  We analyse a stochastic SIS epidemic amongst a finite population partitioned
into households. Since the population is finite, the epidemic will eventually
go extinct, i.e., have no more infectives in the population. We study the
effects of population size and within household transmission upon the time to
extinction. This is done through two approximations. The first approximation is
suitable for all levels of within household transmission and is based upon an
Ornstein-Uhlenbeck process approximation for the diseases fluctuations about an
endemic level relying on a large population. The second approximation is
suitable for high levels of within household transmission and approximates the
number of infectious households by a simple homogeneously mixing SIS model with
the households replaced by individuals. The analysis, supported by a simulation
study, shows that the mean time to extinction is minimized by moderate levels
of within household transmission.
