A practical introduction to ODE modelling in Stan for biological systems

2026-03-20

AI Summary

This study addresses the challenge of parameter estimation and model evaluation for ordinary differential equation (ODE) models in biological systems when combined with time-series data. It proposes a comprehensive Bayesian inference framework implemented in the Stan probabilistic programming language, which seamlessly integrates numerical ODE solvers with Markov chain Monte Carlo (MCMC) sampling. The approach efficiently fits both linear and nonlinear first-order ODE models, as demonstrated through systematic experiments on pedagogical toy examples and real-world biological datasets. Results show that the framework is effective and robust across both synthetic and empirical data, substantially lowering the technical barrier for interdisciplinary researchers seeking to apply Bayesian ODE modeling. This work thus establishes a reproducible and extensible practical paradigm for analyzing dynamical systems in biology.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Bayesian Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

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Abstract

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.

Problem

• ODE modelling
• parameter estimation
• time series data
• mathematical biology
• model evaluation

Innovation

• ODE modeling
• Stan
• Bayesian inference
• parameter estimation
• mathematical biology