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ODE models are ubiquitous in scientific applications, and form a cornerstone of the field of dynamical systems. Often, in applications, ODE models are derived using "rule of thumb" as this is more convenient for practitioners than deriving a mean field deterministic model from explicitly stochastic models. This can, unfortuantely, lead to oversimplified model assumptions. One way modelers have attempted to address this issue is to use classical linear chain trick (LCT), which I will review in this talk. I will also present our generalized linear chain trick (GLCT) which allows modelers to more easily derive systems of ODEs from first principles when framed using continuous time Markov chains. This approach also provides modelers with some additional advantages for interpreting analytical results for such models using stochastic processes theory.

I will illustrate the utility of the GLCT using an example application of this technique to derive, and then find reproduction numbers for, a generalized family of SEIRS models with an arbitrary number of state variables. Reproduction numbers, like the basic reproduction number $R_0$, play an important role in the analysis and application of dynamic models of contagion spread (and parallels exist elsewhere, e.g., in multispecies ecological models). One difficulty in deriving these quantities is that they typically are computed on a model-by-model basis, since it is impractical to obtain and interpret general reproduction number expressions applicable to a family of related models. This is especially true if these models are of different dimensions (i.e., differing numbers of state variables). I will show how a general reproduction number expression for this family of models can be found using the next generation operator approach in conjunction with the GLCT, and how the GLCT 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). These results highlight the utility of the GLCT for the derivation and analysis of mean field ODE models.

Zoom: https://ucdavis.zoom.us/j/92718892837