Systemic Infinitesimal Over-dispersion on General Stochastic Graphical Models

Stochastic models of interacting populations have crucial roles in scientific fields such as epidemiology and ecology, yet the standard approach to extending an ordinary differential equation model to a Markov chain does not have sufficient flexibility in the mean-variance relationship to match data (e.g. \cite{bjornstad2001noisy}). A previous theory on time-homogeneous dynamics over a single arrow by \cite{breto2011compound} showed how gamma white noise could be used to construct certain over-dispersed Markov chains, leading to widely used models (e.g. \cite{breto2009time,he2010plug}). In this paper, we define systemic infinitesimal over-dispersion, developing theory and methodology for general time-inhomogeneous stochastic graphical models. Our approach, based on Dirichlet noise, leads to a new class of Markov models over general direct graphs. It is compatible with modern likelihood-based inference methodologies (e.g. \cite{ionides2006inference,ionides2015inference,king2008inapparent}) and therefore we can assess how well the new models fit data. We demonstrate our methodology on a widely analyzed measles dataset, adding Dirichlet noise to a classical SEIR (Susceptible-Exposed-Infected-Recovered) model. We find that the proposed methodology has higher log-likelihood than the gamma white noise approach, and the resulting parameter estimations provide new insights into the over-dispersion of this biological system.