Integrating dynamical systems models with time series data is a central part of contemporary mathematical biology. With the rich variety of available models and data, numerous methods and computational tools have been developed for these purposes. One such tool is Stan, a freely available and open-source probabilistic programming framework that provides efficient methods for estimating model parameters from data using computational Bayesian inference algorithms. Stan includes built-in mechanisms for working with ordinary differential equation (ODE) models, which are widely used in mathematical biology and related fields to study simulated, experimental, and real-world systems that change over time. Through step-by-step worked examples, including both pedagogical toy models and applications with real data, this article provides a practical, self-contained introduction to performing parameter estimation and model evaluation for first-order linear and nonlinear ODE models in Stan. The article also explains key statistical methods that underpin Stan and discusses computational Bayesian modelling in the context of biological applications.

翻译：将动力系统模型与时间序列数据整合是当代数学生物学的核心内容。面对种类繁多的模型与数据，学界已开发出大量方法与计算工具。Stan便是其中之一：作为免费开源的概率编程框架，它利用计算贝叶斯推断算法提供从数据中估计模型参数的高效方法。Stan内置处理常微分方程模型的功能——这类模型广泛用于数学生物学及相关领域，用于研究随时间演变的模拟、实验及真实系统。本文通过逐步演算实例（涵盖教学性玩具模型与真实数据应用），为在Stan中实现一阶线性和非线性ODE模型的参数估计与模型评估提供实用且自成体系的入门指南。文中还阐释了支撑Stan的关键统计方法，并结合生物学应用场景探讨计算贝叶斯建模。