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

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.

Distributed Control and the Lyapunov Characteristic Exponents in the Model of Infectious Diseases

“…Lemma 5 (see [25]). If (a 4 − k) 2 < 4b, then the second column of the Cauchy matrix of (10) is as follows:…”

“…In order to estimate z(t), we can use the results of the paper [25], where the elements of the Cauchy matrix C(t, s) were constructed.…”

Global Asymptotic Stability and Nonlinear Analysis of the Model of the Square Immunopixels Array Based on Delay Lattice Differential Equations

“…Mathematical modeling of arrays of immunopixels is based on lattice differential equations and delayed differential equations [11][12][13][14]. The lattice differential equations are related to the models with the discrete spatial structure [15].…”

Stability Analysis of a Mathematical Model of Hepatitis B Virus with Unbounded Memory Control on the Immune System in the Neighborhood of the Equilibrium Free Point

“…The combined method with IL-2 therapy was based on the proposed mathematical model described by the system of integro-differential equations. This approach was proven to be efficient in [6][7][8][9][10][11][12][13]. Previous models predicted the success rate of an antiviral cure but without any relation to the immune respond impact [14][15][16].…”