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

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.