An integration-based method for estimating parameters in a system of differential equations

The application of ordinary differential equations to modelling the physical world is extensive and widely studied in many fields including physics, engineering and bioinformatics. Using these models to predict the behaviour of important state variables given particular parameter values has been extensively studied. On the other hand the inverse problem of predicting parameter values that will fit a solution of a differential equation to observed data has traditionally only been considered by using a few methods, many of which approach the problem via a least squares fit method. These methods either can only be applied when the differential equation being studied has a closed form solution or can become very computationally intensive when applying it to a system that can only be solved numerically and hence require optimisation algorithms. We propose an integration-based method that transforms an ordinary differential equation to an algebraic system of equations for which we solve for the unknown parameters in our equation. The method is computationally unintensive, can be extended to systems of differential equations and the number of parameters that can be estimated is not restricted. We demonstrate the method by simulating data, with and without noise, from a number of biological models described by ordinary differential equations and then estimate the parameters via the proposed technique.