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.

💡 Research Summary

The paper introduces a rigorous framework for improving structural identifiability of ordinary differential equation (ODE) models by incorporating known time‑varying forcing functions into parameters. Traditional practice often treats such time‑varying parameters as constants for simplicity, which can create identifiability problems—multiple parameter sets produce identical model outputs, undermining predictive reliability. Existing remedies—re‑parameterization, additional experiments, or adding new inputs—can be costly, reduce biological interpretability, or be infeasible.

The authors propose a different strategy: replace or scale a problematic parameter θᵢ with a known, bounded, non‑constant, Cⁿ‑smooth function û(t) (e.g., environmental temperature, treatment schedule, or contact rate). They first formalize the model class: rational‑function ODE systems with state vector x(t), measured output y(t), known inputs u(t), and parameter vector θ. A forcing function û(t) is introduced either as a multiplier (û(t)·θᵢ) or as a direct replacement (θᵢ ← û(t)).

Using differential algebra, specifically the characteristic set method and Ritt’s pseudo‑division algorithm, the authors eliminate state variables to obtain input‑output equations that relate y (and its derivatives) to the parameters. The coefficients of these equations form a “coefficient map” whose injectivity determines structural identifiability.

Two central theorems are proved. The first shows that adding a forcing function never degrades structural identifiability: the new input‑output relation contains all original monomials (possibly multiplied by û(t) or its derivatives) plus additional terms, but no term removes information about any original parameter. Consequently, the injectivity of the coefficient map is preserved.

The second theorem identifies conditions under which the forcing function actually improves identifiability. If û(t) is non‑zero over a sufficiently rich interval and its derivatives are linearly independent functions, then the monomials involving θᵢ split into distinct terms (e.g., θᵢ·û(t) and θᵢ·û(t)²), reducing the dimension of the parameter combination space. In graph‑theoretic terms, a parameter graph’s connected component becomes “observable” once at least one node is linked to a known forcing input, making all other nodes in that component identifiable.

A practical workflow is outlined: (1) construct the parameter graph to locate unidentifiable components; (2) search for existing data streams that can serve as forcing functions for parameters in those components; (3) augment the model accordingly; (4) recompute the input‑output equations using software such as DAISY; (5) assess structural (global/local) and practical identifiability (e.g., via profile‑wise analysis). This workflow avoids re‑parameterization and leverages already‑collected data, potentially reducing experimental cost.

The theoretical results are illustrated with two concrete examples. In a two‑compartment pharmacokinetic model, scaling the inter‑compartmental transfer rate k₁₂ by a measured temperature profile transforms a locally identifiable but not globally identifiable system into a globally identifiable one, as the temperature‑dependent terms separate the previously conflated parameters. In a seasonally forced disease transmission model, replacing the transmission rate β(t) with a product of a known rainfall series and a baseline β₀ splits the β‑γ interaction, allowing independent estimation of the baseline transmission and recovery rates.

The authors also discuss limitations: the approach assumes the forcing function is known exactly (or measured with negligible error), and the current proofs are limited to rational‑function ODEs. Extending the theory to stochastic models, partial differential equations, or handling noisy forcing data are identified as future research directions.

Overall, the paper provides a mathematically sound, computationally tractable method for turning external time‑varying data streams into assets for structural identifiability, offering modelers a systematic, cost‑effective alternative to traditional remedies while preserving the biological interpretability of parameters.