Stability Analysis of Nonlinear Time–Delayed Systems with Application to Biological Models

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.

Periodic Solution and Stability Behavior for Nonlinear Oscillator Having a Cubic Nonlinearity Time-Delayed

“…Kruthika et al [2] investigate the local stability of a gene-regulatory network and immunotherapy for cancer modeled in a nonlinear time-delay system. Many articles have appeared as collecting theorems homotopy methods for solutions that concerned with the properties of delayed systems [3][4][5][6].…”

Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model

“…The presence of the delay, an unstableness's factor, lead to a complex global stability for the delayed nonlinear systems, even more complicated than the delay free case, and it shares the same use of sufficient conditions with the free delay systems which represents a conservative way to analyze a system stability [6,36], however by optimizing the upper bound terms researchers try to improve results and minimize this conservativeness effect. Also, the stability conditions are splitted to delay dependent [13,23,36] or delay independent techniques [13,18,36].…”

“…Independent delay conditions. Independent delay conditions are the simplest way to investigate a system stability, it relies on resolving LMIs without involving delay terms, which means theoretically, if such conditions exists, that the system will be stable for any delay value [13,18,36], but practically such conditions are hard and sometimes even impossible to find.…”

Duffing Oscillator’s Vibration Control under Resonance with a Negative Velocity Feedback Control and Time Delay

“…The analysis showed that the oscillator vibrational energy could be transferred to the pendulum parametrically when the pendulum natural frequency is equal to one-half of the oscillator natural frequency. Kruthika et al [16] analyzed the local stability of a gene-regulatory network and immunotherapy of cancer. They are modeled as nonlinear time-delayed systems.…”