Warwick Mathematics Institute and Department of Biological Sciences, University of Warwick, Coventry, UK

Microbial Risk Assessment, Health Protection Agency, Emergency Response Department, Porton Down, Wiltshire, UK

Abstract

Background

In the event of a release of a pathogen such as smallpox, which is human-to-human transmissible and has high associated mortality, a key question is how best to deploy containment and control strategies. Given the general uncertainty surrounding this issue, mathematical modelling has played an important role in informing the likely optimal response, in particular defining the conditions under which mass-vaccination would be appropriate. In this paper, we consider two key questions currently unanswered in the literature: firstly, what is the optimal spatial scale for intervention; and secondly, how sensitive are results to the modelling assumptions made about the pattern of human contacts?

Methods

Here we develop a novel mathematical model for smallpox that incorporates both information on individual contact structure (which is important if the effects of contact tracing are to be captured accurately) and large-scale patterns of movement across a range of spatial scales in Great Britain.

Results

Analysis of this model confirms previous work suggesting that a locally targeted 'ring' vaccination strategy is optimal, and that this conclusion is actually quite robust for different socio-demographic and epidemiological assumptions.

Conclusions

Our method allows for intuitive understanding of the reasons why national mass vaccination is typically predicted to be suboptimal. As such, we present a general framework for fast calculation of expected outcomes during the attempted control of diverse emerging infections; this is particularly important given that parameters would need to be interactively estimated and modelled in any release scenario.

Background

Stopping transmission of the smallpox (

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

Three main model types have been used to investigate the spread and control of smallpox

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

Methods

For reasons explained in the introduction above, we chose a modelling approach that uses mathematical techniques to reduce the computational burden. This means that the underlying model has simple underlying assumptions and relatively few parameters, which we outline below, but ultimately rests on specialist mathematical results, which we have included in Additional file

**Supplementary Material**. We have included a supplementary PDF containing mathematical and simulation results necessary to reproduce our work but not necessary for the main thrust of argument in the paper. This can be viewed in a free viewer such as Adobe Acrobat Reader.

Click here for file

Model parameters, together with baseline value and range if varied during analysis.

**Parameter**

**Description**

**Value (Range)**

**Refs**

_{
E
}

Rate of transition from latent to prodromal

1/12 days^{-1}

_{
P
}

Rate of transition from prodromal to infectious

1/2.5 days^{-1}

_{
I
}

Rate of transition from infectious to removed

1/8.6 days^{-1}

Case fatality risk

30%

Probability of case isolation success

90%

_{
Q
}

Rate of transition from isolated to removed

1/20 days^{-1}

ϵ

Probability of contact tracing success

80%

ϵ_{1}

Vaccine efficacy when susceptible

97.5%

ϵ_{2}

Vaccine efficacy when latent

30%

_{
O
}

Rate of transition from observed to removed or vaccinated

1/15 days^{-1}

γ

Proportion of population contraindicated for mass vaccination

30%

_{0}

Number of index cases

10

see text

Number of index cases for secondary outbreak

1

see text

_{trig}

Number of clinical cases prior to detection

4

see text

Number of clinical cases prior to detection for secondary outbreak

1

see text

_{
V
}

Vaccine fatality risk

10^{-5}(0 -- 10^{-5})

_{0}

Basic reproductive ratio

5 (3 -- 7)

_{
P
}

Prodromal type reproductive ratio

0.5 (0.1 -- 1.5)

Movement reductions when infectious

0.9 (0 -- 1)

Rate of mass vaccination

see text

Population size of region

10^{5}(see Figure 3(a))

Region outwardness

0.4 (see Figure 3(d))

Number of regions of same type as outbreak region

n/a (see Figure 3(a))

_{0}

Number of regions at relevant scale initially infected

1 (1 -- 10)

see text

Neighbourhood size (contacts per individual)

17 (5 -- 50)

Clustering coefficient

0 (0 -- 0.5)

see text

Either references are given, or the parameter value is discussed in the main text. Values referenced to

Smallpox natural history

The basic natural history of infection within our model is shown in Figure

Disease states of the model, and processes connecting them

**Disease states of the model, and processes connecting them**. These are: ^{S}^{E}

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

When we come to consider transmission, the most important parameter is the basic reproductive ratio, _{0}, 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 _{0 }= _{P }+ _{I }where _{P }is the expected number of secondary cases caused during the prodromal stage and _{I }the number during the fully infectious stage. Our default assumptions are that _{0 }= 5, _{P }= 0.1 × _{0}, _{I }= 0.9 × _{0 }

Contact network structure

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, _{0}.

The mean number of contacts,

Introducing a notation in which [_{P }_{I}

So prodromal individuals infect susceptible individuals that they are connected to at a rate _{P }_{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 _{0}, _{P}, _{I }discussed above is subtle, and the technical details of our approach are given in Additional file _{0 }(composed of _{I }and _{P}), relates to population-level measurements of incidence and prevalence and their early growth rate, while the second, which we write ℛ_{0 }(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.

Control strategies

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

Case isolation

The first step in outbreak containment is to isolate symptomatic cases as soon as possible, to stop transmission of disease. We model this by taking a proportion _{Q}= (1/20) days^{-1 }when they are assumed to pose no further risk of transmission _{P }+ 0.1 × _{I }= 1 under our baseline assumptions, which is consistent with existing work and expert opinion

Contact tracing

Contact tracing of known (and hence isolated) cases takes place through attempts to find the individuals potentially infected through epidemiologically relevant contacts. Some individuals found in this way will still be susceptible, some will be latent cases, and others will have developed symptoms. We assume that traced individuals displaying both rash and less specific prodromal fever are isolated an enter the quarantined _{T }.

Tracing takes place across network links in a manner analogous to infection, and this is the reason why models that include tracing must explicitly include network structure. We therefore formulate the model rates in the same manner as for transmission of infection above:

For a given isolated individual who is subject to contact tracing, we assume that each contact in the transmission network is discovered independently with a probability of ∈ = _{Q}) = 0.8

Vaccination

Smallpox is a vaccine-preventable illness, meaning that susceptible individuals who are successfully vaccinated acquire long-lasting immunity, and in the model are placed in the class of individuals removed due to vaccination, _{1 }= 0.975 _{2 }= 0.3 _{V }= 10

The rate _{O }= (1/15) days^{-1 }is given by the length of the observation period for observed suspected cases

When we also consider mass vaccination, then we need to introduce the class _{V}, which represents those susceptibles not contraindicated for vaccination (which we take to be around 70% of the population

where ^{-1 }.

Spatial scale

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.

The NUTS classification

We make use of a hierarchy of statistical units in our analysis. These start at Great Britain (GB), and work down through five Nomenclature of Units for Territorial Statistics (NUTS) levels to the whole of Great Britain. The definition of NUTS regions as currently used by the Office for National Statistics is, at the time of writing, available online at

Range of geographical units around Output Area 00CQFP0009, which contains the main campus of the University of Warwick

**Range of geographical units around Output Area 00CQFP0009, which contains the main campus of the University of Warwick**. Numbers shown are distances in kilometres

We then make use of data on population sizes and commuter movements from the 2001 census

Characteristics of seven spatial scales in England, Wales and Scotland (Great Britain) and inferred outwardness and time to vaccinate as a function of region population

**Characteristics of seven spatial scales in England, Wales and Scotland (Great Britain) and inferred outwardness and time to vaccinate as a function of region population**. (a) shows the median and variability in population size at each scale, together with the number of regions. (b) shows the commuting data at each scale, (c) shows the inferred time to vaccinate against population size, and (d) the inferred outwardness against population size.

Ring vaccination

As in the section above on vaccination, the Department of Health provided us with an estimate of the time to vaccinate a district as three days. We then extrapolate to Figure

Interaction between regions

Our approach to spatial interaction between regions considers disease escape from the region around the initial cases through a tractable model known as a

In order to make use of this method, we need to have a measure of the propensity of a region's resident population to leave the area (and hence transfer infection) on a day to day basis. To our knowledge, there is no technical term for this value; we use the term

The rate at which infection escapes from the region around the initial cases is therefore proportional to the outwardness, and depends on the number if prodromal cases [

Here, we parameterise the significantly reduced movement of symptomatic infectious individuals through multiplying their probability of travel by a factor (1 -

Optimisation of response

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 _{V }and

Initial cases and detection threshold

Our first assumption concerning initial cases is that they are concentrated, and the epidemic starts with _{0 }= 10 latent cases in one area. While in principle this could be significantly higher, the more severe epidemic arising from such a situation may favour mass vaccination as an intervention. We therefore choose to consider an initial outbreak size where there is the strong possibility of containment through case isolation and contact tracing alone.

We assume that until a certain number of fully symptomatic cases has been seen and diagnosed, no public response is instigated. The triggering of interventions is modelled by introducing control measures to the system once [_{trig}. We take _{trig}= 4, for the same reason as our relatively conservative choice of _{0 }above.

In the event that secondary areas are infected, we assume that in a situation of heightened awareness and public health response, the secondary epidemic is initialised by

An alternative scenario is to consider _{0 }= 10 maximally dispersed initial cases, each in a distinct region of the size under consideration. In this case, we let the initial regions take 'secondary' parameters; since the outbreaks are simultaneous, this means that the expected national prevalence of clinical infection upon triggering of intervention will be _{0}. This is reasonable since dispersed cases are collectively less likely to trigger intervention.

In terms of interpretation, our baseline assumption for regions of relative small sizes would correspond to an accidental/intentional release from a terror cell operating from a domestic address, and at larger sizes a release involving the resident population of a city or region. The dispersed assumption would, on the other hand, involve the other extreme of a release involving a largely transient population of commuters or shoppers. We note that at country level (Great Britain) the dispersed scenario should be interpreted as identical to the NUTS1 level dispersed scenario.

Minimisation of expected mortality

The decision about what constitutes an optimal outcome is essentially political, and so our focus is on how epidemiological outputs vary with spatial scale and contact network structure. Nevertheless, we also wish to demonstrate how optimisation can be considered and so also calculate the total expected mortality in primary and secondary affected regions as a simply described and calculated quantity for optimisation. The full details of calculation of this quantity are in Additional file

Results and Discussion

Temporal dynamics

The model's dynamical behaviour, for all default choices in Table

Baseline dynamics for network smallpox model

**Baseline dynamics for network smallpox model**. The dynamical variables associated with a free-fall epidemic are shown growing in Pane (a), until the trigger is reached and intervention parameters come to dominate and the outbreak is eventually contained. (b) the cumulative probability of escape grows with untreated cases then levels off. Overall outcomes for the seven spatial scales in England, Wales and Scotland are shown in Panes (c) and (d).

Spatial scale

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

We find, as shown in Figure _{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

Parameter sensitivity

To assess the impact of various parameters on the model conclusions, we consider the population size (^{6}) at which the expected number of deaths (given _{V }= 10^{-5}) is the same both with and without local mass vaccination. Using the interpolations shown in Figure

Effects of varying network and disease parameters on expected deaths

**Effects of varying network and disease parameters on expected deaths**. The effects are shown of modifying (a) neighbourhood size _{0}, (c) prodromal transmission _{P}, and (d) clustering

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 ℛ_{0 }and one inferred from early growth _{0}.

We have found relatively small effects from varying the neighbourhood size _{0 }(Figure _{0 }(Figure _{P }(Figure _{0 }has been estimated from early growth in prevalence or from contact-tracing data has a profound effect on the impact of clustering.

Optimisation and global sensitivity

The optimisation problem of minimising expected total mortality is addressed in Figure _{0 }= 7, _{P }= 1.5, _{0 }= 3, _{P }= 0.1,

Outcomes combined into a trade-off

**Outcomes combined into a trade-off**. (a) predicted full deaths for baseline scenario with all cases contained inside one region. (b) 'worst' parameters cause the outbreak to be uncontrolled by targeted interventions, leading to preference for national mass vaccination (the massive increase in overall mortality in this case necessitates a different

Parameters not varied

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.

Conclusions

In dealing with the optimisation of public-health response to deliberate release of smallpox, we are considering a highly complicated system that is not directly amenable to experimental testing. This means that there will always be a degree of uncertainty associated with conclusions presented, however strongly these manifest themselves in a model.

Despite this general caveat, investigation of the problem for Great Britain suggests that, for a wide variety of parameter choices, and with differing modelling approaches, a vaccination strategy that involves a wider section of the population than the traced contacts of isolated cases but one that stops short of vaccinating the whole country is likely to be optimal. Such a conclusion is, we believe, likely to be robust in the event of a relatively small initial outbreak and given our best estimates of the contagiousness and natural history of

We have generalised on previous work in two main ways. Firstly, we have extended the treatment of contact networks to include the realistic assumption of clustering, which has an extremely important impact on the efficacy of contact tracing. However, the measurement of even basic quantities like the clustering coefficient

Secondly, we have considered human movement patterns across a full range of geographical scales, enabling the calculation of an optimal scale for intervention. Furthermore, the techniques in this paper can be used to analyse situations where the numbers people leaving an area are inferable but the full region to region 'commuting' behaviour is unknown. As such this could be applied to other countries that don't collect detailed workplace location information in their census programmes.

There are many further generalisations of our work that could be considered, which would be of use regarding a response to a wide variety of emerging infectious agents. In particular, the inclusion of higher-order clusters in human contact networks, such as households, workplaces and social groups, is likely to be of significant importance. Also likely to be significant is the modification of 'baseline' patterns of movement and social interaction in response to perceived risk of infection. Consideration should also be made of the logistical constraints on local- and national-scale policies. We hope that the current work presents a useful foundation for the consideration of these and other questions.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

TH carried out the mathematical formulation of the model and coding. IH, LD and MJK provided parameter values, and informed the choice of simulations to undertake. All authors contributed to the writing of the paper.

Acknowledgements

This work was funded by the Department of Health for England and the Home Office Counter Terrorism and Intelligence Directorate through the Health Protection Agency, and also by the UK Medical Research Council (Grant Number G0701256). We would like to thank Joseph Egan, Steve Leach, Judith Legrand and Neil Ferguson for helpful discussions in project meetings during the preparation of these results.

Pre-publication history

The pre-publication history for this paper can be accessed here: