The basic natural history of infection within our model is shown in Figure 1. At the centre of this are five disease states that smallpox cases pass through in the absence of any intervention. At first, individuals are susceptible (S), then following infection they are exposed (E) and asymptomatically incubate the infection. After the latent period, cases become prodromal (P) and exhibit non-specific influenza-like symptoms whilst being able to transmit infection, before quickly developing the characteristic rash of a fully infectious (I) case. Once recovered from infection, individuals play no further role in transmission and are removed (R) from the infection model, which can represent a range of outcomes. We assume that a proportion δ = 0.3 of cases die, and that others become fully recovered and immune.

The progression of disease following infection therefore involves three processes, which are assumed to happen at the following rates 141516:

When we come to consider transmission, the most important parameter is the basic reproductive ratio, R₀, which represents the expected number of secondary cases produced by a typical primary case early in the epidemic. Given that both prodromal and fully infectious individuals can transmit infection, we can split R₀ = R_P + R_I where R_P is the expected number of secondary cases caused during the prodromal stage and R_I the number during the fully infectious stage. Our default assumptions are that R₀ = 5, R_P = 0.1 × R₀, R_I = 0.9 × R₀ 141517.

The full network of human contacts capable of spreading infectious disease is undoubtedly complex, highly structured and dynamic. At the same time, there are many network statistics that are known to be epidemiologically important, however these are all based on the facts that transmission-relevant contacts are finite and non-random (captured by the mean number of contacts per individual, n) and that contacts are often shared by individuals (captured by the clustering coefficient ϕ). These two parameters are, therefore, an appropriate choice for our purposes and we vary them significantly during our analysis. We note that in many cases the heterogeneity in the number of contacts per individual can play an important role; but for the early stages of an outbreak considered here the effects of this heterogeneity can largely be subsumed into the parameterisation of R₀.

The mean number of contacts, n, has been estimated and we take it to be 17 people as a default 1819. The clustering coefficient ϕ, which is equal to the mean proportion of shared contacts between two linked individuals, has yet to be measured in an accurate way, and so our baseline value is ϕ = 0 for comparison with existing unclustered models. Since we know that there are many internally well-connected groups and cliques within human populations, such as households, schools and workplaces, it is most likely that ϕ is far from 0 and so large values are considered when we generalise.

Introducing a notation in which [S] represents the number of susceptible individuals in the population, and [SI] the number of contacts between a susceptible and an infectious individual (and similarly for other disease states) we can now introduce rates for the transmission of infection across the contact network, τ_P and τ_I, which are defined by

So prodromal individuals infect susceptible individuals that they are connected to at a rate τ_P and infectious individuals infect susceptible individuals that they are connected to at a rate τ_I .

If we were not considering a model with an explicit contact network, then the relationship between underlying transmission rates and observables can be dealt with in a rigorous way, such as in 20. On a network, however, the relationship between transmission rates and the reproduction numbers R₀, R_P, R_I discussed above is subtle, and the technical details of our approach are given in Additional file 1. The important conclusion from this technical development is that there are really two types of reproduction number that one should consider. The first, which we write r₀ (composed of r_I and r_P), relates to population-level measurements of incidence and prevalence and their early growth rate, while the second, which we write ℛ₀ (and is composed of ℛ_I and ℛ_P) relates to individual-level measurements made during contact tracing and is more true to the verbal definition. We consider both types of reproduction number in our analysis.

We now consider methods and parameters for the various interventions that are typically considered in smallpox contingency planning.

One of the main aims of our work is to consider the optimal spatial scale for intervention, and so here we present the data and modelling framework within which such assessments can be made.

Our results consider a six-month outbreak, centred on either a single region or number of regions. For each region, we are able to calculate an expected mortality from the final numbers vaccinated and removed, and the respective mortality risks δ_V and δ. We are also able to calculate a probability of escape from the region. Logistical and ethical factors become important in the consideration of which outcomes are most desirable, however, and so our approach is to consider a simply described optimal outcome within two release scenarios.

The model's dynamical behaviour, for all default choices in Table 1, is shown in Figure 4(a,b). The 'baseline' values of population size and outwardness are chosen for simple comparison with other published city-level and district-based models, which operate at a similar population size, but these parameters are varied everywhere else in our analysis. Figure 4(a) shows a rise in symptomatic, prodromal and latent infection until case isolation and other interventions are implemented (at approximated 7 days), when the epidemic becomes controlled. However, even after these measures are implemented, the number of latent and isolated individuals continues to rise for several days or weeks. Figure 4(b) shows the total escape probability rising monotonically over time, with a larger gradient at higher prevalence of infection, as would be expected.

Using the geographical framework discussed above, varying the vaccine mortality risk and mobility of fully symptomatic cases, but otherwise using the baseline parameters of Table 1, gives the results shown in Figure 4(c,d).

We find, as shown in Figure 4(c), that below a particular population size close to two million (if vaccine-induced mortality δ_V is 10 deaths per million vaccinated) local mass vaccination is optimal, with the preferred spatial scale being defined such that there is reasonable confidence that all existing cases are contained at that scale. Above this critical population size, the number of vaccine-induced deaths becomes unacceptably large. Our conclusion is that, even if vaccine-induced mortality is significantly lower, for the small release considered here vaccination of the whole country is likely to be unnecessary given that vaccination at smaller spatial scales and more individually targeted approaches will have brought the epidemic under control.

In terms of escape of infection, Figure 4(d) shows that both the reduced mobility of symptomatic cases and local mass vaccination have a significant impact on final probability. While this is not plotted for clarity, increasing κ still further does not have a large effect. The dominant trend is, in any case, that the outwardness of a region is the main determining factor of magnitude of escape probability.

To assess the impact of various parameters on the model conclusions, we consider the population size (N ≈ 2.4 × 10⁶) at which the expected number of deaths (given δ_V = 10^-5) is the same both with and without local mass vaccination. Using the interpolations shown in Figure 3(c,d), we can then see how modification of a baseline parameter moves us away from the point at which both strategies are equivalent. The results from this analysis are shown for within-region deaths in Figure 5 and for escape probabilities in Additional file 1.

When examining sensitivity to parameter choices, it is important to consider which basic observables of the epidemic should be maintained while the parameters are varied. Here, we have decided to fix the basic reproductive ratio as this is undoubtedly the key observation from any epidemic. For example, as we vary the number of contacts, the transmission rate per contact is varied to compensate for this change. This is analogous to ensuring that all models are fit to the same early epidemic behaviour. One difficulty with fitting to the basic reproductive ratio is that it can be measured in two main ways: either directly by examining the contacts of each infected case, or indirectly by calculating the early growth rate of the epidemic. Unfortunately, for epidemics with complex natural histories and network-based transmission these two methods of calculating the basic reproductive ratio are not in direct agreement, and we therefore show two sets of results in which either measurement is held constant. To make this distinction clear, we denote an individually measured reproduction number ℛ₀ and one inferred from early growth r₀.

We have found relatively small effects from varying the neighbourhood size n at constant R₀ (Figure 5(a)), although varying the reproductive ratio itself has the strong but predictable effect of favouring mass vaccination for larger values of R₀ (Figure 5(b)). What we find significantly more important is the amount of transmission during the prodromal phase of disease, parameterised by R_P (Figure 5(c)). While this is highly unlikely to be over 1, a similar effect would be seen from a delay in isolation of infectious individuals, which can be shown in a general manner using threshold arguments 22. Investigating the effects of clustering (Figure 5(d)) gives strong results, which are both subtle and counter-intuitive, although some theoretical progress has been made in understanding the impact of clustering on contact tracing 6. In particular, we see in Figure 5(d) that whether R₀ has been estimated from early growth in prevalence or from contact-tracing data has a profound effect on the impact of clustering.

The optimisation problem of minimising expected total mortality is addressed in Figure 6. This figure shows the existence of a spatial cutoff above which mass vaccination causes excess total mortality for our baseline parameters in Figure 6(a). Figure 6(b), however, shows that for significantly 'worse' parameters--R₀ = 7, R_P = 1.5, ϕ = 0.5--this conclusion can be reversed and no such cutoff is seen and mass vaccination is always optimal. Although we do not think such a scenario particularly likely, this points to the importance of real-time estimation in the event of an outbreak. On the other hand, the spatial cutoff scale is reduced for 'better' parameters--R₀ = 3, R_P = 0.1, ϕ = 0--as shown in Figure 6(c). When we consider the 'dispersed' assumption in Figure 6(d), delayed intervention causes a generally more severe outbreak but does not in itself remove the existence of a population size above which mass vaccination is suboptimal. Since the dispersed scenario has no interpretation when applied to the whole of Great Britain, we simply duplicate the NUTS1 result at this point.

There are a large number of tasks for realistic, informative smallpox modelling, and our particular focus has been on outbreak severity, the impact of contact-network structure, spatial scale and level of undetected transmission. It is worth considering briefly the likely impact of two other extensions to simple models known to be important for general policy conclusions.

Firstly, we have assumed simple rates of transmission between infectious states rather than a more sophisticated class-age structure. Our methodology allows for the inclusion of extra realism of this form through the method of stages, where clinically distinct disease states are broken down into additional sub-compartments. This brings significant extra computational overheads, however when considering a binary choice about vaccination the current understanding of such realism is that it should not qualitatively modify our results, and would be most important if our approach were extended to an accurate system of real-time estimation.

Secondly, we have assumed homogeneity at the individual level, while risks of death, mixing patterns and spatial location are likely to be highly heterogeneous. Again, given that we were only considering mean behaviour of the system this does not invalidate our approach, but would be important if policy decisions needed to be calibrated to 'reasonable worst case' rather than expected outcomes.