Agrarian
Bulletin
of the Urals

Russian Journal of Agricultural Research

The publication is registered by the Ministry of the Russian Federation
for Affairs of the Press, Television and Radio Broadcasting and Mass Communication Media.
Registration certificate: PI number 77-12831 on May 31, 2002
Subscription index in catalog «Russian Press» - 16356
ISSN 1997 - 4868 (Print)

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Journal is included in the list of VAK (from 25.09.2017), No. 291

ISSN 2307-0005 (Online)
Key title: Agrarnyj vestnik Urala (Online)
Abbreviated key title: Agrar. vestn. Urala (Online)

Аграрный вестник Урала № 12 (142) 2015

Биология и биотехнологии

Красовский А. Н. доктор физико-математических наук, заведующий кафедрой Уральский государственный аграрный университет

Ким А. В. доктор физико-математических наук, руководитель группы функционально-дифференциальных уравнений Институт математики и механики Уральского отделения Российской академии наук

УДК:577.270+517.977

THE SOLUTION TO THE PROBLEM OF EVASION FOR THE MATHEMATICAL HIV-MODEL

 In the article the mathematical model of HIV-process is considered. The problem of the maximum deviation of the model form border model that envelopes fields of death is solving. Under the fields of death or pits we define the condition of controlled object – the patient, appropriate to his death. The control of the model is based on the principle of feedback. The role of control actions is played by efficiencies of two medications. We consider the problem of translation of non-linear controlled object from initial to final position in a fixed time (one year), so as to prevent the hit of field of death. As an informational image of controlled object we choose phase coordinates of controlled object vector, determined by the amount of healthy and infected cells, virus particles and immune effectors. Because of non-linear property of differential equations that describe the HIV-process, the problem is solved in a class of mixed control strategies using the method of extremal shift to the accompanying elements. As a border model graphs of system parameters changes over the time are used. For the stable movement evasion of a real dynamic object from a border model a probability control scheme is used. The final result (evasion) is guaranteed with the probability arbitrarily close to unity. Theoretical results are illustrated by computer simulation of the process with system parameters and data close to real. Getting results continue the author’s researches.


Keywords:

HIV-process, mathematical model, field of death, pit, extermal shift, mixed strategy, border model.


References:

1. Bocharov G., Kim A., Krasovskii A., Chereshnev V. et al. An extremal shift method for control of HIV-infection dynamics // Russian Journal on Numerical Analysis and Mathematical Modeling. 2015. Vol. 30. № 1.

2. Kim A. V., Krasovskii A. N. The mathematical and computer modeling for systems with the time delay. Ekaterinburg : USTU-UPI, 2010.

3. Kim A. V., Krasovskii A. N., Glushenkova V. V. On the control of the mathematical model of hiv -process // Agrarian Bulletin of the Urals. 2015. № 1.

4. Kim A. V., Pimenov V. G. i-Smooth analysis and numerical methods of solving functional-differential equations. Izhevsk : ISC “Regular and Chaotic Dynamics”, 2004.

5. Krasovskii A. N., Krasovskii N. N. control under lack of information. Boston : Birkhauser, 1994.

6. Krasovskii A. N., Ladeischikov A. N., Choi Y. S. Any problems of optimal control under lack of information. Ekaterinburg : USAU, 2014.

7. Krasovskii A. N., Choi Y. S. Stochastic control with the leaders-stabilizers. Ekaterinburg : IMM Ural Branch of RAS, 2001.

8. Krasovskii N. N. Control of the dynamic system. M. : Nauka, 1985.

9. Marchuk G. I. Mathematical models in immunology. M. : Nauka, 1980.

10. Jang T. S., Kwon H.-D., Lee J. Free terminal time optimal control problem. US, 2011


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