Negative binomial distribution

The negative binomial distribution is a discrete probability distribution that models the number of failures before a specified number of successes in a sequence of independent Bernoulli trials. Unlike the Poisson distribution — which assumes that the mean equals the variance — the negative binomial allows the variance to exceed the mean, a property called overdispersion that makes it indispensable for modeling count data in biology, epidemiology, and ecology.

In the context of superspreading, the negative binomial distribution replaces the Poisson assumption of the classical SIR model. Where the Poisson distribution predicts that most individuals produce roughly the average number of secondary infections, the negative binomial captures the reality that most individuals produce few or zero secondary infections while a small fraction produces many. The dispersion parameter k controls the skew: as k decreases, the distribution becomes increasingly heavy-tailed, and superspreading events become more likely.

The negative binomial is a member of the exponential family and can be derived as a gamma-Poisson mixture: a Poisson distribution whose rate parameter is itself drawn from a gamma distribution. This hierarchical structure provides a natural interpretation for overdispersion — the rate varies across individuals or events — and connects the distribution to generalized linear models and Bayesian hierarchical modeling.