Examining the impact of forcing function inputs on structural identifiability.

Accurate parameter estimation in biological models-from optimizing drug dosing schedules to forecasting disease outbreaks-requires that model parameters be uniquely identifiable from available data. When parameters are not uniquely identifiable, estimation yields non-unique solutions, leading to unreliable predictions and misallocated resources. Existing approaches to resolving identifiability problems often involve re-parameterizing models, but this can reduce the biological interpretability of parameters. We present a mathematical framework demonstrating that time-varying parameters, driven by environmental or behavioral data-such as temperature patterns, treatment schedules, or contact rates-can resolve parameter identifiability issues without re-parameterization. We prove that incorporating such forcing functions into model parameters cannot worsen structural identifiability, and establish conditions under which they provably improve it. Our results illustrate how forcing functions can be used to break parameter entanglements, with a single forcing function sometimes resolving multiple identifiability issues simultaneously. We validate this framework with two worked examples, from pharmacokinetics (drug tissue-blood exchange rates) and disease transmission (seasonally varying contact rates), demonstrating that readily available time-varying data can resolve identifiability while preserving parameter interpretability. By connecting rigorous mathematical proofs to practical applications, this work provides modelers with systematic guidance on when, where, and how to leverage existing data streams to improve parameter estimation.