1UMR Modélisation Systémique Appliquée aux Ruminants,INRA,AgroParisTech,Université Paris-Saclay,75005 Paris,France.
3PEGASE,AgroCampus Ouest,INRA,35590 Saint-Gilles,France.
Animal. 2018 Apr;12(4):701-712. doi: 10.1017/S1751731117002774. Epub 2017 Nov 3.
What is a good (useful) mathematical model in animal science? For models constructed for prediction purposes, the question of model adequacy (usefulness) has been traditionally tackled by statistical analysis applied to observed experimental data relative to model-predicted variables. However, little attention has been paid to analytic tools that exploit the mathematical properties of the model equations. For example, in the context of model calibration, before attempting a numerical estimation of the model parameters, we might want to know if we have any chance of success in estimating a unique best value of the model parameters from available measurements. This question of uniqueness is referred to as structural identifiability; a mathematical property that is defined on the sole basis of the model structure within a hypothetical ideal experiment determined by a setting of model inputs (stimuli) and observable variables (measurements). Structural identifiability analysis applied to dynamic models described by ordinary differential equations (ODEs) is a common practice in control engineering and system identification. This analysis demands mathematical technicalities that are beyond the academic background of animal science, which might explain the lack of pervasiveness of identifiability analysis in animal science modelling. To fill this gap, in this paper we address the analysis of structural identifiability from a practitioner perspective by capitalizing on the use of dedicated software tools. Our objectives are (i) to provide a comprehensive explanation of the structural identifiability notion for the community of animal science modelling, (ii) to assess the relevance of identifiability analysis in animal science modelling and (iii) to motivate the community to use identifiability analysis in the modelling practice (when the identifiability question is relevant). We focus our study on ODE models. By using illustrative examples that include published mathematical models describing lactation in cattle, we show how structural identifiability analysis can contribute to advancing mathematical modelling in animal science towards the production of useful models and, moreover, highly informative experiments via optimal experiment design. Rather than attempting to impose a systematic identifiability analysis to the modelling community during model developments, we wish to open a window towards the discovery of a powerful tool for model construction and experiment design.
动物科学中的好（有用）数学模型是什么？对于为预测目的而构建的模型，模型充分性（有用性）的问题传统上是通过应用于模型预测变量的观测实验数据的统计分析来解决的。然而，很少有人关注利用模型方程数学性质的分析工具。例如，在模型校准的背景下，在尝试对模型参数进行数值估计之前，我们可能想知道是否有机会从可用的测量值中成功估计模型参数的唯一最佳值。这个唯一性问题被称为结构可识别性；这是一个仅基于模型结构的数学性质，在由模型输入（刺激）和可观察变量（测量）设定确定的假设理想实验中定义。在控制工程和系统识别中，常规模型结构可识别性分析应用于由常微分方程 (ODE) 描述的动态模型。这种分析需要超出动物科学学术背景的数学技术，这可能解释了结构可识别性分析在动物科学建模中缺乏普遍性的原因。为了填补这一空白，本文从实践者的角度探讨了结构可识别性分析，利用了专用软件工具。我们的目标是：(i) 为动物科学建模界提供结构可识别性概念的全面解释；(ii) 评估可识别性分析在动物科学建模中的相关性；(iii) 激发动物科学建模界在建模实践中使用可识别性分析（当可识别性问题相关时）。我们专注于 ODE 模型。通过使用包括描述奶牛泌乳的已发表数学模型的说明性示例，我们展示了结构可识别性分析如何有助于推动动物科学中的数学建模朝着生产有用模型和通过最佳实验设计提供更具信息性的实验的方向发展。我们不希望在模型开发过程中试图将系统的可识别性分析强加给建模界，而是希望为模型构建和实验设计开辟一条发现强大工具的途径。
