Modelling and finite time stability analysis of psoriasis pathogenesis

A new systems model of psoriasis is presented and analysed from the perspective of control theory. Cytokines are treated as actuators to the plant model that govern the cell population under the reasonable assumption that cytokine dynamics are faster than the cell population dynamics. The analysis of various equilibria is undertaken based on singular perturbation theory. Finite time stability and stabilisation has been studied in various engineering applications where the principal paradigm uses non-Lipschitz functions of the states. A comprehensive study of the finite time stability properties of the proposed psoriasis dynamics is carried out. It is demonstrated that the dynamics are finite time convergent to certain equilibrium points rather than asymptotically or exponentially convergent. This feature of finite time convergence motivates the development of a modified version of the Michaelis-Menten function, frequently used in biology. This framework is used to model cytokines as fast finite time actuators.This work was supported by EPSRC via research grants EP/J018295/1 and EP/J018392/1. Marc Goodfellow gratefully acknowledges the financial support of the EPSRC via grant EP/N014391/1 The contribution of Marc Goodfellow was generously supported by a Wellcome Trust Institutional Strategic Support Award (WT105618MA)