Journal

Journal of Mathematical Biology

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.

Overview

This paper is about structural identifiability for parabolic PDEs in settings where the input parameters (scalar-valued or functions) appear linearly. The main idea is to translate the identifiability question into an existence-and-uniqueness problem for an associated elliptic equation. That turns out to be a very useful point of view, because it lets one bring in more classical tools from elliptic PDE theory rather than treating identifiability as a completely separate problem.

In the homogeneous case, this perspective connects naturally to the Fredholm alternative, which gives a clean way to understand when identifiability holds automatically and when it can fail because of particular choices of initial or boundary data. One point the paper tries to make is that even if some of these failures look a bit pathological at first, they still matter in practice, especially for spatial experiments where the initial condition is often constrained by the setup and not freely chosen by the modeller.

The paper also treats nonlinear reaction terms and shows that, in many cases, uniqueness of the auxiliary elliptic problem corresponds directly to identifiability.

Keywords

structural identifiability; parabolic PDEs; elliptic operators; Fredholm alternative; inverse problems; parameter estimation

MSC 2020

35Q92; 35K10; 35J15; 62F99; 34A12

Links

• DOI / Journal page
• Preprint (arXiv)