Framing structural identifiability in terms of parameter symmetries

A key step in mechanistic modelling of dynamical systems is to conduct a structural identifiability analysis. This entails deducing which parameter combinations can be estimated from a given set of observed outputs. The standard differential algebra approach answers this question by re-writing the model as a higher-order system of ordinary differential equations that depends solely on the observed outputs. Over the last decades, alternative approaches for analysing structural identifiability based on Lie symmetries acting on independent and dependent variables as well as parameters, have been proposed. However, the link between the standard differential algebra approach and that using full symmetries remains elusive. In this work, we establish this link by introducing the notion of parameter symmetries, which are a special type of full symmetry that alter parameters while preserving the observed outputs. Our main result states that a parameter combination is locally structurally identifiable if and only if it is a differential invariant of all parameter symmetries of a given model. We show that the standard differential algebra approach is consistent with the concept of structural identifiability in terms of parameter symmetries. We present an alternative symmetry-based approach for analysing structural identifiability using parameter symmetries. Lastly, we demonstrate our approach on two well-known models in mathematical biology.