Unifying the effective reproduction number, incidence, and prevalence under a stochastic agedependent branching process
Abstract
Renewal equations are a popular approach used in modelling the number of new infections (incidence) in an outbreak. A unified set of renewal equations where incidence, prevalence and cumulative incidence can all be recovered from the same stochastic process has not been attempted. Here, we derive a set of renewal equations from an agedependent branching process with a timevarying reproduction number. Our new derivation utilises a measuretheoretic approach and yields a fully selfcontained mathematical exposition. We find that the renewal equations commonly used in epidemiology for modelling incidence are an equivalent special case of our equations. We show that these our equations are internally consistent in the sense that they can be separately linked under the common back calculation approach between prevalence and incidence. We introduce a computationally efficient discretisation scheme to solve these renewal equations, and this algorithm is highly parallelisable as it relies on row sums and elementwise multiplication. Finally we present a simple simulation example in the probabilistic programming language Stan where we jointly fit incidence and prevalence under a single timevarying reproduction number and generation interval.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.05579
 Bibcode:
 2021arXiv210705579B
 Keywords:

 Quantitative Biology  Populations and Evolution;
 Quantitative Biology  Quantitative Methods;
 Statistics  Applications