# Identifiability of Delay Parameters for Nonlinear Time-delay Systems with Applications in Systems Biology

The concept of parameter identifiability will be introduced briefly, followed by a short description of how this property can be tested for ODE-systems in general by rank calculations. Then, the extension of this analysis to delay systems, recently developed by Xia et al. [1] and Zhang et al. [2] will be reviewed. In these works, the authors use the framework of modules over non-commutative rings to formulate an analogous rank test for parameter identifiability of delay systems with known time-delays. Our ongoing work will then be motivated by a model of cellular signal transduction by Timmer et al. [3], where the sojourn time of STAT-5 in the nucleus is modelled by an unknown time delay which is estimated numerically by Timmer et al.. We will show how the identifiability of the time delay parameter is determined by the form of the external input-output representation of the system. Working in the mathematical framework of [1] and [2], we formulate explicit criteria based on rank calculations for the space spanned by the gradients of the output derivatives. Finally, several examples of biological systems from the literature will be discussed.

Joint work with Bernt Wennberg.

*Author: Milena Anguelova, Chalmers University of Technology and Gothenburg University*