  Inference of the reproduction number through time is of vital importance
during an epidemic outbreak. Typically, epidemiologists tackle this using
observed prevalence or incidence data. However, prevalence and incidence data
alone is often noisy or partial. Models can also have identifiability issues
with determining whether a large amount of a small epidemic or a small amount
of a large epidemic has been observed. Sequencing data however is becoming more
abundant, so approaches which can incorporate genetic data are an active area
of research. We propose using particle MCMC methods to infer the time-varying
reproduction number from a combination of prevalence data reported at a set of
discrete times and a dated phylogeny reconstructed from sequences. We validate
our approach on simulated epidemics with a variety of scenarios. We then apply
the method to a real data set of HIV-1 in North Carolina, USA, between 1957 and
2019.
