Exact identifiability analysis for a class of partially observed near-linear stochastic differential equation models

Stochasticity plays a key role in many biological systems, necessitating the
calibration of stochastic mathematical models to interpret associated data. For
model parameters to be estimated reliably, it is typically the case that they
must be structurally identifiable. Yet, while theory underlying structural
identifiability analysis for deterministic differential equation models is
highly developed, there are currently no tools for the general assessment of
stochastic models. In this work, we extend the well-established differential
algebra framework for structural identifiability analysis to linear and a class
of near-linear, two-dimensional, partially observed stochastic differential
equation (SDE) models. Our framework is based on a deterministic recurrence
relation that describes the dynamics of the statistical moments of the system
of SDEs. From this relation, we iteratively form a series of necessarily
satisfied equations involving only the observed moments, from which we are able
to establish structurally identifiable parameter combinations. We demonstrate
our framework for a suite of linear (two- and n-dimensional) and non-linear
(two-dimensional) models. Most importantly, we define the notion of structural
identifiability for SDE models and establish the effect of the initial
condition on identifiability. We conclude with a discussion on the
applicability and limitations of our approach, and potential future research
directions.