Salmaniw Yurij, Browning Alexander P
Mathematical Institute, University of Oxford, Oxford, United Kingdom.
School of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia.
J Math Biol. 2025 May 29;91(1):4. doi: 10.1007/s00285-025-02225-w.
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.
参数可识别性通常是数学模型在生物学数据解释中有效应用的必要条件，然而适用于偏微分方程研究的理论仍然有限。我们提出了一种新方法，用于对参数线性的完全观测抛物方程进行结构可识别性分析。我们的方法将可识别性构建为一个密切相关的椭圆方程中的存在性和唯一性问题，并针对齐次方程，利用著名的弗雷德霍姆择一性来建立无条件可识别性，以及特定初始条件和边界条件选择导致不可识别性的情况。虽然在某种意义上这是病态的，但我们证明这种结构可识别性的丧失对实际可识别性有影响；这在空间问题中尤为重要，因为初始条件常常受到实验限制。对于具有非线性反应项的情况，辅助椭圆方程解的唯一性对应于可识别性，通常在温和假设下导致无条件全局可识别性。我们对一系列具有各种边界条件的简单标量模型进行了分析，这些模型包括线性（指数）和非线性（逻辑斯蒂）源项，以及一个双物种细胞运动模型的特殊情况。我们通过讨论这种新观点如何使成熟的分析工具推动偏微分方程结构可识别性的基础理论发展来结束本文。
