Computing the final epidemic size distributions of a multi-type Galton--Watson process

Abstract

We provide a theoretical ground to directly compute the final size distribution of a Galton--Watson process (GWP) from the implicit fixed point equations for its probability generating function (PGF). Firstly we show that valid identification of the final size PGF is ensured on any contour inside the unit polydisc. Furthermore, we derive error bounds for the PGF values on the contour, and establish the relationship between the differentiability of the offspring PGF and the final size PGF as a function of parameters that control the offspring distribution. Finally, we provide demonstrations using simulated and real-world data, and discuss potential identifiability issues regarding statistical inference that is associated with likelihoods that are not conditioned on extinction.

arXiv commentSubmitted; under review