Measles and rubella: Scale free distribution of local infection clusters

Abstract

This study examined the size distribution of local infection clusters (referred to as clusters hereafter) of measles and rubella from 2008–2013 in Japan. When the logarithm of the cluster sizes were plotted on the x-axis and the logarithm of their frequencies were plotted on the y-axis, the plots fell on a rightward descending straight line. The size distribution was observed to follow a power law. As the size distribution of the clusters could be equated with that of local secondary infections initiated by 1 patient, the size distribution of the clusters, in fact, represented the effective reproduction numbers at the local level. As the power law distribution has no typical sizes, it was suggested that measles or rubella epidemics in Japan had no typical reproduction number. Higher the population size and higher the total number of patients, flatter was the slope of the plots, thus larger was the proportion of larger clusters. An epidemic of measles or rubella in Japan could be represented more appropriately by the cluster size frequency distribution rather than by the reproduction number.