  This paper is concerned with the analysis of vaccination strategies in a
stochastic SIR (susceptible $\to$ infected $\to$ removed) model for the spread
of an epidemic amongst a population of individuals with a random network of
social contacts that is also partitioned into households. Under various vaccine
action models, we consider both household-based vaccination schemes, in which
the way in which individuals are chosen for vaccination depends on the size of
the households in which they reside, and acquaintance vaccination, which
targets individuals of high degree in the social network. For both types of
vaccination scheme, assuming a large population with few initial infectives, we
derive a threshold parameter which determines whether or not a large outbreak
can occur and also the probability and fraction of the population infected by
such an outbreak. The performance of these schemes is studied numerically,
focusing on the influence of the household size distribution and the degree
distribution of the social network. We find that acquaintance vaccination can
significantly outperform the best household-based scheme if the degree
distribution of the social network is heavy-tailed. For household-based
schemes, when the vaccine coverage is insufficient to prevent a major outbreak
and the vaccine is imperfect, we find situations in which both the probability
and size of a major outbreak under the scheme which minimises the threshold
parameter are \emph{larger} than in the scheme which maximises the threshold
parameter.
