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HIV is one of the major causes of deaths, especially in Sub-Saharan Africa. In this paper, an in vivo deterministic model of differential equations is presented and analyzed for HIV dynamics. Optimal control theory is applied to investigate the key roles played by the various HIV treatment strategies. In particular, we establish the optimal strategies for controlling the infection using three treatment regimes as the system control variables. We have applied Pontryagin’s Maximum Principle in characterizing the optimality control, which then has been solved numerically by applying the Runge-Kutta forth-order scheme. The numerical results indicate that an optimal controlled treatment strategy would ensure significant reduction in viral load and also in HIV transmission. It is also evident from the results that protease inhibitor plays a key role in virus suppression; this is not to underscore the benefits accrued when all the three drug regimes are used in combination.

There is an ever-changing need for new and useful treatment regimes that will provide assistance and relief in all aspects of the human condition. Subsequently, many researchers have embarked on the journey of analyzing the dynamics of various diseases affecting mankind with the aim of improving control and effect and finally eradicating the diseases from the population. Modelling and numerical simulations of the infectious diseases have been used as tools to optimize disease control. This is due to the fact that medical community has insufficient animal models for testing efficacy of drug regimes used in controlling infections. Human immunodeficiency virus (HIV) is one of the major problems that researchers have been working on for over three decades. According to the Joint United Nations Programme on HIV and AIDS (UNAIDS), there were 36.7 million people living with HIV/AIDS in 2016, 1.6 million of which live in Kenya [

Mathematical modelling is one of the many important tools used in understanding the dynamics of disease transmission. It is also used in developing guidelines important in disease control. In HIV, mathematical models have provided a framework for understanding the viral dynamics and have been used in the optimal allocation of the various interventions against the HIV virions [

In the literature, optimal control theory has been applied in the analysis of in-host HIV dynamics as well as in population-based HIV models. For instance, Yusuf and Benyah [

Drugs such as fusion inhibitors (FIs), reverse transcriptase inhibitors (RTIs), and protease inhibitors (PIs) have been developed and applied in the various optimal control problems. Srivastava et al. [

Karrakchou et al. [

As per the literature cited, it is clear that as much as ARTs have been used for viral suppression, the optimal treatment schedule necessary to maintain low viral load is always an approximation. Until the time when HIV cure is found, physicians will try as much as possible to apply the control strategy that will inhibit viral progression while simultaneously holding the side effects of treatment to a minimum. Most of the treatment regimes have many side effects that must be maintained at a low level. For example, long-term use of protease inhibitors is associated with insulin intolerance, cholesterol elevation, and the redistribution of body fat. Therefore, there is a need to establish the optimal treatment strategy, that is, the one which both maximizes the patient’s uninfected

This study has addressed some of the shortcomings noted from the in-host HIV dynamics models by applying three control variables representing the three drug regimes on the market, that is, the fusion inhibitor, reverse transcriptase inhibitors, and the protease inhibitors, in the in vivo HIV model. In addition, the study has incorporated the

In order for us to carry out optimal control processes, it is paramount to formulate a model that describes the basic interaction between the HIV virions and the body immune system. We develop a mathematical model for HIV in-host infection with three combinations of drugs. We define seven variables for the model as follows: susceptible

The parameters for the model are as follows. The susceptible

The summary for the model description is given as follows. The variables, parameters, and the control variables for the in-host model are described in Tables

Variables for HIV in vivo model with therapy.

Variable | Description |
---|---|

| The concentration of the noninfected |

| The concentration of the infected |

| The concentration of latently infected |

| The concentration of HIV virions, copies/mL, at any time |

| The concentration of the immature noninfectious virions, |

| The concentration of the |

| The concentration of the activated |

Parameters for HIV in vivo model with therapy.

Parameter | Description |
---|---|

| The rate at which the noninfected |

| The rate at which the noninfected |

| The rate at which the |

| The death rate of the infected |

| The death rate of the latently infected |

| The rate in which HIV virions are generated from the infected |

| The death rate of the infectious virus. |

| The death rate of the noninfectious virions. |

| The rate at which the infected cells are eliminated by the activated |

| The rate at which the |

| The death rate of the |

| The rate at which the |

| The rate at which the activated defense cells decay. |

Control variables for HIV in vivo model.

Control variable | Description | Purpose |
---|---|---|

| Fusion inhibitors | Are a class of antiretroviral drugs that work on the outside of the host |

| ||

| Reverse transcriptase inhibitors | Are a class of antiretroviral drugs used to treat HIV infection by inhibiting the reverse transcription process. |

| ||

| Protease inhibitors | Are a class of antiviral drugs that are widely used to treat HIV/AIDS by inhibiting the production of protease enzyme necessary for the production of infectious viral particles. |

From Figure

A compartmental representation of the in vivo HIV dynamics with therapy.

Control efforts are carried out to limit the spread of the disease and, in some cases, to prevent the emergence of drug resistance. Optimal control theory is a method that has been widely used to solve for an extremum value of an objective functional involving dynamic variables. In this section, we consider optimal control methods to derive optimal drug treatments as functions of time. The control variables as used in (

The necessary conditions that an optimal control must satisfy come from Pontryagin’s Maximum Principle [

Suppose that the objective function

The existence of the solution can be shown using the results obtained in Fleming and Rishel [

the class of all initial conditions with controls

the right-hand side of system (

by definition, each right-hand side of system (

by definition, the control set

this condition is satisfied by the control set

the integrand which is

there exist constants

this implies that

Since all the above conditions are satisfied, we conclude that there exist optimal controls

We now proceed by applying Pontryagin’s Maximum Principle [

Therefore, Pontryagin’s Maximum Principle gives the existence of adjoint variables that are obtained by differentiating the Lagrangian given by (

The adjoint variables are given by

By maximization of the Lagrangian with respect to the control variables

On the set

Similarly, on the set

Finally, on the set

Consequently, combining all the three cases given by (

On the set

On the set

Finally, on the set

Consequently, combining all the three cases given by (

On the set

On the set

Finally, on the set

Consequently, combining all the three cases given by (

In this section, we investigate the effect of optimal strategy on HIV by applying Runge-Kutta forth-order scheme on the optimality system. The optimality system is obtained by taking the state system together with the adjoint system, the optimal control, and the transversality conditions. The dynamical behaviour of the models in relation to various control is also studied. The optimal strategy is achieved by obtaining a solution for the state system (

Parameters and controls for HIV in vivo model with therapy.

Parameters | Value | Source |
---|---|---|

| 10 cell/mm^{3}/day | Nowak et al. [ |

| 0.01 day^{−1} | Srivastava and Chandra [ |

| 0.000024 mm^{3} vir^{−1} day^{−1} | Alizon and Magnus [ |

| 0.5 day^{−1} | Wodarz and Nowak [ |

| 0.5 day^{−1} | Wodarz and Nowak [ |

| 100 vir. cell^{−1} day^{−1} | Estimate |

| 3 day^{−1} | Mbogo et al. [ |

| 0.06 day^{−1} | Estimate |

| 0.02 day^{−1} | Arruda et al. [ |

| 20 cell/mm^{3}/day | Arruda et al. [ |

| 0.06 day^{−1} | Arruda et al. [ |

| 0.004 day^{−1} | Arruda et al. [ |

| 0.004 day^{−1} | Arruda et al. [ |

| 0-1 variable | Estimate |

| 0-1 variable | Estimate |

| 0-1 variable | Estimate |

The initial values for the variables for HIV in vivo model.

Variable | Values |
---|---|

| ^{3} |

| ^{3} |

| ^{3} |

| ^{3} |

| ^{3} |

| ^{3} |

| ^{3} |

The initial values given in Table ^{3} [

Figure

Simulated control strategies.

Figure

The population of the

It is important to point out that

Figure

The population of the latently infected

Figure

The population of the infected

HIV entry mechanism [

Figure

The population of the HIV virions in various control strategies.

From Figure

The population of the noninfectious HIV virions in various control strategies.

Figure

The population of the

Figure

The population of the activated

In this paper, we have analyzed a seven-dimension in vivo HIV model with inclusion of three drug combinations, that is, FIs, RTIs, and PIs. Optimal control theory is applied to determine the optimal treatment regime. The study applied Pontryagin’s Maximum Principle in deriving the conditions for optimal control, which maximizes the objective function. The systems of ODEs, the state system, and the adjoint system were solved numerically by both forward and backward Runge-Kutta forth-order scheme. Results from the numerical simulations show that FIs and RTIs should be used within the four months and later the doctors should change the drugs and introduce another type, whereas the PIs can be used for a longer period of time without necessarily leading to major side effect. However, the inferiority of monotherapy compared with combination of therapies has been observed in the simulated result, especially in suppression of viral replication,

ARTs have been seen to play a significant role as far as viral suppression is concerned. Therefore, they should be recommended for all patients immediately after one is diagnosed as HIV-positive regardless of the

In the future, it is important to develop the model in such a way that it brings out the relationship between the number of the

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The corresponding author acknowledges the financial support from the DAAD and the National Research Fund from the Kenyan Government.