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

Reproduction numbers, like the basic reproduction number $R_0$, 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 not practical to obtain a general reproduction number expressions applicable to a family of related models, especially if some are of different dimensions. For example, this is the case for SIR-type infectious disease models derived using the linear chain trick (LCT). Here we show how the next generation operator approach for computing reproduction numbers can be used in conjunction with the generalized linear chain trick (GLCT) to find reproduction numbers expressions that apply generally to such families of models of varying dimensions. We further show how the GLCT enables modelers to draw insights from these results by using theory and intuition from continuous time Markov chains (CTMCs) and their absorption time distributions (i.e., the 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 a generalized family of SEIRS models of arbitrary finite dimension, and to a generalized family of predator-prey (Rosenzweig-MacArthur) models of arbitrary finite dimension. 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.