Add a beta-Poisson to the P(I):
https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1000399#pcbi.1000399-Holland1
Sze_comparison_Wells-Riley.pdf
https://www.medrxiv.org/content/10.1101/2023.03.24.23287701v1