Structural identifiability of linear-in-parameter parabolic PDEs through auxiliary elliptic operators

TL;DR

This paper introduces a novel method for analyzing the structural identifiability of linear-in-parameter parabolic PDEs by linking it to elliptic equations, enhancing understanding of parameter recoverability in biological models.

Contribution

It presents a new approach that frames identifiability as an elliptic equation problem, providing conditions for both unconditional and non-identifiability, and extends analysis to nonlinear reaction terms.

Findings

Unconditional identifiability for homogeneous equations using Fredholm alternative.

Non-identifiability cases linked to specific initial and boundary conditions.

Global identifiability often achievable under mild assumptions for nonlinear models.

Abstract

Parameter identifiability is often requisite to the effective application of mathematical models in the interpretation of biological data, however theory applicable to the study of partial differential equations remains limited. We present a new approach to structural identifiability analysis of fully observed parabolic equations that are linear in their parameters. Our approach frames identifiability as an existence and uniqueness problem in a closely related elliptic equation and draws, for homogeneous equations, on the well-known Fredholm alternative to establish unconditional identifiability, and cases where specific choices of initial and boundary conditions lead to non-identifiability. While in some sense pathological, we demonstrate that this loss of structural identifiability has ramifications for practical identifiability; important particularly for spatial problems, where the initial condition is often limited by experimental constraints. For cases with nonlinear reaction terms, uniqueness of solutions to the auxiliary elliptic equation corresponds to identifiability, often leading to unconditional global identifiability under mild assumptions. We present analysis for a suite of simple scalar models with various boundary conditions that include linear (exponential) and nonlinear (logistic) source terms, and a special case of a two-species cell motility model. We conclude by discussing how this new perspective enables well-developed analysis tools to advance the developing theory underlying structural identifiability of partial differential equations.