Optimal reactive vaccination strategies for a foot-and-mouth outbreak in the UK.
ABSTRACT Foot-and-mouth disease (FMD) in the UK provides an ideal opportunity to explore optimal control measures for an infectious disease. The presence of fine-scale spatio-temporal data for the 2001 epidemic has allowed the development of epidemiological models that are more accurate than those generally created for other epidemics and provide the opportunity to explore a variety of alternative control measures. Vaccination was not used during the 2001 epidemic; however, the recent DEFRA (Department for Environment Food and Rural Affairs) contingency plan details how reactive vaccination would be considered in future. Here, using the data from the 2001 epidemic, we consider the optimal deployment of limited vaccination capacity in a complex heterogeneous environment. We use a model of FMD spread to investigate the optimal deployment of reactive ring vaccination of cattle constrained by logistical resources. The predicted optimal ring size is highly dependent upon logistical constraints but is more robust to epidemiological parameters. Other ways of targeting reactive vaccination can significantly reduce the epidemic size; in particular, ignoring the order in which infections are reported and vaccinating those farms closest to any previously reported case can substantially reduce the epidemic. This strategy has the advantage that it rapidly targets new foci of infection and that determining an optimal ring size is unnecessary.
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ABSTRACT: Foot-and-mouth disease (FMD) occurred in five provinces and 24 counties as part of the FMD incursion into Mongolia during 2010. The first detection occurred on 21 April 2010 (confirmed 26 April 2010) with the last detection occurring approximately 8 months later on 13 December 2010. The number of livestock detected in the spring phase of the outbreak was 323 cattle and in the summer phase was 13 485 sheep, 6748 cattle, 5692 goats and 10 camels (total livestock summer phase = 25 935; for spring and summer phases combined = 26 258). Infection of livestock was confirmed by PCR for each affected county but not necessarily for every outbreak cluster involving more than one herder. It is likely that the summer phase of the outbreak was a continuation of the spring event. In the summer phase, the spatio-temporal pattern of spread suggested an extension of infection from the main cluster in the Sukhbaatar county. There was also a number of long-distance clusters established. The relative importance of spread by three potential pathways of gazelle, livestock, animal product and fomite movements has not been determined and will require further study. The estimated dissemination ratio (EDR) did not provide evidence of high rate of transmission of infection between herders; however, the data are limited by the quality of surveillance and the method of calculation which used the date of detection rather than the date of infection.Transboundary and Emerging Diseases 01/2014; · 2.10 Impact Factor
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ABSTRACT: Comparison of control strategies against animal infectious diseases allows determining optimal strategies according to their epidemiological and/or economic impacts. However, in real life, the choice of a control strategy does not always obey a pure economic or epidemiological rationality. The objective of this study was to analyze the choice of a foot and mouth disease (FMD) control strategy as a decision-making process in which the decision-maker is influenced by several stakeholders (government, agro-food industries, public opinion). For each of these, an indicator of epizootic impact was quantified to compare seven control strategies. We then determined how, in France, the optimal control strategy varied according to the relative weights of stakeholders and to the perception of risk by the decision-maker (risk-neutral/risk-averse). When the scope of decision was national, whatever their perception of risk and the stakeholders' weights, decision-makers chose a strategy based on vaccination. This consensus concealed marked differences between regions, which were connected with the regional breeding characteristics. Vaccination-based strategies were predominant in regions with dense cattle and swine populations, and in regions with a dense population of small ruminants, combined with a medium density of cattle and swine. These differences between regions suggested that control strategies could be usefully adapted to local breeding conditions. We then analyzed the feasibility of adaptive decision-making processes depending on the date and place where the epizootic starts, or on the evolution of the epizootic over time. The initial conditions always explained at least half of the variance of impacts, the remaining variance being attributed to the variability of epizootics evolution. However, the first weeks of this evolution explained a large part of the impacts variability. Although the predictive value of the initial conditions for determining the optimal strategy was weak, adaptive strategies changing dynamically according to the evolution of the epizootic appeared feasible.PLoS ONE 01/2014; 9(1):e86323. · 3.53 Impact Factor
© 2006 Nature Publishing Group
Optimal reactive vaccination strategies for a
foot-and-mouth outbreak in the UK
Michael J. Tildesley1, Nicholas J. Savill2,3, Darren J. Shaw4, Rob Deardon2,5, Stephen P. Brooks2,
Mark E. J. Woolhouse3, Bryan T. Grenfell6,7& Matt J. Keeling1
Foot-and-mouth disease (FMD) in the UK provides an ideal
opportunity toexplore optimal control measures for an infectious
disease. The presence of fine-scale spatio-temporal data for the
2001 epidemic has allowed the development of epidemiological
models that are more accurate than those generally created for
other epidemics1–5and provide the opportunity to explore a
variety of alternative control measures. Vaccination was not
used during the 2001 epidemic; however, the recent DEFRA
(Department for Environment Food and Rural Affairs) contin-
gency plan6details how reactive vaccination would be considered
in future. Here, using the data from the 2001 epidemic, we
consider the optimal deployment of limited vaccination capacity
spread to investigate the optimal deployment of reactive ring
vaccination of cattle constrained by logistical resources. The
predicted optimal ring size is highly dependent upon logistical
constraints but is more robust to epidemiological parameters.
Other ways of targeting reactive vaccination can significantly
reduce the epidemic size; in particular, ignoring the order in
which infections are reported and vaccinating those farms closest
to any previously reported case can substantially reduce the
epidemic. This strategy has the advantage that it rapidly targets
new foci of infection and that determining an optimal ring size is
The control of infectious diseases is often a compromise between
the desire for large-scale implementation of control measures and
that control measures are optimally targeted so as to minimize the
adverse population-level impact of a disease. Optimal targeting will
in general depend in a complex nonlinear fashion on disease
dynamics, host demography and logistical constraints. Well-para-
meterized mathematical models of infectious disease spread are
therefore a necessary tool for determining optimal control strategies,
because a vast range of policies can be rapidly tested by simulation.
One important aspect of optimal control is the balance between
maintaining local control of outbreaks while attacking new foci of
During the course of the 2001 FMD epidemic a range of control
measures were implemented to try to reduce the transmission of
infection; these included movement restrictions preventing long-
range transportation of livestock, increased farm bio-security, rapid
and the reactive culling of livestock on ‘at risk’ farms to eliminate
potential new cases. Reactively culled farms fell into two main
categories (see Methods): dangerous contacts (DCs) and contiguous
premises (CPs). Reactive vaccinationwas considered during the later
stages of the 2001 epidemic but was dismissed for several reasons
(detailed in ref. 7), including concerns that the limited vaccination
campaign that was logistically feasible at the time would have little
impact on the epidemic.
The recently published European Union Directive8suggests reac-
tive vaccination as a preferred means of intervention should a foot-
and-mouth disease outbreak occur in any member state. This is
reflected in the UK by the new FMD contingency plan published by
DEFRA6. However, neither document suggests a specific design for
reactive vaccination programmes. Here, we concentrate on develop-
on the 2001 UK epidemic, assuming vaccination-to-live so that
vaccinated animals do not have to be culled. Vaccination of cattle
is combined with localized culling (following EU and DEFRA
recommendations6,8) in which all livestock on infected premises are
culled within 24 hours of reporting and farms considered by
epidemiological investigation to be at increased risk (DCs) are culled
within 48 hours. By default, simulations do not include CP culling.
The model closely follows that developed in refs 1 and 2; we use the
farm census database from June 2000 to incorporate farm-level
are obtained by fitting to the 2001 spatio-temporal epidemic profile
(see Methods and Supplementary Information).
We begin by assuming that vaccination (of cattle only) takes place
within an annulus (defined by an inner and outer radius, Rinnerand
of these farms is then prioritized in the order that the associated IPs
were reported and, for those farms identified on the same day,
priority is given to those furthest from the IP. Thus vaccination
within each annulus is performed from the outside in. Protection of
cattle is influenced by several factors. We assume avaccine efficacy of
90% so that 10% of vaccinated cattle remain susceptible10,11; in
addition there is a four-day delay between vaccination and protec-
tion. Finally, although farms within an annulus are targeted for
vaccination two days after the IP is reported, it may take longer
beforethey are vaccinated owing to the logistical constraint that only
a limited number of cattle can be vaccinated per day.
Varying Rinnerand Router, we seek the vaccination annulus that
minimizes the epidemic impact—defined as the number of farms
that are infected (IPs) or culled as part of the control (DCs þ CPs if
appropriate). For a plausible vaccination capacity (defined as the
1Department of Biological Sciences and Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK.2Statistical Laboratory, Centre for Mathematical
Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK.3Epidemiology Group, Centre for Infectious Diseases, University of Edinburgh, Ashworth
Laboratories, Kings Buildings, West Mains Road, Edinburgh EH9 3JF, UK.4Veterinary Clinical Studies, Royal (Dick) School of Veterinary Studies, University of Edinburgh, Easter
Bush Veterinary Centre, Roslin, Midlothian EH25 9RG, UK.5Cambridge Infectious Diseases Consortium, Centre for Veterinary Science, University of Cambridge, Madingley Road,
Cambridge CB3 0ES, UK.6Center for Infectious Diseases Dynamics, Biology Department 208, Mueller Laboratory, Pennsylvania State University, University Park, Pennsylvania
16802, USA.7Fogarty International Center, National Institutes of Health, Bethesda, Maryland 20892, USA.
Vol 440|2 March 2006|doi:10.1038/nature04324
© 2006 Nature Publishing Group
number of cattle that can be vaccinated per day), the model predicts
that the optimal policy is to vaccinate in a complete ring with the
inner radius set to zero (Fig. 1a). Extensive simulations have shown
that thisistrue for allthesituations examined here,indicating that—
despite the delays involved—vaccination is still in general a locally
effective control measure. This finding could be refined using
predictive vaccination2, where (on a case-by-case basis) the use of
predictive simulations indicates that it may be optimal to ignore
some local farms if they are likely to become infected before
vaccination would confer immunity.
We focus on the behaviour of the model when the inner radius is
fixed at zero (Fig. 1b and c), and observe several important patterns.
Increasing the vaccination capacity reduces the average epidemic
impact (Fig. 1b and c), although there is considerable variation
between simulations. The optimal vaccination ring increases with
vaccination capacity. However, the number of animals within
the optimal ring is not proportional to the number that can be
vaccinated per day; instead it increases more rapidly, compensating
for overlapping vaccination rings. This highlights the fact that
calculation of the optimal ring size is a complex nonlinear problem
that requires detailed spatial models. Finally, we note that optimal
reactive ring vaccination at 35,000 animals per day (together with IP
and DC culling) is a far better strategy, in terms of reducing the total
number of farms lost, than a policy of CP culling (together with IP
promptly (Fig. 1b).
Armed with these insights, we then explore the effect of varying
Figure 1 | Epidemic impact (number of farms infected and culled).
a, Epidemic impact as inner vaccination radius (Rinner) and width of the
vaccination annulus (Router– Rinner) vary. The white dot marks the optimal
strategy. b, Epidemic impact for various outer radii (Rinner¼ 0); points
representindividualsimulations,lines givea smoothedaveragefor different
vaccination capacities with large dots marking the optimal ring size. The
culling but no vaccination. c, Epidemic impact (red) and optimal radius
(blue) as the daily vaccination capacity is varied (with 95% confidence
intervals for each point).
Figure 2 | The optimal vaccination radius and the associated epidemic
impact. Results were determined (together with 95% confidence intervals)
using an outside-in vaccination strategy and vaccination capacity of 35,000
cattle per day. a, The number of farms infected before vaccination begins is
varied—capturing early or late detection and start of vaccination. b, The
total number of doses of vaccine available is varied. c, The efficacy of the
vaccine is varied. Rinneris zero in all simulations. Each point represents the
results of 40,000 epidemic simulations. Similar results for the shortest-
distance prioritization are given in the Supplementary Information.
NATURE|Vol 440|2 March 2006
© 2006 Nature Publishing Group
other logistical and biological factors on the optimal vaccination
radius and corresponding epidemic impact (Fig. 2). Figure 2a
measures the delay before the disease is first detected combined
with the delay before the onset of vaccination in terms of the average
number of farms infected before vaccination begins. Prompt detec-
tion of the epidemic and a rapid decision to vaccinate allows larger
vaccination rings to be implemented around each IP and substan-
tially reduces the epidemic size, echoing more general principles1,12.
Any supply of vaccine will be limited because of the impracticality of
storing large amounts of vaccine for each virus type; also, vaccine
efficacy will vary between strains and may be considerably reduced if
there is a discrepancy between the invading strain and the closest
available vaccine. Both of these have implications for the success of
any vaccination campaign13–15. When the supply of vaccine is below
three million doses, the vaccine ring must be reduced in size to
prevent premature exhaustion of this supply, which in turn increases
the epidemic impact (Fig. 2b). In contrast, although a reduction in
vaccine efficacy (which may act as a surrogate for the farm-level
more limited effect on the optimal ring size (Fig. 2c).
The epidemiological characteristics within the model have been
parameterized using the data from the 2001 epidemic. However,
should foot-and-mouth disease enter the UK in future, it will not
necessarily be the same strain and so its epidemiological character-
istics could be substantially different. We therefore calculate the
optimal vaccination ring (for the simple outside-in prioritization)
while varying the susceptibility and transmissibility of cattle and
sheep, and the width of the dispersal kernel, by a factor of two
(Table 1). Although the average epidemic size is highly dependent
upon these epidemiological parameters, the effect on the optimal
because the true variation in strain characteristics has yet to be
quantified, but do reflect the general robustness of the optimal ring
size. We note, however, that strains that are effectively transmitted
as air-borne plumes may give rise to dramatic changes to the
transmission kernel, and thus could invalidate the use of localized
Ring vaccination is a form of targeted control that uses spatial
proximity as a risk factor, but other refinements to the targeting
could be used16,17. Table 2 shows that whether vaccination is priori-
tized outside-in or inside-out within each day has very little effect on
theoptimalringsize or theassociatedepidemic impact.Surprisingly,
if all prioritization is completely ignored and farms within a given
radius are vaccinated at random there is a slight improvement over
outside-in. This is because local herd-immunity (the threshold
population level of immunity needed to prevent disease spread)
around each IP is reached more quickly as it does not require the
vaccination of all cattle. Identifying farms within a given radius, but
prioritizing vaccination towards those farms with the largest num-
bers of cattle or livestock (rather than prioritizing in chronological
order) again increases the optimal ring size and decreases the
epidemic impact. This improvement mirrors the general epidemio-
logical tenet that targeting of control measures can provide a
substantial increase in effectiveness.
Avariety of previous models have shown the success of CP culling
in the absence of vaccination1,4,5and this is confirmed by our
simulations. However, for all vaccination priorities investigated
here the inclusion of additional CP culling leads to the total loss of
more farms (though not necessarily more livestock). We postulate
that this loss of efficiency is because a high proportion of farms that
are CP culled are (at least partially) protected by vaccination, and
therefore the effect of CP culling on the spread of the infection is
diminished comparedto a situationwith novaccination. Thismeans
that the reduction in cases owing to CP culling is smaller than the
number of additional culls performed, and we therefore conclude
that CP culling is not advantageous when combined with
We next turned to new strategies for optimizing the balance
between managing existing and new foci of infection. A simple and
very successful strategy is that of prioritizing farms for vaccination
purely by their proximity to any previously infected premises, while
vaccinating at capacity every day. The resultant epidemic impact is
smaller than for all other prioritizations investigated (Table 2). This
strategy has two main advantages. Firstly, an optimal ring size does
not have to be determined, because this strategy utilises the full
logistical resources that are available on any day. Therefore any
change in logistical constraints does not necessitate a change in the
a new IP is rapid, thus focusing control on those farms that are at the
most immediate risk. The benefits gained from this targeting
approach are not particularly sensitive to the precise ordering of
vaccination, the number of cattle that can be vaccinated per day or
the initial seeding of the epidemic. A further improvement is
achieved if the prioritization is in terms of the shortest distance to
Table 1 | Effect of strain variation on optimal vaccination radius
Optimal ring size (km)
Kernel width (keeping total transmission
The effect of varying epidemiological parameters, mimicking variation due to alternative
strains, upon the optimal position of the outer vaccination ring. 95% confidence intervals are
shown in parentheses.
Table 2 | Effect of control strategies on optimal vaccination radius
Vaccination priorityOptimal ring size (km) IPs þ DCs þ CPs
Total animals culled (millions)
Earliest IP reporting date (outside-in)
Earliest IP reporting date (inside-out)
Largest cattle farms
Largest mixed farms
Shortest distance to any IP
Shortest distance to any IP or DC
Shortest distance to any IP or DC identified within previous ten days
No vaccination (þCP culling)
Earliest IP reporting date (þCP culling)
Largest mixed farms (þCP culling)
Shortest distance to IP (þCP culling)
The effect of vaccination priority and additional control strategies upon the optimal position of the outer vaccination ring together with the associated epidemic impact (IPs, and also DCs and
CPs if appropriate) and the total number of animals culled. 95% confidence intervals are shown in parentheses.
NA, not applicable.
NATURE|Vol 440|2 March 2006
© 2006 Nature Publishing Group
any IP or DC identified within the past ten days, as this also targets
vaccination around DCs that are suspected of being infected and
ignores regions of the country that no longer pose any risk.
We have shown that reactive vaccination could be a very powerful
tool for combating FMD, and potentially other locally transmitted
pathogens, when combined with efficient IP and DC culling and
animal movement restrictions. As ever, rapid implementation is
distance to any reported infection or DC has strong epidemiological
benefits over traditional fixed-radius policies, especially in terms of
balancing local and regional control and making full use of the daily
vaccination capacity. It would also be easy to achieve practically,
because optimization for a particular set of logistical constraints is
unnecessary. This specific result echoes a more general trend, that
detailed data-driven models are becoming an increasingly important
tool in determining optimal epidemic control policies.
is given by the equation:
where S(i)and T(j)refer to the susceptibility and transmissibility of uninfected
farm i and infectious farm j, respectively. These farm-level measures are in turn
calculated from species-level parameters where sAand tArefer to the suscepti-
bility and transmissibility of species A, and nðiÞ
farm i. These animal-level parameters are estimated from the 2001 data by
matching to the cumulative number of cases and culls in five distinct regions
(Cumbria, Devon, Wales, Scotland and the rest of England). The best-fit
parameters for Cumbria exemplify the general pattern: tcow ¼ 7.7 £ 1027,
tsheep ¼ 5.1 £ 1027, scow¼ 10.5, ssheep¼ 1 (see Supplementary Information
for full parameters and confidence intervals). Pigs have been ignored in this
modelfor simplicity becausethereis insufficientdatafrom the2001epidemicto
determine precise parameter values or even to show that the parameters are
significantly different from zero. K is the infection kernel and dijis the distance
between susceptible farm i and infectious farm j. The kernel is calculated from
the contact tracing performed by DEFRA during 2001 after movement restric-
tions, which provides a distribution of distances between infecting farms2. We
note that at present in the UK, only the residential position of the owner of the
farm is recorded, which could be separate from the farm itself and also distant
fromwherelivestockis kept. However, largediscrepanciesinpositionaffectonly
a small fraction of the total data and sensitivity analysis shows that the general
results are unaffected by small errors in the positional data.
This epidemiological framework is coupled with the prompt culling of
livestock on IPs (within 24 hours) and on DCs (within 48 hours). The model
therefore mimics the rapid speed and significant levels of local control that were
could be achieved from the detection of an epidemic in the future. The
simulations are initialized with the exact conditions of 23 February 2001
(including infected, infectious and reported farms); in any future outbreak it
is hoped that the disease will be detected earlier so that the spread is less
disseminated (Fig. 2a).
Calculation of dangerous contacts and contiguous premises. In practice, DCs
are identified for each IP on a case-by-case basis, and are based on veterinarian
In our model, DCs are determined stochastically, accounting for infection risk
factors and known transmission events (see Supplementary Information) and
parameterized to match the best sustainable levels achieved during 2001.
CPs are determined by a comprehensive knowledge of the farm geography
and are defined as farms that share a common boundary—in practice, this is
again determined on a case-by-case basis using local maps and knowledge.
However, the data on farms within the UK only identifies the location of the
owner of the farm. We approximate CPs by tessellating around each point
location, taking into account the known area of each farm, to obtain a surrogate
set of adjacent farms. Clearly this set of farms will not necessarily be the set of
true CPs1,2, but this approximation will capture many of the elements of local
Ais the number of species A on
throughout the paper about the characteristics of vaccination. First, we assume
that only cattle are vaccinated—vaccination of other species, such as sheep,
would substantially limit the number of farms that could be vaccinated per day
and therefore limit the effect of vaccination (see Supplementary Information).
Unless otherwise stated, the vaccine efficacy is taken to be 90%, and a four-day
delay fromvaccination to immunity is assumed. Therefore, after four days, 90%
of cattle on a given farm become totally immune and the remainder are totally
susceptible. We make the pessimistic assumption that during the four days
between vaccination and immunity, all cattle are completely susceptible and if
infected, the disease progresses in the same way as for non-vaccinated cattle. A
farm in which 90% of the cattle are immune has the same transmission and
susceptibility properties as a farm with 10% of the number of cattle. In general,
we assume that 35,000 animals can be vaccinated per day (unless otherwise
stated) and there is no limit on the number of farms that can be vaccinated per
shortest-distance ring vaccination. The latest DEFRA estimates suggest that a
maximum of 300 farms could be vaccinated per day, so our limit of 35,000
animals perday is reasonable. Vaccinationis assumed to begin five daysafter the
disease is first detected, in agreement with the DEFRA contingency plan6.
Received 16 June; accepted 29 September 2005.
1. Keeling, M. J. et al. Dynamics of the 2001 UK foot and mouth epidemic:
stochastic dispersal in a heterogeneous landscape. Science 294, 813– -817
Keeling, M. J., Woolhouse, M. E. J., May, R. M., Davies, G. & Grenfell, B. T.
Modelling vaccination strategies against foot-and-mouth disease. Nature 421,
136– -142 (2003).
Ferguson, N. M., Donnelly, C. S. & Anderson, R. M. The foot-and-mouth
epidemic in Great Britain: pattern of spread and impact of interventions.
Science 292, 1155– -1160 (2001).
Ferguson, N. M., Donnelly, C. S. & Anderson, R. M. Transmission intensity and
impact of control policies on the foot and mouth epidemic in Great Britain.
Nature 413, 542– -548 (2001).
Morris, R. S., Wilesmith, J. W., Stern, M. W., Sanson, R. L. & Stevenson, M. A.
Predictive spatial modelling of alternative control strategies for the foot-and-
mouth disease epidemic in Great Britain, 2001. Vet Reco ´rd 149, 137– -144
DEFRA Foot and Mouth Disease Contingency Plan khttp://www.defra.gov.uk/
Anderson, I. Foot and Mouth Disease 2001: Lessons to be Learned Inquiry (The
Stationery Office, London, 2002).
Council Directive 2003/85/EC. Official J. Eur. Union L306, 46, 1– -87 (2003).
Muller, J., Schonfisch, B. & Kirkilionis, M. Ring vaccination. Math. Biol. 41,
143– -171 (2000).
10. Barnett, P. V. & Carabin, H. A review of emergency foot-and-mouth disease
(FMD) vaccines. Vaccine 20, 1505– -1514 (2002).
11. Woolhouse, M. E. J., Haydon, D. T., Pearson, A. & Kitching, R. P. Failure of
vaccination to prevent outbreaks of foot-and-mouth disease. Epidem. Infect.
116, 363– -371 (1996).
12. Ferguson, N. M. et al. Planning for smallpox outbreaks. Nature 425, 681– -685
13. McLean, A. R. & Blower, S. M. Modelling HIV vaccination. Trends Microbiol. 3,
458– -463 (1995).
14. Williams, J. R., Nokes, D. J. & Anderson, R. M. Targeted hepatitis B
vaccination—a cost effective immunisation strategy for the UK? J. Epidemiol.
Comm. Health 50, 667– -673 (1996).
15. Woolhouse, M. E. J., Haydon, D. T. & Bundy, D. A. P. The design of veterinary
vaccination programmes. Vet. J. 153, 41– -47 (1997).
16. Woolhouse, M. E. J. et al. Heterogeneities in the transmission of infectious
agents: implications for the design of control programs. Proc. Natl Acad. Sci.
USA 94, 338– -342 (1997).
17. Matthews, L. et al. Neighbourhood control policies and the spread of infectious
disease. Proc. R. Soc. Lond. Ser. B 270, 1659– -1666 (2003).
Supplementary Information is linked to the online version of the paper at
Acknowledgements This research was supported by the Wellcome Trust.
Author Contributions M.J.T. and M.J.K. were responsible for the model
formulation and analysis of results; N.J.S. provided helpful discussions
throughout; D.J.S. generated cleaned demographic and epidemic data; R.D. and
S.P.B. provided vital statistical input; M.E.J.W. and B.T.G. were instrumental in
the initial development of the project. All authors contributed to the writing of
Author Information Reprints and permissions information is available at
npg.nature.com/reprintsandpermissions. The authors declare no competing
financial interests. Correspondence and requests for materials should be
addressed to M.J.K. (email@example.com).
NATURE|Vol 440|2 March 2006