  Renewal processes are a popular approach used in modelling infectious disease
outbreaks. In a renewal process, previous infections give rise to future
infections. However, while this formulation seems sensible, its application to
infectious disease can be difficult to justify from first principles. It has
been shown from the seminal work of Bellman and Harris that the renewal
equation arises as the expectation of an age-dependent branching process. In
this paper we provide a detailed derivation of the original Bellman Harris
process. We introduce generalisations, that allow for time-varying reproduction
numbers and the accounting of exogenous events, such as importations. We show
how inference on the renewal equation is easy to accomplish within a Bayesian
hierarchical framework. Using off the shelf MCMC packages, we fit to South
Korea COVID-19 case data to estimate reproduction numbers and importations. Our
derivation provides the mathematical fundamentals and assumptions underpinning
the use of the renewal equation for modelling outbreaks.
