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

Abstract

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 impractical to obtain general reproduction number expressions applicable to a family of related models, especially if these are of different dimensions (i.e., differing numbers of state variables). For example, this is typically the case for SIR-type infectious disease models derived using the linear chain trick. Here we show how to find general reproduction number expressions for such model 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.

Keywords: Coxian distribution; Erlang distribution; Gamma chain trick; Linear chain trick; Phase-type distribution.

Fig. 1

Diagrams for SEIR-type models with substate structures resulting from the application of the generalized (and classic) linear chain trick. (a) An SEIR model with Erlang latent period distribution and a Coxian infectious period distribution (see eq. (6) in section 2.1); (b) An SEIRD model with Erlang latent period and an infectious period that is a mixture of two Erlang distributions to model two different courses of infection (e.g., symptomatic and asymptomatic individuals) with potential for recovery or death; and (c) An SEIR type model used to model the COVID-19 epidemic by the Center for the Ecology of Infectious Diseases at the University of Georgia (Drake et al 2020), which models asymptomatic, symptomatic undetected and symptomatic detected tracks, and hospitalizations, within the infectious state