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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.
Highlights
In this paper we consider the Marchuk model of infection diseases dt = βV ( t ) − γF ( t ) V ( t )
The goal of this paper is to demonstrate new possibilities of distributed control in the model of infection diseases through analysis of integro-differential systems
We propose here a simple method of analysis and estimation based on a reduction of integro–differential systems to ones of ordinary differential equations
Summary
From the medical point of view, our results could be interpreted as follows: supporting the immune system we transform infection disease to a stable state of “almost healthy” body After getting this stable state we do not stop the use of corresponding medicine allowing to hold antibody concentration rate on the higher level than in the normal conditions of a healthy body. In all these stages it is important to estimate influence of many additional factors in order to hold the process in a corresponding zone.
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