TY - GEN
T1 - Marchuk’s Models of Infection Diseases
T2 - 7th International Conference on Functional Differential Equations and Applications, FDEA 2019
AU - Volinsky, Irina
AU - Domoshnitsky, Alexander
AU - Bershadsky, Marina
AU - Shklyar, Roman
N1 - Publisher Copyright: © 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
PY - 2021
Y1 - 2021
N2 - We consider mathematical models of infectious diseases built by G. I. Marchuk in his well-known book on immunology. These models are in the form of systems of ordinary delay differential equations. We add a distributed control in one of the equations describing the dynamics of the antibody concentration rate. Distributed control looks here naturally since the change of this concentration rather depends on the corresponding average value of the difference of the current and normal antibody concentrations on the time interval than on their difference at the point t only. Choosing this control in a corresponding form, we propose some ideas of the stabilization in the cases, where other methods do not work. The main idea is to reduce the stability analysis of a given integro-differential system of the order n, to one of the auxiliary systems of the order n+ m, where m is a natural number, which is “easy” for this analysis in a corresponding sense. Results for these auxiliary systems allow us to make conclusions for the given integro-differential system of the order n. We concentrate our attempts in the analysis of the distributed control in an integral form. An idea of reducing integro-differential systems to systems of ordinary differential equations is developed. We present results about the exponential stability of stationary points of integro-differential systems using the method based on the presentation of solution with the help of the Cauchy matrix. Various properties of integro-differential systems are studied by this way. Methods of the general theory of functional differential equations developed by N. V. Azbelev and his followers are used. One of them is the Azbelev W-transform. We propose ideas allowing to achieve faster convergence to stationary point using a distributed control. We obtain estimates of solutions using estimates of the Cauchy matrices.
AB - We consider mathematical models of infectious diseases built by G. I. Marchuk in his well-known book on immunology. These models are in the form of systems of ordinary delay differential equations. We add a distributed control in one of the equations describing the dynamics of the antibody concentration rate. Distributed control looks here naturally since the change of this concentration rather depends on the corresponding average value of the difference of the current and normal antibody concentrations on the time interval than on their difference at the point t only. Choosing this control in a corresponding form, we propose some ideas of the stabilization in the cases, where other methods do not work. The main idea is to reduce the stability analysis of a given integro-differential system of the order n, to one of the auxiliary systems of the order n+ m, where m is a natural number, which is “easy” for this analysis in a corresponding sense. Results for these auxiliary systems allow us to make conclusions for the given integro-differential system of the order n. We concentrate our attempts in the analysis of the distributed control in an integral form. An idea of reducing integro-differential systems to systems of ordinary differential equations is developed. We present results about the exponential stability of stationary points of integro-differential systems using the method based on the presentation of solution with the help of the Cauchy matrix. Various properties of integro-differential systems are studied by this way. Methods of the general theory of functional differential equations developed by N. V. Azbelev and his followers are used. One of them is the Azbelev W-transform. We propose ideas allowing to achieve faster convergence to stationary point using a distributed control. We obtain estimates of solutions using estimates of the Cauchy matrices.
KW - Cauchy matrix
KW - Exponential stability
KW - Functional differential equations
UR - http://www.scopus.com/inward/record.url?scp=85125253155&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/978-981-16-6297-3_10
DO - https://doi.org/10.1007/978-981-16-6297-3_10
M3 - منشور من مؤتمر
SN - 9789811662966
T3 - Springer Proceedings in Mathematics and Statistics
SP - 131
EP - 143
BT - Functional Differential Equations and Applications - FDEA-2019
A2 - Domoshnitsky, Alexander
A2 - Rasin, Alexander
A2 - Padhi, Seshadev
Y2 - 22 August 2019 through 27 August 2019
ER -