Reproduction Numbers for ODE Models of Arbitrary Finite Dimension: An Application of the Generalized Linear Chain Trick

The Generalized Linear Chain Trick (GLCT) is a conceptually and practically useful approach for deriving mean field ODE models, since it describes how the structure of mean-field ODE models (and quantities like the basic reproduction number) reflect the assumptions of an often unspecified underlying continuous-time, stochastic state-transition model. In this talk, I will first describe how to generalize an existing ODE model -- such as the SEIR model or Rosenzweig-MacArthur consumer-resource model -- using the GLCT to incorporate non-exponential dwell times (e.g., latent periods in SEIR models, or predator maturation times in consumer-resource models) that are Erlang distributed or, more generally, are phase-type distributed. The phase-type family of distributions are the absorption time distributions for continuous time Markov chains, and include exponential, Erlang, generalized Erlang, and Coxian distributions. Second, I will show how the structure of the resulting ODE model, which is of arbitrary finite dimension, can be exploited to obtain a general expression for the (basic) reproduction number. These results illustrate the utility of the GLCT, not just for model derivation, but also for model analysis and interpretation.