Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Approaches to assess the so-called structural identifiability of models are well-established for ordinary differential equation models, yet there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture spatial features inherent to many phenomena. The differential algebra approach to structural identifiability has recently been demonstrated to be applicable to several specific PDE models. In this brief article, we present general methodology for performing structural identifiability analysis on partially observed linear reaction-advection-diffusion (RAD) PDE models. We show that the differential algebra approach can always, in theory, be applied to linear RAD models. Moreover, despite the perceived complexity introduced by the addition of advection and diffusion terms, identifiability of spatial analogues of non-spatial models cannot decrease structural identifiability. Finally, we show that our approach can also be applied to a class of non-linear PDE models that are linear in the unobserved variables, and conclude by discussing future possibilities and computational cost of performing structural identifiability analysis on more general PDE models in mathematical biology.

翻译：数学模型的参数可辨识性是确保其有效解释生物数据并做出准确预测的关键前提。针对常微分方程模型的结构可辨识性评估方法已相当成熟，但对于捕捉众多现象中空间特征所需的偏微分方程模型，目前尚无通用的评估方法。本文简要介绍了一种将微分代数方法应用于结构可辨识性分析的新进展，证明了该方法可推广至部分可观测的线性反应-平流-扩散偏微分方程模型。理论分析表明，微分代数方法原则上总能应用于线性反应-平流-扩散模型。值得注意的是，尽管平流项与扩散项的引入增加了模型复杂度，但空间对应模型的可辨识性不会低于非空间模型。此外，本文进一步证明该方法同样适用于未观测变量呈线性关系的一类非线性偏微分方程模型，最终讨论了数学生物学中更一般偏微分方程模型结构可辨识性分析的未来发展方向与计算代价。