Stopping transmission of the smallpox (variola) virus amongst the human population was one of the greatest public health triumphs of the twentieth century; and yet since the events of 11 September 2001 the possibility of its reemergence has been under increased study . Effective contingency planning for such a scenario requires knowledge of the optimal deployment and use of case isolation, contact tracing and vaccination. Since a series of controlled experiments to inform such decisions is not possible, we are forced to rely on mathematical modelling to improve the evidence base for emergency preparedness and response. Recent simulation models [2–5] typically conclude that, provided a given smallpox outbreak can be well controlled, the emphasis should be on case isolation, contact tracing and targeted vaccination rather than immediate country-level mass vaccination, due to the frequency of adverse effects from vaccine and the likely high efficacy of other measures. Targeting of vaccination can either be exclusively towards suspected cases found by contact tracing, or can also involve a policy of local 'ring vaccination' of the population in the vicinity known cases. Evidence exists that such ring vaccination may provide additional benefit over individually targeted measures . However, two questions remain and form the main considerations of this paper: the exact conditions for which ring vaccination is optimal, and the scale at which such vaccination should take place--in particular we focus on the interaction between ring vaccination and contact tracing in a clustered network of contacts.
When considering optimal ring vaccination in any epidemiological context, the key issue is the spatial scale for intervention. This presents a technical challenge, since spatial models of disease transmission are typically much more complex than non-spatial models. Local vaccination is also not applied in isolation, but is deployed together with other control measures that must be captured by the model. One of the most widely used additional measures is contact tracing, which seeks to identify infected individuals before they become fully symptomatic by tracing the contacts (and therefore potential secondary cases) from each identified case. Theoretical work shows that failure to account for the underlying structure of the contact network when considering contact tracing can cause severe, qualitative errors in model results . We therefore find that our model choice is constrained by the epidemiology, dynamics, and types of control measure that we wish to capture.
Three main model types have been used to investigate the spread and control of smallpox . Mean-field models [, §8.3] have the advantage of being a relatively tractable set of differential equations, but suffer from the underlying assumption that transmission occurs at random within the entire population. Metapopulation models  are based on the observation that most infectious individuals create the majority of their secondary cases within their local environment; in these metapopulation models the population is therefore aggregated in local administration units with most transmission occurring within each unit and at random. Individual-based simulation models [2–4] are generally considered the most realistic and aim to capture the full contact structure between all individuals in the population, often assigning individuals to households, schools and workplaces where there is known to be a higher risk of transmission. Here we define a model that includes mixtures from all three of the above methods: it is based on differential equation and therefore is rapid to simulate allowing rigorous sensitivity analysis; it retains the metapopulation concept of local and longer-range transmission so can be used to consider regional (ring) vaccination; and it incorporates elements of local contact structure and can therefore reliably capture the effects of contact tracing.
Individual-based simulation can in principle incorporate any population structure and interventions necessary, and as such there will always be an important role in contingency planning for computationally intensive models that aim for maximum realism. Nevertheless, there are limitations to these approaches that motivate our methodology of developing a new, more parsimonious model. Most important for our purposes are the problems of parameterisation, numerical tractability and transparency. With such individual-based simulations there is always the temptation to include ever more complexity, however any model is necessarily a caricature of reality and so the quest for ever greater realism can never be fulfilled. While computers and algorithms continue to improve, increasing the number of individuals that can be directly simulated and decreasing the time to perform a simulation, considering the whole population of England, Wales and Scotland still involves significant computational resources. While baseline results can be obtained at these population sizes reasonably quickly, applications that require large number of model realisations such as comprehensive sensitivity analysis, real-time parameter fitting and determination of optimal strategies quickly become highly time-consuming. The development of models and methods to implement such applications remains necessary, however complementary approaches that are more abstract can significantly reduce the computational burden.
A model that incorporates synthetic data on individuals and their contacts almost by definition involves many more parameters than can be measured directly, and as such assumptions have to be made on the basis of available data. For example, if an explicit network is generated as in , then there is really a network parameter for every pair of individuals in the population modelled. Given the gap between available data on human contacts and the information contained in an explicit network, a complementary approach to individual-based simulation is to develop mathematical approaches that give epidemiological outputs based on statistical properties of contact networks. Such network properties can often be measured directly, and the uncertainty in them estimated systematically. One published methodology for this, applied to smallpox, is to use a branching-process model . Such an approach does not, however, allow the consideration of clustering--the possibility that an individual's contacts also contact each other. Work not directly related to smallpox, based on pairwise models, has shown that clustering has a major impact on transmission and tracing dynamics [6, 11, 12]. This leads us to extend pairwise models to enable smallpox to be modelled whilst considering clustering. A thorough treatment of this methodology is available in [, Chapter 7] and .