A Brief Tutorial on Quadratic Stability of Linear Parameter-Varying Model for Biomathematical Systems

Many biological phenomena are investigated using models formulated in terms on nonlinear differential equations and, in general, the global stability is assured point-to-point by Lyapunov's theory. However, many times the stability analysis becomes a hard task due to the complex interactions between variables and parameters. Furthermore, when some parameters vary over time, it is not possible to employ the conventional methods of stability analysis regarding linear time-invariant (LTI) theory. However, several of those nonlinear models can be rewritten as linear systems with time-varying parameters lying inside bounded and known intervals for which there exist in the control theory literature convex optimization procedures based on linear matrix inequality (LMI) conditions to test the asymptotic stability. Knowing that the Robust LMI Parser (ROLMIP) and YALMIP are openly distributed toolboxes for MATLAB
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that allow easy programming of LMI conditions for systems with uncertain (possibly time-varying) parameters, this paper presents a implementation tutorial of a local stability analysis methodology that can be applied in linear parameter varying (LPV) models of biomathematical systems. A numerical example based on infectious disease is investigated.