Trajectory enclosures for nonlinear systems with uncertain initial conditions and parameters

In this paper we provide a novel method to deal with uncertainties in initial value problems based on solving positivity conditions, by means of semidefinite programmes and sum of squares decompositions. More specifically, given a nonlinear dynamical system with uncertainties in initial conditions and parameter values, our method provides enclosures for state trajectories. Due to the fact that we compute ellipsoids rather than boxes to estimate the enclosures, our method is less affected by conservation effects. We demonstrate the proposed method to models from biology, on the tasks of parameter estimation, model robustness and model selection. These problems that arise from biological applications are particularly challenging due to the paucity of data as well as the presence of significant data noise. We illustrate the applicability of our method on a simple model of gene regulation and the mitogen activated protein kinase signalling pathway.