  Stochastic models for collections 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. To handle that, over-dispersed Markov chains have previously been
constructed using gamma white noise on the rates. 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 graphical structure, for which ecological and
epidemiological models provide motivating examples. We demonstrate our approach
on a widely analyzed measles dataset, adding Dirichlet noise to a classical
SEIR (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.
