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Around the Model of Infection Disease: The Cauchy Matrix and Its Properties

Department of Mathematics, Ariel University, 40700 Ariel, Israel
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Symmetry 2019, 11(8), 1016; https://doi.org/10.3390/sym11081016
Received: 27 June 2019 / Revised: 29 July 2019 / Accepted: 31 July 2019 / Published: 6 August 2019
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Abstract

In this paper the model of infection diseases by Marchuk is considered. Mathematical questions which are important in its study are discussed. Among them there are stability of stationary points, construction of the Cauchy matrices of linearized models, estimates of solutions. The novelty we propose is in a distributed feedback control which affects the antibody concentration. We use this control in the form of an integral term and come to the analysis of nonlinear integro-differential systems. New methods for the study of stability of linearized integro–differential systems describing the model of infection diseases are proposed. Explicit conditions of the exponential stability of the stationary points characterizing the state of the healthy body are obtained. The method of the paper is based on the symmetry properties of the Cauchy matrices which allow us their construction. View Full-Text
Keywords: integro–differential systems; Cauchy matrix; exponential stability; distributed control integro–differential systems; Cauchy matrix; exponential stability; distributed control
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Domoshnitsky, A.; Volinsky, I.; Bershadsky, M. Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Symmetry 2019, 11, 1016.

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