Title: Generalizations of the 'Linear Chain Trick': Incorporating more flexible dwell time distributions into mean field ODE models,
Abstract: Mathematical modelers have long known of a "rule of thumb" referred to as the
Linear Chain Trick (LCT; aka the Gamma Chain Trick): a technique used to
construct mean field ODE models from continuous-time stochastic state
transition models where the time an individual spends in a given state (i.e.,
the dwell time) is Erlang distributed (i.e., gamma distributed with integer
shape parameter). Despite the LCT's widespread use, we lack general theory to
facilitate the easy application of this technique, especially for complex
models. This has forced modelers to choose between constructing ODE models
using heuristics with oversimplified dwell time assumptions, using time
consuming derivations from first principles, or to instead use non-ODE models
(like integro-differential equations or delay differential equations) which can
be cumbersome to derive and analyze. Here, we provide analytical results that
enable modelers to more efficiently construct ODE models using the LCT or
related extensions. Specifically, we 1) provide novel extensions of the LCT to
various scenarios found in applications; 2) provide formulations of the LCT and
it's extensions that bypass the need to derive ODEs from integral or stochastic
model equations; and 3) introduce a novel Generalized Linear Chain Trick (GLCT)
framework that extends the LCT to a much broader family of distributions,
including the flexible phase-type distributions which can approximate
distributions on $\mathbb{R}^+$ and be fit to data. These results give modelers
more flexibility to incorporate appropriate dwell time assumptions into mean
field ODEs, including conditional dwell time distributions, and these results
help clarify connections between individual-level stochastic model assumptions
and the structure of corresponding mean field ODEs.