  Renewal equations are a popular approach used in modelling the number of new
infections, i.e., incidence, in an outbreak. We develop a stochastic model of
an outbreak based on a time-varying variant of the Crump-Mode-Jagers branching
process. This model accommodates a time-varying reproduction number and a
time-varying distribution for the generation interval. We then derive
renewal-like integral equations for incidence, cumulative incidence and
prevalence under this model. We show that the equations for incidence and
prevalence are consistent with the so-called back-calculation relationship. We
analyse two particular cases of these integral equations, one that arises from
a Bellman-Harris process and one that arises from an inhomogeneous Poisson
process model of transmission. We also show that the incidence integral
equations that arise from both of these specific models agree with the renewal
equation used ubiquitously in infectious disease modelling. We present a
numerical discretisation scheme to solve these equations, and use this scheme
to estimate rates of transmission from serological prevalence of SARS-CoV-2 in
the UK and historical incidence data on Influenza, Measles, SARS and Smallpox.
