Exact quadratization of nonlinear systems: a useful tool in mathematical biology

Last modified: 2014-03-31

#### Abstract

It is widely established that nontrivial phenomena under investigation in life science undergo a nonlinear behavior, examples coming from reaction chemical kinetics, mathematical modeling of physiological systems, systems biology. It is as well known that linear approximations use to work according to little displacements from the linearization working point, that in fact strongly reduces the reliability of their applications in real cases, especially when the final purpose of the mathematical model is to provide a rigorous support to design a suitable control law (e.g. for the artificial pancreas, where most of the model-based control laws available in the literature are based on the linearization of the nonlinear model equations). Steps forward the first-order approximation have been done according to the Carleman polynomial approximation, providing the embedding of the original nonlinear system into a bilinear one on an infinite dimensional state space: as a matter of fact, a finite dimensional approximation will be required, truncating the higher order terms of the polynomial. This contribute aims to introduce a different viewpoint, providing the exact quadratization of the original nonlinear model on a finite dimensional state space. The class of nonlinear system where the proposed methodology can be applied is the one of Ordinary Differential Equations (ODE) nonlinear algebraic systems in n-dimension, that means ODE systems with each scalar derivative described by the product of the powers of the components of the state vector. The algorithm providing the exact quadratization allows to build the new ODE equations on an extended (but finite-dimensional) state space. Nevertheless, extensions can be build also for saturating functions (e.g. the Hill functions) that so frequently come up in life science frameworks. The general setting of application of the proposed exact quadratization seems in fact promising to face a wide range of problems, spanning from stability analysis to pharmacokinetics/pharmacodynamics problems, to control system design. This talk will show the application of this mathematical tool in a rather recent model of the glucose-insulin system, which reveals to be a nonlinear Delay Differential Equations (DDE) system, since the pancreatic insulin delivery rate properly models the physiological delay between the glucose-induced stimulus and the corresponding endogenous insulin secretion.

#### Keywords

Nonlinear systems