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HIV dynamics: analysis and robust multirate MPC-based treatment schedules. (English) Zbl 1166.92025

Summary: Analysis and control of human immunodeficiency virus (HIV) infections have attracted the interests of mathematicians and control engineers during the recent years. Several mathematical models exist and adequately explain the interactions of the HIV infection and the immune system up to the stage of clinical latency, as well as viral suppression and immune system recovery after treatment therapy. However, none of these models can completely exhibit all that is observed clinically and account the full course of infection. Besides model inaccuracies that HIV models suffer from, some disturbances/uncertainties from different sources may arise in the modelling.

We study the basic properties of a 6-dimensional HIV model that describes the interactions of HIV with two target cells, $CD{4}^{+}$ T cells and macrophages. The disturbances are modelled in the HIV model as additive bounded disturbances. Highly Active AntiRetroviral Therapy (HAART) is used. The control input is defined to be dependent on the drug dose and drug efficiency. We developed treatment schedules for HIV infected patients by using a robust multirate Model Predictive Control (MPC)-based method. The MPC is constructed on the basis of an approximate discrete-time model of the nominal model. We established a set of conditions, which guarantee that the multirate MPC practically stabilizes the exact discrete-time model with disturbances.

The proposed method is applied to the stabilization of the uninfected steady state of the HIV model. The results of simulations show that, after initiation of HAART with a strong dosage, the viral load drops quickly and can be kept under a suitable level with mild dosage of HAART. Moreover, the immune system is recovered with some fluctuations due to the presence of disturbances.

##### MSC:
 92C50 Medical applications of mathematical biology 93C95 Applications of control theory 34D23 Global stability of ODE 34D05 Asymptotic stability of ODE 93B25 Algebraic theory of control systems 37N25 Dynamical systems in biology
##### Keywords:
robust MPC; HIV/AIDS; feedback stabilization