Session S34 - Symbolic and Numerical Computation with Polynomials

Wednesday, July 14, 15:30 ~ 16:00 UTC-3

Weak identifiability of differential algebraic equation systems

Gabriela Jeronimo

Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Mathematical models of biochemical systems often describe the dynamics of the species concentrations by means of autonomous systems of ordinary differential equations given by polynomials or rational functions depending on parameters with unknown values. Parameter structural identifiability mainly addresses the question of deciding whether the parameter values can be uniquely determined from noise-free data on the input-output behaviour of the system. This problem has been broadly studied for several notions of identifiability using a variety of tools and under different perspectives, including Taylor series, Gröbner bases, differential algebra and numerical algebraic geometry based approaches.

In this talk we will introduce a notion of weak identifiability that extends the structural identifiability considered by G. Craciun and C. Pantea (2008), for systems arising from biochemical reaction networks under mass-action kinetics, to the general class of ODE systems with finitely many arbitrary algebraic outputs. More precisely, we consider differential algebraic equation systems of the form \[\Sigma := \left\{ \begin{array}{lll} \dot {\mathbf{x}} & = & \mathbf{F}(\mathbf{x},\theta,\mathbf{u})\\ \mathbf{y} & = & \mathbf{G}(\mathbf{x},\theta,\mathbf{u}) \end{array} \right. \] where $\mathbf{F}$ and $\mathbf{G}$ are two families of rational functions, $\mathbf{x}$, $\mathbf{u}$ and $\mathbf{y}$ are sets of variables representing the states, inputs and outputs of the system respectively, and $\theta$ are the unknown parameters. We adopt a multi-experiment approach, allowing for as many experiments as needed with different initial conditions.

Our notion of weak identifiability requires the injectivity of a map induced by the given system $\Sigma$. We will show that it can be effectively checked with finitely many successive derivatives of the input equations by studying the injectivity of a certain rational map of the parameters $\theta$ defined from the coefficients of the polynomials involved in the representation of the rational functions giving those derivatives. Finally, we will compare the notion of weak identifiability with other identifiability definitions given in previous works, including the approach via input-output equations.

Joint work with Pablo Solernó (Universidad de Buenos Aires, Argentina).

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