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.

翻译：随机性在许多生物系统中扮演着关键角色，这需要通过校准随机数学模型来解释相关数据。为了可靠地估计模型参数，这些参数通常必须在结构上是可辨识的。然而，尽管确定性微分方程模型的结构可辨识性分析理论已高度发展，但目前尚无用于一般性评估随机模型的工具。在本工作中，我们将用于结构可辨识性分析的成熟微分代数框架扩展到线性和一类近线性、二维、部分观测的随机微分方程（SDE）模型。我们的框架基于一个确定性递推关系，该关系描述了SDE系统统计矩的动力学。从这一关系出发，我们迭代地构建一系列仅涉及观测矩且必然满足的方程，并由此能够确定结构上可辨识的参数组合。我们针对一系列线性（二维及$n$维）和非线性（二维）模型展示了我们的框架。最重要的是，我们定义了SDE模型的结构可辨识性概念，并确立了初始条件对可辨识性的影响。最后，我们讨论了本方法的适用性与局限性，以及潜在的未来研究方向。