# Deriving and analyzing ODE models using the generalized linear chain trick

Speaker: Associate Professor Paul Hurtado

Affiliation: University of Nevada, Reno, USA

## Abstract

ODE models are widely used, and often derived or interpreted as a mean-field approximation of some (often unspecified) continuous-time stochastic model. Such ODE models often implicitly assume that, under the corresponding stochastic model, the time individuals spend in a given state is exponentially distributed. The linear chain trick (LCT) is a well-known technique for replacing exponentially distributed dwell times with Erlang distributions (i.e., gamma distributions with integer shape parameters). We have recently extended this technique beyond Erlang distributions to the much broader family of univariate, matrix exponential distributions known as phase-type distributions. These are the absorption time distributions for continuous-time Markov chains, and include exponential, Erlang, and Coxian distributions, among others. This generalized linear chain trick (GLCT) helps clarify connections between individual-level stochastic model assumptions and the structure of corresponding mean-field ODE models, and serves as a bridge allowing for the application of tools and concepts from Markov chain theory in the analysis and interpretation of mean-field ODE models. In this talk, I will (1) introduce the GLCT framework and some related concepts from Markov chain theory; (2) describe a practical procedure for using the GLCT to quickly generalize or approximate an existing ODE, DDE, or distributed delay equation model; and (3) illustrate some benefits of viewing ODE models from the perspective of the GLCT.

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