Optimal vaccination schemes for epidemics among a population of households, with application to variola minor in Brazil

This paper is concerned with stochastic models for the spread of an epidemic among a community of households, in which individuals mix uniformly within households and, in addition, uniformly at a much lower rate within the Population at large. This two-level mixing structure has important implications for the threshold behaviour of the epidemic and, consequently, for both the effectiveness of vaccination strategies for controlling an outbreak and the form of optimal vaccination schemes. A brief introduction to optimal vaccination schemes in this setting is provided by presenting a unified treatment of the simplest and most-studied case, viz. the single-type SIR (susceptible -> infective -> removed) epidemic. A reproduction number R*(,) which determines whether a trace of initial infection can give rise to a major epidemic, is derived and the effect of a vaccination scheme on R* is studied using a general model for vaccine action. In particular, optimal vaccination schemes which reduce R* to its threshold value of one with minimum vaccination coverage are considered. The theory is illustrated by application to data on a variola minor outbreak in Sao Paulo, which, together with other examples, is used to highlight key issues related to vaccination schemes.