We treat a panmictic host population interacting with a virus. The virus is transmitted both horizontally and vertically. We modify the Moran model to describe the stochastic dynamics of individual host and viral lineages. For a sample of individuals from the population, the model gives rise to a branching and coalescing graph that contains the combined host and viral genealogies as a subgraph. The associated diffusion process, obtained in the limit of large host population, is related to the Neuhauser-Krone selection graph process.
We consider two study populations: cougars infected with FIV and a UK cohort of HIV patients. We fit the joint host-virus process to viral sequence data and known host pedigrees (which are trivial in the human case). We use MCMC to average over the variable dimension parameter space of labeled graphs.