Parameter range reduction for ODE models using cumulative backward differentiation formulas

We consider fitting an ODE model to time series data of the system variables. We assume that the parameters of the model have some initial range of possible values and the goal is to reduce these ranges to produce a smaller parameter region from which to start a global nonlinear optimization algorithm. We introduce the class of cumulative backward differentiation formulas (CBDFs) and show that they inherit the accuracy and stability properties of their generating backward differentiation formulas (BDFs). Discretizing the system with these CBDFs and applying consistency conditions results in reductions of the parameter ranges. We show that these reductions are better than can be obtained simply using BDFs. In addition CBDFs inherit any monotonicity properties with respect to the parameters that the vector field possesses, and we exploit these properties to make the consistency checking more efficient. We illustrate with several examples, analyze some of the behavior of our range reduction method, and discuss how the method could be extended and improved.