Generalizations of the ‘Linear Chain Trick’: incorporating more flexible dwell time distributions into mean field ODE models

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 and can be fit to data.

Generalized differential equation compartmental models of infectious disease transmission

“…The requirement that the duration of infection follows an exponential distribution is often a source of weakness with regards to the biological validity of differential equation compartmental models. While the extension of such models through the linear chain trick [9,31,32] to a duration of infection that follows the Erlang distribution alleviates this weakness to some degree, it does so at the cost of inflating the size of the compartmental model, and thereby increasing the computational complexity of the system. Our new class of models avoids this inflation, while retaining the benefits of having a distribution of infection that follows an Erlang distribution.…”

“…Subsequently by applying the 'linear chain trick' [9,31,32] (Supplemental Materials) to the system of three equations, we can illustrate the effects of a duration of infection that is Erlang distributed:…”

“…Next, we consider our model with a duration of infection that follows an Erlang distribution. We illustrate how this choice of distribution has a representation as either an ODE system of 3 equations, regardless of the Erlang distribution parameters, or as an ODE system that features a chain of infected individual equations, as is typical from the linear chain trick [9,31,32]. Finally, we consider a duration of infection that is Pearson distributed.…”

Modeling the efficiency of filovirus entry into cells in vitro: Effects of SNP mutations in the receptor molecule

“…The linear chain trick is method to change from continuous-time stochastic state transition model in which an individual's time spent in a given state lasts to ODE models [26]. This method allows us to only represent not delayed infectious phase state but also applicable for numerical simulation [18,27,28].…”

A dynamically-structured matrix population model based on renewal processes for accumulative development under variable environmental conditions

“…Numerous methods have been proposed to represent developmental heterogeneity. According to the way they structure a population, two main categories emerge: age-structured (1, 8–10) and pseudo-stage-structured models (4, 8, 11, 12).…”