Parameter uncertainty in dynamical models: a practical identifiability index
Abstract
Ordinary differential equation models are widely used to describe how complex systems change over time, but reliable conclusions depend on how well their parameters can be estimated from limited, noisy data. Common measures of uncertainty, such as confidence intervals and coefficients of variation, do not provide a single, easy-to-interpret scale for comparing uncertainty across parameters, models, error structures, and data-collection designs. We introduce the Practical Identifiability Index (PII), a unit-free, scale-invariant measure of uncertainty for individual positive-valued parameters, defined as the base-10 logarithm of the ratio between the upper and lower bounds of a confidence interval; a value of 1 corresponds to a tenfold ratio between the interval bounds. We study PII using parametric bootstrap experiments in growth and compartmental epidemic models. PII decreases as longer data windows are used for model fitting and tends to increase with observation noise (although not uniformly across settings) and when several parameters must be estimated together. PII remains larger for parameters associated with latent or indirectly observed processes, while adding more measured quantities substantially improves estimation of progression and recovery parameters. We then apply PII to daily influenza incidence from the 1918 San Francisco epidemic. As a benchmark, PII and normalized confidence-interval width (NCIW) with equivalent thresholds give the same classification in 134 of 135 model, scenario, parameter, and error-structure combinations (99.3% agreement), while PII expresses uncertainty directly as a multiplicative range, without normalization by the parameter estimate. PII is meant to complement, rather than replace, other identifiability and uncertainty analyses, and should be considered together with empirical coverage and other diagnostic measures.
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BibTeX
@misc{karami2026parameter,
title = {{Parameter uncertainty in dynamical models: a practical identifiability index}},
author = {Hamed Karami and Alexandra Smirnova and Sunmi Lee and Gerardo Chowell},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.08475},
eprint = {2606.08475},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.08475},
}
RIS
TY - UNPB
TI - Parameter uncertainty in dynamical models: a practical identifiability index
AU - Hamed Karami
AU - Alexandra Smirnova
AU - Sunmi Lee
AU - Gerardo Chowell
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2606.08475
UR - https://arxiv.org/abs/2606.08475
AB - Ordinary differential equation models are widely used to describe how complex systems change over time, but reliable conclusions depend on how well their parameters can be estimated from limited, noisy data. Common measures of uncertainty, such as confidence intervals and coefficients of variation, do not provide a single, easy-to-interpret scale for comparing uncertainty across parameters, models, error structures, and data-collection designs. We introduce the Practical Identifiability Index (PII), a unit-free, scale-invariant measure of uncertainty for individual positive-valued parameters, defined as the base-10 logarithm of the ratio between the upper and lower bounds of a confidence interval; a value of 1 corresponds to a tenfold ratio between the interval bounds. We study PII using parametric bootstrap experiments in growth and compartmental epidemic models. PII decreases as longer data windows are used for model fitting and tends to increase with observation noise (although not uniformly across settings) and when several parameters must be estimated together. PII remains larger for parameters associated with latent or indirectly observed processes, while adding more measured quantities substantially improves estimation of progression and recovery parameters. We then apply PII to daily influenza incidence from the 1918 San Francisco epidemic. As a benchmark, PII and normalized confidence-interval width (NCIW) with equivalent thresholds give the same classification in 134 of 135 model, scenario, parameter, and error-structure combinations (99.3% agreement), while PII expresses uncertainty directly as a multiplicative range, without normalization by the parameter estimate. PII is meant to complement, rather than replace, other identifiability and uncertainty analyses, and should be considered together with empirical coverage and other diagnostic measures.
ER -