Systemic infinitesimal over-dispersion on graphical dynamic models

Stochastic models for collections of interacting populations have crucial roles in many scientific fields such as epidemiology, ecology, performance engineering, and queueing theory, to name a few. However, 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. To handle that, we develop new approaches using Dirichlet noise to construct collections of independent or dependent noise processes. This permits the modeling of high-frequency variation in transition rates both within and between the populations under study. Our theory is developed in a general framework of time-inhomogeneous Markov processes equipped with a general graphical structure. We demonstrate our approach on a widely analyzed measles dataset, adding Dirichlet noise to a classical Susceptible–Exposed–Infected–Recovered model. Our methodology shows improved statistical fit measured by log-likelihood and provides new insights into the dynamics of this biological system.