of Pontryagin’s Maximum Principle.

Belonging to the family of retroviruses, the Human Immunodeficiency Virus (HIV) is responsible for AIDS. HIV infection results in a chronic, progressive disease that can lead to the destruction of the immune system. The disease is characterized by a high rate of viral replication, which results in the emergence of more virulent variants. HIV infection is currently characterized by the count of CD4+ T cells, by the amount of viral particles in the blood (viral load) and also by the clinical symptoms. Not all patients develop every stage of the disease, and the time elapsed between the infection and the manifestation of different clinical symptoms is highly variable, even though the causes of such a variation remain partly unknown.

To reproduce, HIV joins the membrane of the T4 cell, which is vital to the immune response. The virus releases its RNA and an enzyme, which produces the DNA of the virus. Then, the DNA of the virus enters the nucleus and joins to the DNA of the cell, taking full control. The result of this union is the pro-viral DNA, that produces the messenger RNA, which contains the genetic code of the virus. The messenger RNA then reaches the cytoplasm and produces virions, which leave the host cell as newly formed HIV’s. Thus, when joined to a T4 cell, a single virus produces many potential threats to other cells.

By making quantification possible and unfolding non-trivial equilibrium points, the analysis of viral load in HIV infection has facilitated the management of the disease. It turns out that an exponential decrease in viral levels in plasma can be attained by the reverse transcriptase inhibitors and protease that are included in Anti- retroviral Therapy (ART) [

When HIV viruses invade the human body, they attack the CD4+ T cells in their way. When attacked, these auxiliary cells signal the presence of an invader to other immune cells (CD8+ T cells). The CD8+ T cells then respond to this signal and become Cytotoxic T Lymphocytes (CTL) by attempting to destroy the infected cells [

This work proposes a simple mathematical model to describe the dynamics of the HIV taking into account the immune response. The proposed model introduces changes to several existing models in the literature [

Even though ART has produced undisputed advances in the treatment of HIV infection, it has been argued that the inhibitors that comprise ART may cause adverse effects; see for example [

This work is organized as follows. Section 2 introduces the model, while Section 3 derives the trivial equi- librium point and analyzes the stability of the model. Section 4 introduces the optimal control problem, which derives the optimal medication levels. In Section 5, numerical experiments are proposed to illustrate the proposed approach and shed light into the behavior of the controlled system. Finally, Section 6 concludes the paper.

Let x represent the susceptible cells, i.e. the cells that can be infected with HIV, and define y as the cells that are already infected. In addition, assume that v represents the free viruses in the body, z denotes the defense cells (CD8+ T and B) and

with initial conditions:

The parameters in the above equations are described in

The virus replication mechanism is depicted in

In the model, one can notice the introduction of two variables z and

It is worth point out that, in the proposed model, the immune CTL cells are activated directly by the free viruses. In contrast, some previous works in the literature assume they are activated by infected cells, healthy cells and also CTL cells [

For a non-HIV-infected person,

By Hartman-Grobman’s theorem [

The Jacobian matrix of the system (1),

evaluated at values corresponding to the equilibrium points.

The Jacobian matrix evaluated at the point of trivial equilibrium (2) is given by:

The eigenvalues of the matrix

It can be noted that

i.e.,

Applying Routh-Hurwitz criterion [

Lets assume that one virus infects one person. During its average survival time

countering and infecting one target cell

viruses produced by one virus infecting a cell in a completely susceptible population of CD4+ T cells.

So, if

Equating to zero all the differential equations given in (1), one comes up to the following general formulation for the equilibrium points of the dynamical system:

where:

or

It is possible to observe that when

where the coefficients are:

The coefficients of Equation (12) are positive when

In order to study the effect of treating HIV infection with the use of a cocktail of drugs, we introduce two control variables (

The initial conditions were obtained by running the uncontrolled model from Equation (1) for a period of 365 days. Note that the second and fourth equations are decoupled.

We point out that

Alternatively, doctors could prescribe average values of the sequences

Observe that, if the dosages

Note that the terms multiplying

In the remainder of this paper, we analyze the system’s dynamics with

It is worth reinforcing that, in relation to typical HIV models that simulate the production rates and infect- ability of the virus, the proposed model includes two new variables in the dynamics: protected uninfected cells

The introduction of the new variables, namely protected uninfected cells

Observing the model we note that the defense cells CD8+ T (z) produce the activated cells CTL

System (17) is now optimized with respect to control parameters

In the expression above,

To find the optimal control variables

Control variables | Label |
---|---|

Reverse transcriptase reverse, integrase and entrance inhibitors | |

Protease inhibitors | |

New state variables | |

Protected susceptible cells due to treatment with inhibitors | |

Infected cells blocked due to treatment with inhibitor |

principle [

where

1) Considering the set:

It follows from Pontryagin’s maximum principle that the optimal control variables

Then,

Thus, isolating

As in this case we necessarily have

2) Considering the set:

It follows from Equation (21), that

Since, by definition,

Therefore, to ensure that

Hence, the optimal control for Problem (20) is characterized by (25).

The necessary conditions of the Pontryagin’s maximum principle [

Finally, we analyze the conditions of transversality. In our case, there is no terminal value for the state variables. Therefore, the transversal conditions to the adjoint variables are given by

Observe that the optimal control values in (25) depend directly on the co-state variables and, through these variables, on the dynamics described in (17). Hence, they cannot be analytically determined. Consequently, to solve Problem (20), one searches for optimal control values

For the numerical simulations we have used the dataset described in

In

State variables | Label | Value |
---|---|---|

CD4+ T cells in body (susceptible) | x | |

CD4+ T cells infected by HIV | y | |

Free HIV in the body | v | |

Defense cells CD8+ T HIV specific | z | |

Activated defense cells |

Parameters and constants | Variable | Value |
---|---|---|

Mortality of susceptible cells | ||

Mortality of infected cells | ||

Mortality of the virus | ||

Mortality of defense cells | ||

Average number of free virus from infected cells | 360 | |

Activation of immunologic response rate | ||

Virus infection rate | ||

Infected cells destruction rate | ||

Virus destruction rate | ||

Susceptible cells supply rate | ||

Defense cells supply rate |

periods, these variables stabilize and gradually approach a non-trivial equilibrium point. As previously men- tioned, the proposed model describes the typical behavior of the immune system in the presence of HIV. The difference is that it also allows a separate analysis of the activated defense cells. In

The optimal control problem in Equation (20) was solved for a one year treatment period. The initial condition for the state variables was obtained by running the uncontrolled system in Equation (1) for a one year period. The last value of each variable (after 365 days) was taken as the initial value for the same variable in the controlled problem. In this work, we first consider Case 1, with

Note in

In this section we examine the effects of the parameters of the optimal control problem in Equation (20). We vary the parameters

objective is to verify the implications of different compromises between immune response and side effects into the medication levels as time elapses.

In Case 2, we examine the influence of parameter

In Case 3 we make

In Case 4 we make

This paper introduces a novel model of HIV dynamics. In contrast to typical HIV dynamics models, the pro- posed model explicitly describes the protected CD4+ T cells and the HIV-specific CD8+ T cells. That allows us to explicitly understand and quantify these effects, thus providing a better understanding of the system’s dynamics.

The dynamics of the proposed model can be influenced by the use of Anti-retroviral Therapy (ART), which has produced significant advances in the treatment of HIV infection. Such a therapy, however, is associated to side effects, which should be avoided whenever possible. To take account of the side effects and produce a desirable compromise between treatment effectiveness and side effects, we propose an optimal control approach

which prescribes an optimal treatment aimed at maximizing the benefits of the treatment, while also minimizing the side effects. The optimal control problem is solved by means of Pontryagin’s maximum principle and the optimal medication levels are numerically found by a standard gradient algorithm.

The proposed optimal control formulation is analyzed in the light of selected numerical examples, which provide insight into the behavior of the system under different compromises between medication effectiveness and side effects. We evaluate the sensitivity of the method with respect to the parameters in the optimal control functional by means of a series of examples, which shed light on the optimal trajectories of the variables with respect to different compromises between efficiency and side-effects. We analyze four different cases with distinct levels of side effect introduced by drugs

The authors would like to thank FAPESP (Projeto temático Fapesp, 09/15098-0), and the Carlos Chagas Filho Foundation for Research Support of the State of Rio de Janeiro, FAPERJ, for supporting the research by means of grants APQ1 E-26/111.940/2012, E-26/110.379/2014 and E-26/110.087/2014. This work was partially sup- ported by the Coordination for the Improvement of Higher Education Personnel of Brazil―CAPES, under grant No. BEX2025/14-0, and by the National Council for Scientific and Technological Development―CNPq, under grant No.302716/2011-4.