Non-Lipschitz Growth Functions as a Natural Way of Modelling Finite Time Behaviour in Auto-immune Dynamics

Abstract

Abstract This paper presents an analysis of healthy immune system dynamics from the perspective of finite time stability. Finite time stability and stabilisation have been well-studied in various engineering applications. The principal paradigm uses non-Lipschitz functions of the states. Recent studies on robustness of closed-loop non-Lipschitz dynamical systems are based on Lyapunov functions and show that finite time stability is both a special case of and superior to the asymptotic stability of the system. Such a finite time convergence property has not been studied extensively in the area of biological systems where it is evident that the dynamics are finite time convergent to certain equilibrium points rather than asymptotically or exponentially convergent. Furthermore, there are examples such as a healthy immune system response where robustness is found to the state of auto-immune disease. These inherent features of robustness and finite time convergence motivate the development of a modified version of the Michaelis-Menten function, which is frequently used in biology, based on the finite time stability paradigm. The paper hypothesizes a potential connection between finite time stability and the dynamics of a healthy immune system.