Building Mean Field State Transition Models Using The Generalized Linear Chain Trick and Continuous Time Markov Chain Theory

TL;DR

This paper introduces the Generalized Linear Chain Trick (GLCT), extending the classical LCT to phase-type distributions, enabling more flexible mean field ODE models derived from stochastic assumptions, with practical applications to ecological and epidemiological models.

Contribution

The paper develops the GLCT, broadening the LCT to phase-type distributions, and demonstrates its application in constructing and solving mean field ODE models from stochastic processes.

Findings

GLCT extends LCT to phase-type distributions.

Efficiently builds mean field ODE models from stochastic assumptions.

Improves numerical solutions for ecological and epidemiological models.

Abstract

The well-known Linear Chain Trick (LCT) allows modelers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these states is the sum of $k$ exponentially distributed random variables, and is thus gamma (Erlang) distributed. The Generalized Linear Chain Trick (GLCT) extends this technique to the much broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Intuitively, phase-type distributions are the absorption time distributions for continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build mean field ODE models from underlying stochastic model assumptions. We generalize the Rosenzweig-MacArthur and SEIR models and show the benefits of using the GLCT to compute numerical solutions. These results highlight some practical benefits, and the intuitive nature, of using the GLCT to derive ODE models from first principles.