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This paper is concerned with the analysis of vaccination strategies in a stochastic SIR (susceptible ? infected ? 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.

This paper is concerned with SIR (susceptible--infected--removed) household epidemic models in which the infection response may be either mild or severe, with the type of response also affecting the infectiousness of an individual. Two different models are analysed. In the first model, the infection status of an individual is predetermined, perhaps due to partial immunity, and in the second, the infection status of an individual depends on the infection status of its infector and on whether the individual was infected by a within- or between-household contact. The first scenario may be modelled using a multitype household epidemic model, and the second scenario by a model we denote by the infector-dependent-severity household epidemic model. Large population results of the two models are derived, with the focus being on the distribution of the total numbers of mild and severe cases in a typical household, of any given size, in the event that the epidemic becomes established. The aim of the paper is to investigate whether it is possible to determine which of the two underlying explanations is causing the varying response when given final size household outbreak data containing mild and severe cases. We conduct numerical studies which show that, given data on sufficiently many households, it is generally possible to discriminate between the two models by comparing the Kullback-Leibler divergence for the two fitted models to these data.

This paper considers a stochastic SIR (susceptible-infective-removed) epidemic model in which individuals may make infectious contacts in two ways, both within 'households' (which for ease of exposition are assumed to have equal size) and along the edges of a random graph describing additional social contacts. Heuristically-motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak, given few initial infectives, and the expected proportion of the population who are ultimately infected by such a major outbreak. These approximate results are shown to be exact as the number of households tends to infinity by proving associated limit theorems. Moreover, simulation studies indicate that these asymptotic results provide good approximations for modestly-sized finite populations. The extension to unequal sized households is discussed briefly.

We consider a stochastic SIR (susceptible → infective → removed) epidemic model with several types of individuals. Infectious individuals can make infectious contacts on two levels, within their own ‘household’ and with their neighbours in a random graph representing additional social contacts. This random graph is an extension of the well-known configuration model to allow for several types of individuals. We give a strong approximation theorem which leads to a threshold theorem for the epidemic model and a method for calculating the probability of a major outbreak given few initial infectives. A multitype analogue of a theorem of Ball et al. (2009) heuristically motivates a method for calculating the expected size of such a major outbreak. We also consider vaccination and give some short numerical illustrations of our results.