Estimating the transmission potential of supercritical processes based on the final size distribution of minor outbreaks

Use of the final size distribution of minor outbreaks for the estimation of the reproduction numbers of supercritical epidemic processes has yet to be considered. We used a branching process model to derive the final size distribution of minor outbreaks, assuming a reproduction number above unity, and applying the method to final size data for pneumonic plague. Pneumonic plague is a rare disease with only one documented major epidemic in a spatially limited setting. Because the final size distribution of a minor outbreak needs to be normalized by the probability of extinction, we assume that the dispersion parameter (k) of the negative-binomial offspring distribution is known, and examine the sensitivity of the reproduction number to variation in dispersion. Assuming a geometric offspring distribution with k=1, the reproduction number was estimated at 1.16 (95% confidence interval: 0.97-1.38). When less dispersed with k=2, the maximum likelihood estimate of the reproduction number was 1.14. These estimates agreed with those published from transmission network analysis, indicating that the human-to-human transmission potential of the pneumonic plague is not very high. Given only minor outbreaks, transmission potential is not sufficiently assessed by directly counting the number of offspring. Since the absence of a major epidemic does not guarantee a subcritical process, the proposed method allows us to conservatively regard epidemic data from minor outbreaks as supercritical, and yield estimates of threshold values above unity.