Identifiable specializations for ODE models

The parameter identifiability problem for a dynamical system is to determine whether the parameters of the system can be found from data for the outputs of the system. Verifying whether the parameters are identifiable is a necessary first step before a meaningful parameter estimation can take place. Non-identifiability occurs in practical models. To reparametrize a model to achieve identifiability is a challenge. The existing approaches have been shown to be useful for many important examples. However, these approaches are either limited to linear models and scaling parametrizations or are not guaranteed to find a reparametrization even if it exists. In the present paper, we prove that there always exists a locally identifiable model with the same input–output behavior as the original one obtained from a given one by a partial specialization of the parameters. Our result applies to parametric rational ODE models with or without input, and our algorithm can find non-scaling reparametrizations. As an extra feature of our approach, the resulting (at least) locally identifiable reparametrization has the same shape: the monomials in the new state variables in the new model are formed in the same way as in the original model. Furthermore, we give a sufficient observability condition for the existence of a state space transformation from the original model to the new one. Our proof is constructive and can be translated to an algorithm, which we illustrate by several examples, with and without inputs.