Building mean field ODE models using the generalized linear chain trick & Markov chain theory

The well-known Linear Chain Trick (LCT) allows modelers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. time spent in these states is the sum $k$ exponentially random variables, and thus distributed. Generalized (GLCT) extends this technique much broader phase-type family distributions, which includes exponential, Erlang, hypoexponential, Coxian distributions. Intuitively, distributions are absorption for continuous Markov chains (CTMCs). Here we review CTMCs then illustrate how use GLCT efficiently build ODE models from underlying stochastic model assumptions. We generalize Rosenzweig-MacArthur SEIR show benefits using compute numerical solutions. These results highlight some practical benefits, intuitive nature, first principles.