• We are sorry, but NCBI web applications do not support your browser and may not function properly. More information
Logo of pnasPNASInfo for AuthorsSubscriptionsAboutThis Article
Proc Natl Acad Sci U S A. Aug 15, 2006; 103(33): 12637–12642.
Published online Aug 7, 2006. doi:  10.1073/pnas.0510873103
PMCID: PMC1567931
From the Cover
Population Biology

Smallpox transmission and control: Spatial dynamics in Great Britain

Abstract

Contingency planning for the possible deliberate reintroduction of smallpox has become a priority for many national public health organizations in recent years. We used an individual-based spatial model of smallpox transmission in Great Britain and census-derived journey-to-work data to accurately describe the spatiotemporal dynamics of an outbreak of smallpox in the community. A Markov chain Monte-Carlo algorithm was developed to generate sociospatial contact networks that were consistent with demographic and commuting data. We tested the sensitivity of model predictions to key epidemiological parameters before choosing three representative scenarios from within the range explored. We examined the spatiotemporal dynamics for these illustrative scenarios and assessed the efficacy of symptomatic case isolation, contact tracing with vaccination, and reactive regional mass vaccination as policy options for control. We conclude that case isolation and contact tracing with vaccination would be sufficient to halt ongoing transmission rapidly, unless policy effectiveness was compromised by resource or other constraints. A slight reduction in the expected size and duration of an outbreak could be achieved with regional mass vaccination, but these benefits are small and do not justify the high numbers of vaccine doses required and their associated negative side effects.

Keywords: epidemiology, mathematical model, network, infectious diseases

Before its eradication, smallpox was a directly transmitted respiratory disease caused by the variola viruses. Variola major was highly pathogenic: The mortality rate among the unvaccinated in India in 1974 was 26.5% (1). Despite there being only two recognized laboratories storing variola major virus, an unknown number of covertly held samples of this virus likely remain in existence, raising concern in recent years over the potential for the virus to be used as a bioterrorist weapon. Even a single confirmed case would cause substantial levels of public concern and social disruption. However, the epidemiological consequences of a small number of deliberately introduced infections in a population such as that of Great Britain (GB) (i.e., England, Scotland, and Wales) is not well understood. A number of mathematical modeling studies in recent years have attempted to address this issue, but their results are equivocal (25). Furthermore, the likely geographic spread of the disease within contemporary populations has not been modeled. Consequently, uncertainties remain as to the relative effectiveness of different public health interventions that might be implemented in response to a smallpox outbreak.

Conceptually, the transmission dynamics of an infectious disease and the potential for control of that disease can be understood in terms of R0 and θ. The basic reproductive number, R0, for a pathogen is defined to be the average number of secondary infections generated by a typically infectious individual in an otherwise susceptible population. In this definition, the word “typically” implies exponential growth that can be described accurately by a next generation operator (6). The condition R0 > 1 is required for a pathogen to successfully invade a host population, and, in general, diseases with higher values of R0 are more difficult to control. The proportion of infections that occur before the onset of symptoms is defined to be θ (7). Interventions that rely on the recognition of symptoms to reduce transmission, such as case isolation and contact tracing, are more likely to be successful when θ is small than when it is large. Here, when quantifying transmissibility, we use an approximation to R0, which we call R0*, the average number of secondary cases. We define R0* to be the expected number of secondary infections generated by a randomly chosen infectious individual in an otherwise susceptible population. This quantity, which shares the R0* < 1 condition for disease extinction, can be calculated precisely for the model used here (ref. 8 and see Supporting Text, which is published as supporting information on the PNAS web site).

Results

Sensitivity Analyses.

A range of estimates of R0 and θ are available for smallpox (7, 9). However, not all key uncertainties can be examined by varying these high-level parameters. Therefore, we investigated the sensitivity of results from an individual-based transmission model (see Methods and Fig. 1) to two types of lower-level parameters: those that determined the topology of the contact network and those that determined the infectiousness of the individual. Because generation of the social network required substantially more computer run-time than model simulation, we investigated transmission uncertainty in two stages. The first stage of the sensitivity analysis was to generate nine network types by using all combinations of three values for peer-group neighborhood size (nPG) (8, 10, and 12) and three values for the proportion of peer-group members who were contacts (pWP) (0.25, 0.5, and 0.75). In the second stage, we used Latin hypercube sampling to randomly select 100 different combinations of the six parameters that determine individual-level infectiousness (sampling range in parentheses; see also Table 2, which is published as supporting information on the PNAS web site): a, the household attack rate, which we defined to be the proportion of susceptible household members infected by a single infectious individual (43–88% (1)); hr/f, infectiousness of an individual with rash relative to infectiousness with fever (1–20); hh/w, infectiousness of an individual to household members relative to infectiousness to peer-group contacts (1–10); μER, the average time that people with smallpox rash continued to function outside the home (0–4 days); αER, shape parameter for the γ-distributed times that people with smallpox rash continued to function outside the home (1–20; variance μER2ER); and R0NT, the average number of community (nontraceable) infections generated by an infectious individual in an otherwise susceptible population (0.01–1). To investigate plausible ranges of smallpox transmission, each of the total of 900 parameter sets (nine network types and 100 combinations of individual-level parameters) were simulated 100 times for the first four generations. Simulations were seeded by using ten individuals from households within 20 km of central London. Seeds were reselected at random for each of the 100 simulations.

Fig. 1.
Model assumptions and inputs. (A) The natural history of smallpox with k states (see Methods) given explicitly. The six disease states were as follows: ki = 0, susceptible; ki = 1, latent (infected but not yet infectious or symptomatic); ki = 2, fever ...

The upper bound from our sensitivity analyses for the average number of secondary cases, R0*, was 5.3 (Fig. 2A and Methods). This higher value was achieved with an average peer-group neighborhood size of nPG = 12. For lower neighborhood sizes of 10 and 8, the maximum values of R0* were 4.8 and 4.3, respectively. For a given peer-group neighborhood size, the key determinants of transmission were the household attack rate a and the average duration of the early rash phase μER (Fig. 2B). With few exceptions, values of R0* > 3, were only achieved when the household attack rate a was >70% and the average duration of the early rash phase μER was >2 days.

Fig. 2.
Sensitivity analyses. (A) The variation in values of R0* (average number of secondary cases) versus nPG (average size of a peer group). High values of R0* were only consistent with high average peer-group sizes. (B) The variation between R0*, μ ...

The highest average number of cumulative infections in the first four generations (including the initial ten seeds as the first generation) was 2,077. This number was obtained with a proportion of peer-group members who were contacts pWP = 0.25 (Fig. 2C). The maximum average numbers of infections in the first four generations for pWP = 0.5 and pWP = 0.75 were 1,886 and 1,783, respectively. Note that for the sensitivity analyses, we selected values of pWP independently of values for contact network neighborhood size, nPG, by controlling the average size of the peer groups, sPG (by using nPG = sPG pWP). Therefore, the three subsets of sensitivity results with different values of pWP (Fig. 2C) all contain the same distribution of samples of R0*, because R0* is determined by nPG. However, the dynamics beyond the first generation of infection were strongly affected by pWP because increases in pWP caused the contact network to be more clustered and, therefore, increased local saturation of infection. Hence, for high pWP, the effective reproduction number dropped significantly in the first few generations of infection. When pWP was small, the network was less well clustered and local saturation was a minor effect, meaning that the reproduction number could remain close to R0* for all four generations of infection.

We defined the spatial extent of an outbreak to be the maximum distance from the center of the seeding circle (central London) to the household of an infected individual. Fig. 2D shows the relationship between the average cumulative number of infections up to the fourth generation and the average spatial extent of the outbreak. Spatial extent grew rapidly up to 100 cumulative infections but more slowly thereafter. The difference was likely because of spatial saturation at different scales. Even a small outbreak shows fast spatial growth in the densely populated Southeast. However, beyond a certain point, spatial extent can only grow further via “long jumps” to other dense areas of population that are less well connected to the Southeast.

Representative Scenarios.

We selected three representative transmission scenarios from those generated during sensitivity analyses: low (scenario A, R0* = 1.5, θ = 4%), medium (scenario B, R0* = 2.6, θ = 7%), and high (scenario C, R0* = 5.4, θ = 20%) to investigate spatiotemporal dynamics and the efficacy of different intervention options. The average dynamics of scenarios A, B, and C, in the absence of any interventions, are shown in Fig. 3 for the first 75 days of an outbreak. Example individual realizations are shown in Movie 1, which is published as supporting information on the PNAS web site, and the time evolution of the ensemble averages is shown in Movie 2, which is published as supporting information on the PNAS web site. These scenarios are discussed in detail in the Supporting Text and defined precisely in Table 2.

Fig. 3.
Spatiotemporal dynamics. Low transmission (A), medium transmission (B), and high transmission (C) scenarios (see also text and Table 2). The density maps show expected cumulative infections for the first 75 days of the outbreak averaged over 100 model ...

Interventions.

In the event of an outbreak of smallpox, the primary objective of the public health response will be to ensure that widespread community transmission does not occur. We defined the loss of control to be when >1,000 cumulative infections (from ten initial seeds) occurred during the first 365 days of the outbreak. In the event of an outbreak, if it becomes clear that the outbreak will be controlled, priorities will shift to reducing the total number of infections and the duration of the outbreak. We defined achieving control to be when there were no infected individuals present in the population. The first type of policy we considered was rash-motivated case isolation, i.e., infected individuals with a rash sought medical care, were successfully recognized as smallpox cases, and were isolated (Table 1, policies 2, 3, and 4). We assumed that the delay from the onset of rash to successful isolation followed a γ distribution. Case isolation with an average delay of 2 days was highly effective in controlling scenario A. Reducing that delay to 0.5 days allowed the policy also to control scenario B. However, this intervention on its own was not capable of controlling scenario C. Note that the efficacy of rash-motivated self-isolation was not obviously sensitive to the shape of the distribution of times from onset of rash to isolation.

Table 1.
Comparison of the efficacy of intervention policies on transmission scenarios

The second type of policy we considered (Table 1, policies 5 and 6) was adding contact tracing with vaccination (so-called “ring” vaccination) to rash-motivated case isolation. When an individual reported to the healthcare system, we assumed that they were immediately interviewed and asked to name their contacts. We assumed that they successfully named 90% of their household contacts, a substantial proportion of their neighbors on the peer-group contact network but none of their community infections. Named individuals were contacted and vaccinated with delays of either 1 day (household) or 2 days (peer-group contacts). To be conservative, we assumed that if contacts were vaccinated after infection, the vaccine was completely ineffective. Crucially, we assumed that individuals who had been contact-traced considered themselves to be at risk of infection. If at-risk individuals developed a fever, we assumed that they immediately sought health care and were successfully isolated with an average delay of 0.5 days. This intervention strategy successfully controlled all transmission scenarios, including scenario C, even when contact tracing of peer-group contacts had only a 65% success rate.

The third type of policy we considered was the addition of regional mass vaccination to rash-motivated isolation and contact tracing with vaccination (Table 1, policies 7 and 8). We assumed that when a case was quarantined, all members of the population living within a certain distance of that individual would be eligible for vaccination. The overall maximum number of doses that could be administered per day was 1 million and not >75% of the population of any small spatial area could be vaccinated on a given day. Although these values represent an aggressive vaccination policy, they are consistent with rapid coverage achieved in a short time in New York and in Yugoslavia near the end of the global eradication campaign (1). We did allow 100% of people in recruited regions to be vaccinated over multiple days. This latter assumption may be somewhat over optimistic but ensures that we do not underestimate the efficacy of regional mass vaccination. For medium and high transmission scenarios (B and C), the addition of regional mass vaccination was significantly better than contact tracing with vaccination alone, but there was no benefit in using a 50-km radius rather than a 15-km radius. However, even with a 15-km radius, the number of vaccine doses used was large. For scenario B, on average, an additional 9 million doses of vaccine prevented only 6 infections (Table 1, policy 7 compared with policy 6). For scenario C, an additional 13.7 million doses prevented 77 infections. These large numbers of additional vaccinations, and their associated negative side effects, do not justify the infections prevented: During 2003, after the inoculation of 37,901 people in the United States, three deaths, two permanent disabilities, and ten life-threatening illnesses were attributed to vaccination (10).

A country would face a rather different public health problem if smallpox had been reintroduced elsewhere in the world, namely, that of a trickle of infected people potentially traveling to the country from the source of the outbreak. If a country faces multiple introductions dispersed in time, immediate mass vaccination may become a more attractive intervention strategy. We used the model to estimate the number of deaths that would arise from repeated independent introductions (under each transmission scenario, A, B and C) with rash-motivated isolation, contact tracing, and vaccination in effect (policy 6 of Table 1 implemented from day 0; see Fig. 4, which is published as supporting information on the PNAS web site), and we compared these results with the expected number of deaths from mass vaccination. We show that for high-transmission scenario C, under which the case for mass vaccination would be strongest, 1,408 independent introductions of smallpox are required for mass vaccination to be justified by a reduction in expected mortality (Fig. 5, which is published as supporting information on the PNAS web site). Note that our estimate for this threshold is based on the assumption that there is one death per 35,643 vaccinations. This rate is the lowest with a 95% confidence interval consistent with ref. 10 but is still much higher than historical estimates (1). If the rate were one death per million, the threshold for mass vaccination would drop to 50 infectious importations under scenario C.

Discussion

Smaller household sizes imply that smallpox may not be very transmissible if reintroduced into GB. However, rigorous sensitivity analyses cannot rule out scenarios where infectiousness during the presymptomatic phase is relatively high, peer-group networks are not highly connected, and there is a significant period during which symptomatic individuals infect those outside their household. Under these conditions, the average number of secondary cases per random primary case, R0*, might be as high as 5.4. We do not believe that levels of transmission higher than 5.4 are plausible, based on past outbreaks. Use of census-derived population density and travel data suggests that spatial correlation between cases would be significant in GB for a prolonged period if the outbreak were to persist: It is likely that it would take a long time for the outbreak to reach remote populations. We predict that rapid rash-motivated case isolation would bring plausible medium- and low-transmission scenarios under control. Improving rapid recognition and diagnosis of symptoms, perhaps via information campaigns aimed at both clinicians and the public, would therefore be a priority in any outbreak.

Contact tracing and vaccination of contacts would be required to control some high-transmission scenarios and would improve the speed and efficacy of control of all plausible scenarios. The efficacy of contact tracing depends on the topology of the smallpox-specific transmission network, with a tracing policy being more effective in densely connected networks. Therefore, we have been conservative in including purely spatial community transmission and low connectivity networks. However, more data on the connectivity of social networks relevant to smallpox would clearly be useful. Regional or national mass vaccination only needs to be considered if it becomes obvious that levels of transmission are at the highest end of the plausible range we have highlighted here. Even then, the tradeoff between infections averted and the negative side effects of vaccines may well not justify such nontargeted intervention policies. In this sense, regional vaccination must be considered nontargeted. Similarly, reductions in the duration of the outbreak are limited even with aggressive vaccination programs.

The model structure presented here allows us to reconcile the different conclusions regarding the transmissibility of smallpox and optimal strategies for control that are present in the literature. Our range of estimates for the basic reproductive number in GB, which is based on local household data and historical attack rates, is somewhat lower than other estimates (3, 9), largely because we adjust for the decrease in household sizes that has occurred in developed countries in the last few decades. Other studies have predicted that contact tracing and vaccination would be less effective than predicted here (4, 5). Our use of an appropriate parametric form for the incubation distribution (11) ensures that the efficacy of contact tracing and vaccination is not underestimated. High values for the presymptomatic proportion of transmission would also tend to underestimate the efficacy of contact tracing and vaccination (4, 5). Although higher values were included in our broad sensitivity analyses, we have restricted ourselves to values of 20% or lower for our representative scenarios, which are consistent with recent estimates based on data from historical outbreaks (3). Our results are in agreement with other studies that have made similar assumptions (12, 13). In addition, by extending traditional compartmental models to include realistic spatial structure at a national scale, we have been able to show that regional mass vaccination, an intermediate policy between contact tracing and mass vaccination, is not likely to be as efficient as contact tracing with vaccination alone for GB. The increase in scale from city-level models (13) to national-level models is made possible by the use of travel data with network and spatial transmission kernels. This combined approach will likely be extended to continental and global models.

Although the results presented here are specific to GB, we expect that our conclusions would hold for other populations. We tested the sensitivity of our results for regional mass vaccination to substantially wider and narrower pair-wise choice kernels (Table 3, which is published as supporting information on the PNAS web site) and found that the additional vaccination requirements of this policy remained high, even with substantially different commuting behavior. Because estimates of a commuting choice kernel for Thailand (14) are similar to those presented here for GB, the robustness of our results to narrower and wider choice kernels is encouraging. We note that for the spatial results presented here, we assumed that substantial proportions of the population do not attempt to flee the area of an outbreak. Also, we have assumed throughout that journey-to-work data are a reasonable proxy for the human behavior that generates sociospatial contact networks for directly transmitted respiratory diseases. Recent spatial analyses of seasonal influenza and journey-to-work data for the United States supports this assumption (15).

Methods

We used an individual-based model of disease progression and transmission in a population of N individuals. We defined ki to be the disease state of the ith individual where i ranges from 1 to N (Fig. 1). The structure of the model allowed for transmission in three settings: household, peer-group network, and community. During any period of time, Δt, the probability that individual i in disease state ki infected individual j in disease state kj = 0 was defined as follows:

equation image

The terms in Eq. (1) are defined in the remainder of this section.

Household and Peer-Group Network Transmission.

The first two terms of Eq. (1) correspond to household and peer-group network transmission respectively: IHH (i, j) was an indicator function equal to 1 only if i and j shared a household and equal to 0 otherwise. Similarly, IPGN (i, j) was the indicator function for i and j sharing a peer-group network link. β was the underlying hazard of infection for transmission within the household and across the peer-group network during the fever stage (ki = 2). hk was the stage specific relative hazard of infection and was equal to 1 when individual i was in the fever stage (ki = 2), equal to hr/f when in either rash stages (ki = 3 or 4) and equal to 0 otherwise. The parameter β could be expressed as a function of a, the household attack rate (Supporting Text). Therefore, we used a as the basic parameter of transmission in the household and over the peer-group network.

Peer-Group Network Generation.

Households were of average size sh, and their sizes were distributed as 1 + P(sh −1), where P(μ) is the Poisson distribution with mean μ. Each household had a location consistent with the 1991 GB census (www.statistics.gov.uk), hereafter referred to as census data, chosen by using a simple accept-reject algorithm with a 1-km2 resolution. Data from the 1991 census was used because the journey-to-work data from the 2001 census was not sufficiently spatially resolved. A peer group was a spatial location where individuals spent a substantial proportion of their time outside the home and where they tended, on average, to meet the same people. Note that peer groups did not necessarily correspond to distinct commercial addresses. The location of peer groups was chosen by using a simple accept-reject algorithm, to be consistent with the destination portion of the journey-to-work section of the census data. The number of peer groups was equal N/sPG with sPG the average size of a peer group. Initially, individuals were assigned to peer groups at random. A Markov chain Monte Carlo algorithm was then used to match the allocation of individuals to peer groups with census data (see Supporting Text and Fig. 5). This algorithm relied on a pair-wise choice kernel κPW (d) rather than the actual distribution of journeys. The kernel κPW (d) was defined such that the probability that an individual would chose a peer-group distance d1 away from their home relative to one d2 away from their home was equal to κPW (d1)/κPW (d2). We used this pair-wise approach, rather than directly matching the population-level distribution of commutes, to permit the use of the Metropolis–Hastings algorithm (16) for a Markov chain defined over the space of all possible individual/peer-group allocations.

Transmission did not occur uniformly between all members of a peer group. The generation of the actual contact network over which transmission occurred, defined by IPGN (i, j), was itself a random process based on peer-group membership. The probability that individual i was a peer-group contact of individual j was equal to pWP if they shared a peer group and equal to 0 otherwise. This selective connectivity within peer groups allowed transmission networks with similar peer-group sizes but very different average neighborhood sizes. The peer-group neighborhood size, nPG, was the per person average number of peer-group contacts.

Community Transmission.

The final term in Eq. (1) corresponds to nontraceable community transmission, which was assumed to occur only during the fever and early rash stages of infection (ki = 2 or ki = 3). It represented infections between people who did not know each other or, more specifically, people who could not name each other during a contact tracing interview. We assumed that the travel behavior that determined choice of peer group, described by the pair-wise choice kernel κPW (d), would also determine the extent of everyday movements during which “random” community transmission could occur. Therefore, we defined κNT (d) to be the two-dimensional convolution of κPW (d) with itself by using κPW (d) as an isotropic two-dimensional kernel, invariant in the angular dimension. In effect, we made the relative probability of a community contact between two people living distance d apart be proportional to the sum of the probabilities of them meeting at any location, assuming they both choose locations as per κPW (d). The function b(i, j, Δt) was random and was defined implicitly by the algorithm used to implement community transmission (Supporting Text) such that the following two conditions held; the expected number of secondary infections per person in an otherwise susceptible population was Poisson-distributed with mean R0NT, and the expected number of secondary community infections was independent of local population density.

Supplementary Material

Supporting Information:

Acknowledgments

We thank R. M. Anderson, C. A. Donnelly, J. E. Trustcott, and C. Fraser for discussions and two anonymous reviewers for helpful comments. We acknowledge funding from Howard Hughes Medical Institute (S.R. and N.M.F.), the Research Fund for the Control of Infectious Diseases of the Health, Welfare, and Food Bureau of the Government of Hong Kong Special Administrative Region of the People’s Republic of China (S.R.), Great Britain Department of Health (N.M.F.), and the European Union Framework 6 Programme (N.M.F.).

Footnotes

Conflict of interest statement: No conflicts declared.

This paper was submitted directly (Track II) to the PNAS office.

See Commentary on page 12221.

References

1. Fenner E., Henderson D. A., Arita L., Jezek Z., Ladnyi I. D. Smallpox and Its Eradication. Geneva: W.H.O.; 1988.
2. Bozzette S. A., Boer R., Bhatnagar V., Brower J. L., Keeler E. B., Morton S. C., Stoto M. A. N. Engl. J. Med. 2003;348:416–425. [PubMed]
3. Eichner M., Dietz K. Am. J. Epidemiol. 2003;158:110–117. [PubMed]
4. Halloran M. E., Longini I. M., Nizam A., Yang Y. Science. 2002;298:1428–1432. [PubMed]
5. Kaplan E. H., Craft D. L., Wein L. M. Proc. Natl. Acad. Sci. USA. 2002;99:10935–10940. [PMC free article] [PubMed]
6. Heesterbeek J. A. P. Acta Biotheor. 2002;50:189–204. [PubMed]
7. Fraser C., Riley S., Anderson R. M., Ferguson N. M. Proc. Natl. Acad. Sci. USA. 2004;101:6146–6151. [PMC free article] [PubMed]
8. Britton T., Becker N. G. Biostatistics. 2000;1:389–402. [PubMed]
9. Gani R., Leach S. Nature. 2001;414:748–751. [PubMed]
10. Casey C. G., Iskander J. K., Roper M. H., Mast E. E., Wen X. J., Torok T. J., Chapman L. E., Swerdlow D. L., Morgan J., Heffelfinger J. D., et al. J. Am. Med. Assoc. 2005;294:2734–2743. [PubMed]
11. Wearing H. J., Rohani P., Keeling M. J. PLoS Med. 2005;2:621–627.
12. Eichner M. Am. J. Epidemiol. 2003;158:118–128. [PubMed]
13. Eubank S., Guclu H., Kumar V. S. A., Marathe M. V., Srinivasan A., Toroczkai Z., Wang N. Nature. 2004;429:180–184. [PubMed]
14. Ferguson N. M., Cummings D. A., Cauchemez S., Fraser C., Riley S., Meeyai A., Iamsirithaworn S., Burke D. S. Nature. 2005;437:209–214. [PubMed]
15. Viboud C., Bjornstad O. N., Smith D. L., Simonsen L., Miller M. A., Grenfell B. T. Science. 2006;312:447–451. [PubMed]
16. Hastings W. K. Biometrika. 1970;57:97–109.

Articles from Proceedings of the National Academy of Sciences of the United States of America are provided here courtesy of National Academy of Sciences
PubReader format: click here to try

Formats:

Related citations in PubMed

See reviews...See all...

Cited by other articles in PMC

See all...

Links

Recent Activity

Your browsing activity is empty.

Activity recording is turned off.

Turn recording back on

See more...