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Paper details
Number 1 - March 2017
Volume 27 - 2017
Stability analysis of nonlinear time-delayed systems with application to biological models
H.A. Kruthika, Arun D. Mahindrakar, Ramkrishna Pasumarthy
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.
Keywords
time-delay, cancer immunotherapy, gene-regulatory network, sum of squares