Parameter uncertainty in dynamical models: a practical identifiability index
Ordinary differential equation models are widely used to understand and forecast complex dynamical systems, but their predictive value depends on reliable parameter estimation. Structural identifiability assesses whether parameters can be uniquely recovered from ideal observations, whereas practical identifiability depends on finite, noisy and partially observed data. We introduce the Practical Identifiability Index (PII), a marginal uncertainty-width metric based on the logarithmic span of confidence intervals. Expressed on an order-of-magnitude scale, the PII summarises how tightly individual positive-valued parameters are constrained by available observations, enabling comparison across parameters, models, error structures and observation designs. The PII is intended as a complementary diagnostic, not a standalone identifiability test, and should be interpreted alongside coverage, profile likelihoods, posterior summaries, sensitivity analysis or structural identifiability results. Using parametric bootstrap experiments across growth and compartmental epidemic models, we identify consistent principles: uncertainty decreases as calibration windows become more informative, increases with observation noise and parameter coupling, and remains high for latent or indirectly observed processes. Parameters governing early observable dynamics become constrained sooner, while additional observables can improve constraint for latent progression and recovery parameters. The PII provides a simple, reportable summary of marginal parameter uncertainty for dynamical modelling.