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

  • Yuta Okada
  • Hiroshi Nishiura
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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.

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Cite

BibTeXRIS
BibTeX
@misc{okada2026computing,
  title = {{Computing the final epidemic size distributions of a multi-type Galton--Watson process}},
  author = {Yuta Okada and Hiroshi Nishiura},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.03007},
  eprint = {2606.03007},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.03007},
}
RIS
TY  - UNPB
TI  - Computing the final epidemic size distributions of a multi-type Galton--Watson process
AU  - Yuta Okada
AU  - Hiroshi Nishiura
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2606.03007
UR  - https://arxiv.org/abs/2606.03007
AB  - 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.
ER  -