Parameter uncertainty in dynamical models: a practical identifiability index

2026-06-07

AI Summary

This study addresses the challenge of practical identifiability in parameter estimation for ordinary differential equation models of dynamical systems, which often arises due to limited, noisy, and partially observable data. To this end, the authors propose a Practical Identifiability Index (PII) that quantifies marginal parameter uncertainty via the logarithmic span of positive-parameter confidence intervals, offering a concise measure of how strongly observational data constrain each parameter. The PII enables consistent, order-of-magnitude comparisons across parameters, models, and experimental designs and complements existing approaches such as coverage probability and profile likelihood. Validated through parametric bootstrapping, sensitivity analysis, and structural identifiability theory on growth and compartmental epidemic models, the framework demonstrates that parameter uncertainty diminishes with more informative calibration windows, lower noise levels, and weaker parameter coupling; early-dynamics observability accelerates convergence, and additional measurements substantially enhance identifiability of latent-variable-associated parameters.

Abstract

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.

Problem

• practical identifiability
• parameter uncertainty
• dynamical models
• ordinary differential equations
• confidence intervals

Innovation

• Practical Identifiability Index
• parameter uncertainty
• dynamical systems
• confidence interval
• ODE models