Scale-Free Distribution of Local Infection Clusters of Measles, Rubella, Syphilis, and HIV/AIDS: Correlation with Scale-Free Distribution of Municipality Population by Size

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The size distribution of a local infection cluster (LIC), a group of infections reported from the same prefecture without interruption in successive weeks, was scale-free for infections that are transmitted from person-to-person (e.g., measles, rubella, syphilis, and HIV/AIDS). For infections that do not spread from person-to-person, the distribution was entirely random. The size distribution for measles, rubella, syphilis, and HIV/AIDS could be simulated successfully by random coin tossing with probabilities that were higher for highly populated prefectures. The size distribution of the population in large municipalities (>120,000), as well as that of LICs, was found to be scale free. As the number of patients per prefecture was correlated with the equation P = kNm, where m was 1.38 for syphilis, 1.63 for HIV/AIDS, and 2 for measles or rubella, the frequency distribution of N1.38, N1.63, and N2, where N was population of municipalities, was compared with the frequency distributions of LIC sizes of syphilis, HIV/AIDS, measles, and rubella. The frequency distribution of LICs, particularly those of measles and rubella during the years when the epidemic was more severe, was close to the frequency distribution of Nm. The analysis suggested that LICs were products of stochastic events under the influence of municipality population size.

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