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This study develops a theoretical framework for computing the final size distribution of multi-type Galton-Watson processes, which model branching phenomena like disease outbreaks. The authors demonstrate that the final size probability generating function can be reliably identified within specific mathematical contours, derive error bounds for these calculations, and establish connections between the differentiability properties of offspring and final size distributions. They validate their approach using both simulated data and real-world examples, while highlighting potential statistical inference challenges when likelihoods are not conditioned on extinction.
Why it matters
This work provides mathematically rigorous tools for computing epidemic final size distributions, which is crucial for understanding outbreak dynamics and predicting total case numbers in infectious disease modeling. The identification of potential statistical inference problems has important implications for improving the accuracy of epidemic predictions and public health decision-making.
⚠️ Preprint – Noch nicht peer-reviewed
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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.
Source: Computing the final epidemic size distributions of a multi-type Galton–Watson process