ODE models (e.g., Lotka-Volterra and SIR models) are widely used to model the number of organisms or other entities as they enter, leave, and/or transition among various states. These models are often thought of as mean-field equations for underlying stochastic models, and as such they typically (implicitly) assume an individual’s dwell time in a given state is exponentially distributed (or more generally, follows a Poisson Process 1st event time distribution). ODEs have been criticized for this inability to incorporate other dwell-time distributions. One solution is the Linear Chain Trick (LCT; aka Gamma Chain Trick) for incorporating gamma distributed dwell-times, however there is no general statement of the LCT that allows its implementation in complex scenarios. Modelers must choose between (a) making overly strong simplifying assumptions, (b) abandoning ODEs and their associated computational and analytical tools for other frameworks like integrodifferential equations, and/or (c) they must derive ODEs from intermediate integrodifferential equations that are themselves derived from a stochastic model. This lack of tools for efficiently building ODEs with the desired dwell-time assumptions has led to widespread use of ODE models with overly-simple dwell-time distribution assumptions, sometimes to the detriment of our scientific understanding.
Results/Conclusions
We present novel generalizations of the Linear Chain Trick that (a) allow modelers to assume much more flexible dwell-time distributions than just exponential and gamma distributions, and (b) allow modelers to efficiently construct ODE models directly from explicit stochastic model assumptions, without deriving ODEs from intermediate integrodifferential equations models.
We also present novel extensions of the underlying mathematical machinery that allows for the derivation of ODEs from integrodifferential equations models, as well as a unifying Generalized Linear Chain Trick (GLCT) framework for constructing ODEs from a broad class of stochastic state transition models that includes various scenarios not addressed explicitly in our more specific results.
Our GLCT framework also builds nicely upon existing statistical methods that can be used to infer ODE model structures that approximate empirically derived (or named) dwell-time distributions.