  Estimation of epidemiological and population parameters from molecular
sequence data has become central to the understanding of infectious disease
dynamics. Various models have been proposed to infer details of the dynamics
that describe epidemic progression. These include inference approaches derived
from Kingman's coalescent theory. Here, we use recently described coalescent
theory for epidemic dynamics to develop stochastic and deterministic coalescent
SIR tree priors. We implement these in a Bayesian phylogenetic inference
framework to permit joint estimation of SIR epidemic parameters and the sample
genealogy. We assess the performance of the two coalescent models and also
juxtapose results obtained with BDSIR, a recently published
birth-death-sampling model for epidemic inference. Comparisons are made by
analyzing sets of genealogies simulated under precisely known epidemiological
parameters. Additionally, we analyze influenza A (H1N1) sequence data sampled
in the Canterbury region of New Zealand and HIV-1 sequence data obtained from
known UK infection clusters. We show that both coalescent SIR models are
effective at estimating epidemiological parameters from data with large
fundamental reproductive number $R_0$ and large population size $S_0$.
Furthermore, we find that the stochastic variant generally outperforms its
deterministic counterpart in terms of error, bias, and highest posterior
density coverage, particularly for smaller $R_0$ and $S_0$. However, each of
these inference models are shown to have undesirable properties in certain
circumstances, especially for epidemic outbreaks with $R_0$ close to one or
with small effective susceptible populations.
