Related Experiment Video
Updated: Jun 7, 2026

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
Published on: September 26, 2016
Quantifying uncertainty, variability and likelihood for ordinary differential equation models
Andrea Y Weisse1, Richard H Middleton, Wilhelm Huisinga
1Hamilton Institute, National University of Ireland, Maynooth, Co, Kildare, Ireland. andrea.weisse@ed.ac.uk
Background:
In many applications, ordinary differential equation (ODE) models are subject to uncertainty or variability in initial conditions and parameters. Both, uncertainty and variability can be quantified in terms of a probability density function on the state and parameter space.
Results:
The partial differential equation that describes the evolution of this probability density function has a form that is particularly amenable to application of the well-known method of characteristics. The value of the density at some point in time is directly accessible by the solution of the original ODE extended by a single extra dimension (for the value of the density). This leads to simple methods for studying uncertainty, variability and likelihood, with significant advantages over more traditional Monte Carlo and related approaches especially when studying regions with low probability.
Conclusions:
While such approaches based on the method of characteristics are common practice in other disciplines, their advantages for the study of biological systems have so far remained unrecognized. Several examples illustrate performance and accuracy of the approach and its limitations.
Related Concept Videos
Modeling with Differential Equations
Uncertainty: Overview
Propagation of Uncertainty from Random Error
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
Introduction to Differential Equations
Mechanistic Models: Compartment Models in Individual and Population Analysis