  Reproduction numbers, like the basic reproduction number $\mathcal{R}_0$,
play an important role in the analysis and application of dynamic models,
including contagion models and ecological population models. One difficulty in
deriving these quantities is that they must be computed on a model-by-model
basis, since it is typically impractical to obtain general reproduction number
expressions applicable to a family of related models, especially if these are
of different dimensions. For example, this is typically the case for SIR-type
infectious disease models derived using the linear chain trick (LCT). Here we
show how to find general reproduction number expressions for such models
families (which vary in their number of state variables) using the next
generation operator approach in conjunction with the generalized linear chain
trick (GLCT). We further show how the GLCT enables modelers to draw insights
from these results by leveraging theory and intuition from continuous time
Markov chains (CTMCs) and their absorption time distributions (i.e., phase-type
probability distributions). To do this, we first review the GLCT and other
connections between mean-field ODE model assumptions, CTMCs, and phase-type
distributions. We then apply this technique to find reproduction numbers for
two sets of models: a family of generalized SEIRS models of arbitrary finite
dimension, and a generalized family of finite dimensional predator-prey
(Rosenzweig-MacArthur type) models. These results highlight the utility of the
GLCT for the derivation and analysis of mean field ODE models, especially when
used in conjunction with theory from CTMCs and their associated phase-type
distributions.
