Reproduction Numbers and Critical Immunity Levels for Epidemics in a Community of Households

Abstract

Epidemie threshold parameters, also called reproduction numbers, play a central role in computing the vaccinatlon coverage required to prevent epidemies. It is possible to define several different reproduction numbers for infectives in a community of households. To illustrate this we compute four distinct reproduction numbers for infectives for a community consisting of a large number of households of size three, using assumptions similar to the socalled general epidemie model. It is found that when individuals are selected independently for immunization the proportion that needs to be immunized so as to prevent epidemics is v = 1 - 1/R_30), where R_30 is one of these reproduction numbers. When a proportion of households is seleeted and every member of each selected household is imrnunized, then the proportion of households that needs to be immunized is v = 1 - 1/R_40, where R_40 is another of the basie reproduction numbers. The result for v applies for an arbitrary household distribution and a disease with an arbitrary infectivity function. However, the result for v is more complicated for a community containing larger households.