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Open Access Highly Accessed Open Badges Methodology article

Quantifying uncertainty, variability and likelihood for ordinary differential equation models

Andrea Y Weiße123*, Richard H Middleton1 and Wilhelm Huisinga14*

Author Affiliations

1 Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland

2 Department of Mathematics & Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany

3 Centre for Systems Biology at Edinburgh, University of Edinburgh, Edinburgh EH9 3JD, UK

4 Institute of Mathematics, University of Potsdam, Am Neuen Palais 10, D-14469 Potsdam, Germany

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BMC Systems Biology 2010, 4:144  doi:10.1186/1752-0509-4-144

Published: 28 October 2010



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.


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.


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.