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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 inderiving these quantities is that they must be computed on a model-by-modelbasis, since it is typically impractical to obtain general reproduction numberexpressions applicable to a family of related models, especially if these areof different dimensions. For example, this is typically the case for SIR-typeinfectious disease models derived using the linear chain trick (LCT). Here weshow how to find general reproduction number expressions for such modelsfamilies (which vary in their number of state variables) using the nextgeneration operator approach in conjunction with the generalized linear chaintrick (GLCT). We further show how the GLCT enables modelers to draw insightsfrom these results by leveraging theory and intuition from continuous timeMarkov chains (CTMCs) and their absorption time distributions (i.e., phase-typeprobability distributions). To do this, we first review the GLCT and otherconnections between mean-field ODE model assumptions, CTMCs, and phase-typedistributions. We then apply this technique to find reproduction numbers fortwo sets of models: a family of generalized SEIRS models of arbitrary finitedimension, and a generalized family of finite dimensional predator-prey(Rosenzweig-MacArthur type) models. These results highlight the utility of theGLCT for the derivation and analysis of mean field ODE models, especially whenused in conjunction with theory from CTMCs and their associated phase-typedistributions.

Finding Reproduction Numbers for Epidemic Models & Predator-Prey Models of Arbitrary Finite Dimension Using The Generalized Linear Chain Trick