Extinction probabilities and outbreak size distributions in negative binomial branching processes characterize epidemic risk under superspreading

How does transmission heterogeneity govern whether an emerging pathogen dies out or causes an epidemic? In negative binomial branching processes at reproduction number 2.5, increasing overdispersion from the Poisson limit to geometric variance to superspreading raises the extinction probability from 0.107355 to 0.4 to 0.860685. Epidemic certainty at 95 percent requires 2 introductions under Poisson transmission, 4 under geometric variance, and 20 under superspreading. Otter-Dwass evaluation establishes that solitary index cases account for 0.721943 extinction probability, or percent of extinctions. Cluster size summation through cases recovers the extinction probability with an absolute residual of , confirmed across simulations. Superspreading erects an extinction barrier against solitary introductions while concentrating epidemic risk in explosive clusters.