Generalizations of the 'Linear Chain Trick': Incorporating more flexible dwell time distributions into mean field ODE models

TL;DR

This paper extends the Linear Chain Trick (LCT) to broader distributions, including phase-type, enabling more flexible and accurate mean field ODE models for stochastic processes with diverse dwell time distributions.

Contribution

It introduces a Generalized Linear Chain Trick (GLCT) framework that broadens the applicability of LCT to complex, data-fitting distributions, simplifying model derivation.

Findings

Extended LCT to various application scenarios.

Formulations that avoid integral or stochastic derivations.

GLCT framework for phase-type distributions.

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.