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A nonlinear PI-type control strategy is designed in order to minimize the HIV concentration in blood plasma, via medical drug injection, under the framework of bounded uncertain input disturbances. For control design it is considered a simplified mathematical model of the virus infection as a benchmark. The model is based on mass balances of healthy cells, infected cells, and the virus concentrations. The proposed controller contains a nonlinear feedback PI structure of bounded functions of the regulation error. The closed-loop stability of the system is analyzed via Lyapunov technique, in which robustness against system disturbances is demonstrated. Numerical experiments show a satisfactory performance of the proposed methodology as a HIV therapy, in which the virion particles and the infected CD4^{+}T cells are minimized and, as an interesting result, the drug dosage can be suspended, thus avoiding drug resistance from the virus. Finally, the proposed controller is compared to a standard sliding-mode and hyperbolic tangent controllers showing better performance.

The Human Immunodeficiency Virus (HIV) causes the named Acquired Immune Deficiency Syndrome (AIDS). This virus is distributed through the mucous membranes and is often transmitted through bodily fluids. It destroys the immune system, by infecting CD4^{+}T cell and the macrophages. Consequently, a poor immune system makes it difficult for the organism to fight against opportunistic infections as pneumonia, meningitis, or tuberculosis causing death. A healthy person has a CD4^{+}T count ranging from 500 to 1000 cells mm^{−3}. A person with a count of 200 cells mm^{−3} or less of CD4^{+}T is considered to have AIDS. There are rare patients who are infected with HIV, but they are able to suppress the virus without drugs therapy and hence do not develop AIDS; this status is known as long-term nonprogressor (LTNP) [

The HIV replication cycle can be summarized in six steps: binding and entry; no coating; reverse transcription; provirus integration; virus protein synthesis and assembly; and finally budding [^{+}T cells to fight off HIV and other infections, keeping the immune system as healthy as possible. From experience, these drugs alone or in combination are not capable of dropping the viral load to be undetectable for sustained periods of time [

One way to analyze the evolution of the HIV dynamic is the mathematic modeling considering the underlying infection mechanisms of the HIV/AIDS to understand and anticipate the spread and evaluate the potential effectiveness of different therapies to bring the infection under control [^{+}T cell counts and rise in peripheral virus seen in AIDS [

Additionally, the HIV mathematical modeling has allowed the analysis of the dynamic behavior of the virus infection, which has helped to propose regulation strategies to diminish indirectly the viral concentration [^{+}T cells and the number of virion particles are considered; the control law employed a reduced HIV model, comprising two nested loops, a linearized feedback and a LQ regulator. It showed that it is not possible to completely eliminate the infection, but it can drive the virion particles to a number below the threshold specification (50 virus copies/mm^{3} in plasma are undetectable in HIV patients). Also they showed that the controlled system is stable to moderate uncertainties in the HIV model parameters.

In this work a novel nonlinear PI-type controller is proposed in order to provide systematic drug-scheduling which drives the HIV virus concentration to minimal values in simulation tests. Unlike the previous therapies based on control algorithms, the proposed control strategy has the advantage of interrupting the drug dosing permanently. The proposed control law contains a class of continuous bounded functions, in order to compensate the dynamic behavior of the virion concentration in plasma and possible unknown disturbances onto the system. Numerical experiments show the successful performance of the proposed methodology as an idealized HIV therapy.

The mathematical model to describe the dynamics of the HIV and CD4^{+}T cells during the primary infection stage was taken from [^{+}T cells ^{−1} as follows:^{+}T cells by the virus and ^{−1}) [^{−1} d^{−1}, ^{−1}, ^{−1} d^{−1}, ^{−1}, ^{−1}, and ^{−1}; all of these kinetic constants are nonnegative and ^{+}T cell generation;

Let us consider a generalized space state representation of the system (

Now, let us consider the following hypothesis:

The control input

Consider the dynamical system (

The proposed controller is inspired on the framework of the sliding-mode theory, which provides some robustness properties against external disturbances [

Therefore, the proposed controller includes two bounded sigmoid functions as described by (

Consider the Lyapunov candidate function:

Therefore,

Now, considering the hypotheses (H1) to (H3),

In particular, the integral term is considered to compensate the effect of the additive bounded disturbance

Numerical experiments were done in order to show the open-loop and closed-loop behavior of the model for HIV infection. From the above, a PC computer with Intel Core i7 processor was used and the ordinary differential equations systems solver libraries, in particular, 23s ODE solver from MATLAB^{−1} and ^{−2}. In Figure

Open-loop behavior of the HIV dynamics.

Phase portrait of the open-loop system.

Closed-loop dynamics of the HIV concentration.

Closed-loop behavior of the uncontrolled concentrations.

Closed-loop phase portrait of HIV dynamics.

Control’s effort for the proposed controller.

Performance index for the considered controllers.

In this work an alternative therapy based on a nonlinear PI feedback control law to minimize the HIV concentration in blood plasma was presented. The proposed methodology contains a class of bounded functions in order to provide a satisfactory performance in the regulation of the number of virion particles and infected CD4^{+}T cells using a single input model with unknown disturbances. This approach shows by simulation an effective diminishment of the HIV concentration under the drug dosing scheme, which drives the cells near to health, improving the quality of life and longevity. Also this approach can avoid the drug resistance due to its minimal drug dosing values. Moreover, a better performance of the current methodology was achieved in contrast to other control laws.

The authors have declared no competing interests.

Rigel Valentín Gómez-Acata wishes to acknowledge the CINVESTAV and the CONACyT for their support.