Examining the impact of forcing function inputs on structural identifiability

For mathematical and experimental ease, models with time varying parameters are often simplified to assume constant parameters. However, this simplification can potentially lead to identifiability issues (lack of uniqueness of parameter estimates). Methods have been developed to algebraically and numerically determine the identifiability of a model, as well as resolve identifiability issues. This specific type of simplification presents an alternate opportunity to instead use this information to resolve the unidentifiability. Given that re-parameterizing, collecting more data, and adding inputs can be potentially costly or impractical, this could present new alternatives. We present a method for resolving unidentifiability in a system by introducing a new data stream correlated with a parameter of interest. First, we demonstrate how and when non-constant input data can be introduced into any rational function ODE system without worsening the model identifiability. Then, we prove when these input functions improve structural and potentially also practical identifiability for a given model and relevant data. By utilizing pre-existing data streams, these methods can potentially reduce experimental costs, while still answering key questions. By connecting mathematical proofs to application, our framework removes guesswork from when, where, and how researchers can best introduce new data to improve model outcomes.