Generalizing the differential algebra approach to input-output equations in structural identifiability

Structural identifiability for parameter estimation addresses the question of whether it is possible to uniquely recover the model parameters assuming noise-free data, making it a necessary condition for successful parameter estimation for real, noisy data. One established approach to this question for nonlinear ordinary differential equation models is via differential algebra, which uses characteristic sets to generate a set of input-output equations which contain complete identifiability information for the model. This paper presents a generalization of this method, proving that identifiability may be determined using more general solution methods such as ad hoc substitution, Gröbner bases, and differential Gröbner bases, rather than via characteristic sets. This approach is used to examine the structural identifiability of several biological model systems using different solution methods (characteristic sets, Gröbner bases, differential Gröbner bases, and ad hoc substitution). It is shown that considering a range of approaches can allow for faster computations, which makes it possible to determine the identifiability of models which otherwise would be computationally infeasible.