Real-time estimation of the reproduction number has become the focus of
modelling groups around the world as the SARS-CoV-2 pandemic unfolds.
One of the most widely adopted means of inference of the reproduction
number is via the renewal equation, which uses the incidence of infection
and the generation time distribution. In this paper, we derive a multi-type
equivalent to the renewal equation to estimate a reproduction number
which accounts for heterogeneity in transmissibility including through
asymptomatic transmission, symptomatic isolation and vaccination. We
demonstrate how use of the renewal equation that misses these heterogeneities
can result in biased estimates of the reproduction number. While the
bias is small with symptomatic isolation, it can be much larger with asymptomatic
transmission or transmission from vaccinated individuals if these
groups exhibit substantially different generation time distributions to unvaccinated
symptomatic transmitters, whose generation time distribution is
often well defined. The bias in estimate becomes larger with greater population
size or transmissibility of the poorly characterized group. We apply
our methodology to Ebola in West Africa in 2014 and the SARS-CoV-2 in
the UK in 2020–2021.