Generalizations of the \u2018Linear Chain Trick\u2019: incorporating more flexible dwell time distributions into mean field ODE models

Adam S Kirosingh,Paul J Hurtado

doi:10.1007/s00285-019-01412-w

Adam S Kirosingh, Paul J Hurtado

Open Access

https://doi.org/10.1007/s00285-019-01412-w

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Abstract

In this paper we generalize the Linear Chain Trick (LCT; aka the Gamma Chain Trick) to help provide modelers more flexibility to incorporate appropriate dwell time assumptions into mean field ODEs, and help clarify connections between individual-level stochastic model assumptions and the structure of corresponding mean field ODEs. The LCT is 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. Modelers must therefore 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 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 provide (1) novel LCT extensions for various scenarios found in applications, including conditional dwell time distributions; (2) formulations of these LCT extensions that bypass the need to derive ODEs from integral equations; and (3) a novel Generalized Linear Chain Trick (GLCT) framework that extends the LCT to a much broader set of possible dwell time distribution assumptions, including the flexible phase-type distributions which can approximate distributions on {mathbb {R}}^+ and can be fit to data.

Highlights

IntroductionDt where x(t) ∈ Rn, parameters θ ∈ Rp, and f : Rn → Rn is smooth
Many scientific applications involve systems that can be framed as continuous time state transition models, and these are often modeled using mean field ordinary differential equations (ODE) of the form dx = f (x, θ, t), (1)dt where x(t) ∈ Rn, parameters θ ∈ Rp, and f : Rn → Rn is smooth
X0 to intermediate sub-state XI, or should the XI dwell time be conditioned on time already spent in X0 so that the X0 →XI transition does not alter the overall dwell time in state X? How do these different assumptions alter the structure of the corresponding mean field ODEs? We answer these question in Sect. 3.6 where we describe how to apply the Linear Chain Trick (LCT) in scenarios with intermediate states, assuming in Sect. 3.6.1 that the dwell time distribution for XI is independent of the amount of time spent in X0, and assuming in Sect. 3.6.2 that the overall dwell time for X is unaffected by transitions from X0 to XI

Summary

Introduction

Dt where x(t) ∈ Rn, parameters θ ∈ Rp, and f : Rn → Rn is smooth The abundance of such applications, and the accessibility of ODE models in terms of the analytical techniques and computational tools for building, simulating, and analyzing ODE models [e.g., straightforward simulation methods and bifurcation analysis software like MatCont (Dhooge et al 2003) and XPPAUT (Ermentrout 1987, 2019)], have made ODEs one of the most popular modeling frameworks in scientific applications. This model can be thought of as the mean field model for some underlying stochastic state transition model where a large but finite number of individuals transition from state S to I to R [see the Appendix, Kermack and McKendrick (1927) for a derivation, and see Armbruster and Beck (2017), Banks et al (2013), and references therein for examples of the convergence of stochastic models to mean field ODEs]

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