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

Abstract

Epidemic threshold parameters, also called reproduction numbers, play a central role in computing the vaccination coverage required to prevent epidemics. 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 so-called general epidemic 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_I* = 1 – 1/R30, where R ₃₀ is one of these reproduction numbers. When a proportion of households is selected and every member of each selected household is immunized, then the proportion of households that needs to be immunized is v_H* = 1 – 1/R ₄₀where R ₄₀is another of the basic reproduction numbers. The result for v_H* applies for an arbitrary household distribution and a disease with an arbitrary infectivity function. However, the result for v_I* is more complicated for a community containing larger households.