Stability analysis of nonlinear time-delayed systems with application to biological models

H.a. Kruthika; Arun D. Mahindrakar; Ramkrishna Pasumarthy

International Journal of Applied Mathematics and Computer Science (2017)

  • Volume: 27, Issue: 1, page 91-103
  • ISSN: 1641-876X

Abstract

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In this paper, we analyse the local stability of a gene-regulatory network and immunotherapy for cancer modelled as nonlinear time-delay systems. A numerically generated kernel, using the sum-of-squares decomposition of multivariate polynomials, is used in the construction of an appropriate Lyapunov-Krasovskii functional for stability analysis of the networks around an equilibrium point. This analysis translates to verifying equivalent LMI conditions. A delay-independent asymptotic stability of a second-order model of a gene regulatory network, taking into consideration multiple commensurate delays, is established. In the case of cancer immunotherapy, a predator-prey type model is adopted to describe the dynamics with cancer cells and immune cells contributing to the predator-prey population, respectively. A delay-dependent asymptotic stability of the cancer-free equilibrium point is proved. Apart from the system and control point of view, in the case of gene-regulatory networks such stability analysis of dynamics aids mimicking gene networks synthetically using integrated circuits like neurochips learnt from biological neural networks, and in the case of cancer immunotherapy it helps determine the long-term outcome of therapy and thus aids oncologists in deciding upon the right approach.

How to cite

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H.a. Kruthika, Arun D. Mahindrakar, and Ramkrishna Pasumarthy. "Stability analysis of nonlinear time-delayed systems with application to biological models." International Journal of Applied Mathematics and Computer Science 27.1 (2017): 91-103. <http://eudml.org/doc/288096>.

@article{H2017,
abstract = {In this paper, we analyse the local stability of a gene-regulatory network and immunotherapy for cancer modelled as nonlinear time-delay systems. A numerically generated kernel, using the sum-of-squares decomposition of multivariate polynomials, is used in the construction of an appropriate Lyapunov-Krasovskii functional for stability analysis of the networks around an equilibrium point. This analysis translates to verifying equivalent LMI conditions. A delay-independent asymptotic stability of a second-order model of a gene regulatory network, taking into consideration multiple commensurate delays, is established. In the case of cancer immunotherapy, a predator-prey type model is adopted to describe the dynamics with cancer cells and immune cells contributing to the predator-prey population, respectively. A delay-dependent asymptotic stability of the cancer-free equilibrium point is proved. Apart from the system and control point of view, in the case of gene-regulatory networks such stability analysis of dynamics aids mimicking gene networks synthetically using integrated circuits like neurochips learnt from biological neural networks, and in the case of cancer immunotherapy it helps determine the long-term outcome of therapy and thus aids oncologists in deciding upon the right approach.},
author = {H.a. Kruthika, Arun D. Mahindrakar, Ramkrishna Pasumarthy},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {time-delay; cancer immunotherapy; gene-regulatory network; sum of squares},
language = {eng},
number = {1},
pages = {91-103},
title = {Stability analysis of nonlinear time-delayed systems with application to biological models},
url = {http://eudml.org/doc/288096},
volume = {27},
year = {2017},
}

TY - JOUR
AU - H.a. Kruthika
AU - Arun D. Mahindrakar
AU - Ramkrishna Pasumarthy
TI - Stability analysis of nonlinear time-delayed systems with application to biological models
JO - International Journal of Applied Mathematics and Computer Science
PY - 2017
VL - 27
IS - 1
SP - 91
EP - 103
AB - In this paper, we analyse the local stability of a gene-regulatory network and immunotherapy for cancer modelled as nonlinear time-delay systems. A numerically generated kernel, using the sum-of-squares decomposition of multivariate polynomials, is used in the construction of an appropriate Lyapunov-Krasovskii functional for stability analysis of the networks around an equilibrium point. This analysis translates to verifying equivalent LMI conditions. A delay-independent asymptotic stability of a second-order model of a gene regulatory network, taking into consideration multiple commensurate delays, is established. In the case of cancer immunotherapy, a predator-prey type model is adopted to describe the dynamics with cancer cells and immune cells contributing to the predator-prey population, respectively. A delay-dependent asymptotic stability of the cancer-free equilibrium point is proved. Apart from the system and control point of view, in the case of gene-regulatory networks such stability analysis of dynamics aids mimicking gene networks synthetically using integrated circuits like neurochips learnt from biological neural networks, and in the case of cancer immunotherapy it helps determine the long-term outcome of therapy and thus aids oncologists in deciding upon the right approach.
LA - eng
KW - time-delay; cancer immunotherapy; gene-regulatory network; sum of squares
UR - http://eudml.org/doc/288096
ER -

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