Predicting infected but undetected infections during the
2007 Foot-and-Mouth Disease outbreak
Jewell CP, Keeling MJ, Roberts GO
April 24, 2008
Active disease surveillance during epidemics is of utmost importance in detecting
and eliminating new cases quickly, and targetting such surveillance to high-risk
individuals is generally more efficient than applying a random strategy 7;12 . Contact tracing has been used as a form of at-risk targetting, and a variety of mathematical models have indicated that it is likely to be highly efficient 4;14 . However
for fast-moving epidemics, resource constraints limit the ability of the authorities
to perform, and follow up, contact-tracing effectively. Here, as an alternative,
we present a real-time Bayesian statistical methodology to determine currently
undetected (occult) infections 10 . For the UK Foot-and-Mouth epidemic of 2007,
we use real-time epidemic data synthesised with previous knowledge of FMD
outbreaks in the UK to predict which premises might have been infected, but
remained undetected, at any point during the outbreak. This provides both a
framework for targetting surveillance in the face of limited resources, and also
an indicator of the current severity and spatial extent of the epidemic. Our
results demonstrate how statistical approaches to real-time prediction can yield
meaningful results in the face of small amounts of data and, therefore, a high
degree of statistical uncertainty. This methodology thus provides a compromise
between targetted manual surveillance and random or spatially targetted strategies. We anticipate such methods to be of substantial benefit in future outbreaks
of livestock diseases, providing a rigorously quantitative base for the assessment
of control measures, targetting of disease surveillance, and assessment of the
current extent of the outbreak.
Epidemic control essentially focuses on reducing the risk of transmission between susceptible
and infected individuals. Traditional methods such as vaccination or pre-emptive culling
serve to decrease the size of the susceptible population and may be used with varying success 1;13;22 . An alternative strategy is to focus on infected individuals, either quarantining
or culling to prevent onward transmission. In such cases, early detection is vital to reduce
the time for which the individual is actively shedding infection 5;9;22 . When infection occurs
significantly before the onset of symptoms, mechanisms of contact tracing become vital with
the aim of indentifying at-risk individuals by their interaction with known infectious cases 19 .
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Contact tracing has traditionally and successfully been used for a variety of human diseases,
including the 2003 SARS outbreak when high-risk individuals were quarantined and a variety of sexually transmitted infections where at-risk contacts are tested and treated 2;3 . For
foot-and-mouth disease, contact tracing from known Infected Premises (IPs) is manditory,
generating Dangerous Contacts (DCs) whose livestock must be culled. During the 2001 Footand-Mouth outbreak in the UK, contact tracing only was deemed insufficient to control the
epidemic – the high number of cases and the limited veterinary resources meant that contacts
could not be identified and removed quickly enough 5;6;13 . Indeed, subsequent studies have
demonstrated that for epidemics where the number of cases and/or contacts is high, contact
tracing loses its efficacy due to resource constraints 4;11;14 . Given these resource constraints,
the Contiguous Premises (CP) cull policy (where animals on premises neighbouring IPs were
culled) was implemented as a cude but rapid method of identifying and removing potentially
at-risk premises 8;21 . During the 2007 FMD outbreak in Surrey, however, the number of cases
was small and policy better developed in the wake of 2001, so that effective contact tracing
was implemented. This lead to a large number of dangerous contacts being identified and
subsequently culled; to what extent this would have been kept up if the epidemic had been
larger is unclear.
Here we analyse the 2007 FMD outbreak, and as the epidemic unfolds we focus on predicting
which premises may be infected but currently undetected – we term such premises occult
infections. Using prior information determined from the 2001 FMD outbreak in the UK 15 ,
we use recently developed Bayesian methodology designed for making quantitative riskpredictions in real-time as epidemics progress 10;17;18 . This methodology therefore addresses
two major epidemiological concerns: it predicts the uncertain scale of the current epidemic;
and it rapidly identifies high-risk premises which can be targetted for further epidemiological
investigation or control.
The model treats each premises as an individual which can be Susceptible, Infected, Notified
(Reported), or Removed. We adopt a similar model to Keeling et al. 13 which assumes
heterogeneity in infectivity and susceptibility according to the number and type of species
present on each premises, as well as using a spatial kernel to allow the probability of infection
to vary over distance. However, in contrast to Keeling’s model, we prefer to make direct
inference on a parametric distance function, rather than to use an empirical kernel derived
from contact tracing data obtained in the 2001 outbreak (see Methods).
Our analysis of the 2007 FMD outbreak began once it became apparent that the disease
was concentrating around the Surrey town of Egham. Throughout the epidemic, the MCMC
algorithm was run daily to give updated predictions. Figure 1(a) shows the number of
infected individuals at each stage in the epidemic, as well as the median predicted number
of undetected infections. For selected timepoints, Figure 2 then gives the probabilities that
supposedly susceptible farms are in fact infected. This gives an estimate of the current extent
of the epidemic given the uncertainties in parameter estimates and the routes of transmission.
Figure 2 shows that at each stage of the epidemic, the greatest occult probabilities were
associated with premises close to the known IPs, and were those with large numbers of
cattle or sheep – in agreement with qualitative understanding. Taken together with Figure
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1(a), this figure also demonstrates how the algorithm learns about the epidemic as data is
gathered from the field. The first map (13th September) is drawn based on only 4 data points.
The dataset here contains very little information on spatial transmission since only two farms
are closer than 10km, and results in a very diffuse prediction which is heavily influenced by
the priors (cf the fast-moving FMD 2001 outbreak). As the epidemic progresses and the
number of datapoints increase so does the amount of statistical information, leading to a
more precise estimate (see Posteriors in Supplementary Information). This is shown most
strikingly by the difference between 13th September and 17th September where just the
addition of one data point drastically reduces the median number of predicted undetected
infections, and correspondingly reduces the extent of the distribution of high probability
premises. Moreover, knowing in hindsight the time-frame of the epidemic we see how the
spatial distribution of high-risk farms localises towards the end, reflecting the fact that the
epidemic was being controlled effectively.
In disease control terms, it is necessary to have an idea on the extent of the resources needed
to cope with the outbreak. To answer this question, Figure 1(b) shows the predicted number
of new infections in a period of 7 days from the observation time. Consistent with the priors
and large amount of uncertainty, this prediction is high early in the epidemic with wide
95% credibility intervals ranging from 0 to 1196 new cases. However, this is soon refined
as the epidemic progresses to give the small value consistent with the final outcome of the
epidemic. We postulate that, taken together, these results give an overview of the extent
of the epidemic, the short-term risk posed by the presence of undetected infections, and the
resource required for control measures.
From this ability to predict occult infections, we can consider whether the Baysian occult
probabilities provide a better guide for targeted control than other measures. Table 1 gives
the occult ranking of the last four Infected Premises for dates in September before their
detection. For the Baysian analysis this ranking simply sorts the undetected farms in order
of highest probability, and this is compared to ranking farms by their proximity to known
Infected Premises. This table reflects the same patterns that were observed in Figure 2;
early predictions (based on just 4 IPs) by Baysian are highly diffuse, but once the fifth
infected premise is identified Baysian analysis outperforms simple spatial measures of risk.
However, the practical use of the Baysian occult ranking requires some careful consideration
in that many more factors should also influence effective surveillance, such as the centrality
of occults in known contact networks. Such model based assessment will, of course, never be
a replacement for the detailed contact-tracing performed by veterinarians when identifying
Dangerous Contacts, but may help to direct additional resources.
The “epidemiological triangle” is an important concept in studying the spread of infectious
diseases and states that the characteristics of an outbreak depend on the host, pathogen,
and environment in which they co-exist 16 . Changes over time in one or more of the three
vertices are therefore likely to alter the dynamics of outbreaks of the same disease recurring
in the population. This is demonstrated in the differences between FMD outbreaks in 1967
and 2001 where large differences in farming practices accounted for very different patterns
of transmission (Defra FMD information http://www.defra.gov.uk/footandmouth), and
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also in the recent 2007 outbreak which occurred in a sparsely populated area of the country.
This shows that by basing predictions of epidemic spread simply on past information, it is
impossible to quantify how relevant they are to the current outbreak. On the other hand,
early in the epidemic, there is inherently very little data from the current epidemic on
which to base a prediction. A Bayesian analysis provides a framework for learning about
a current disease outbreak. At the beginning of an epidemic, the prediction process begins
with prior information based on our past experience and expert opinion and then updates
this by incorporating data obtained from the field during the epidemic under study. Using
this methodology therefore provides a very natural way to make as accurate a prediction
as possible, whilst quantifying the amount of uncertainty we have about the quantities on
which the predictions are based.
We have shown, therefore, that a Bayesian approach to real-time inference on disease epidemics is feasible in the agricultural context. This provides a framework for monitoring
disease-control efforts, and provides information to adapt policy in the event that the outbreak does not behaving as expected based on past experience. Since this work demonstrates
its capability in probabilistically identifying occult cases, it is anticipated that such techniques will form the basis of statistical contact tracing to provide a targetted surveillance
strategy even in the face of limited contact-tracing resource.
Acknowledgements We would like to thank the following individuals for their assistance
with this work: Dr Mike Tildesley, Dr Theodore Kypraios. We also thank Defra for the
provision of data and funding for this research.
Methods
The methodology used in this analysis is based on recently developed Bayesian techniques 10 .
The model treats each premises as an individual which can be Susceptible, Infected, Notified,
or Removed. We define a Notification as the moment that, according to current FMD
control policy, a premises is placed under restrictions, and a Removal as the time when the
subsequent herd or flock cull is completed. Thus we allow for Infected and Notified premises
to be infective, the latter albeit at a potentially different level due to the upregulation of
biosecurity demanded by policy. In contrast to 2001, a national movement ban was imposed
in 2007 swiftly after the first case was detected. For this reason we adopt a similar model
to Keeling et al. 13 which assumes heterogeneity in infectivity and susceptibility according
to the species present on each premises, as well as using a spatial function to allow the
probability of infection to vary over distance. We thus parameterise the disease transmission
rate between infective (i) and susceptible (j) premises as:
βij = (β1 nci ψ + nsi ψ ) · (β2 ncj ψ + nsj ψ ) · β3
β52
ρ2ij + β52
i ∈ I, j ∈ S
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βij? = (β1 nci ψ + nsi ψ ) · (β2 ncj ψ + nsj ψ ) · β4
β52
ρ2ij + β52
i ∈ N, j ∈ S
(1)
where nc and ns represent the number of cattle and sheep respectively on premises i and
j , and ρij the Euclidean distance between them. β1 is then interpreted as the relative
infectivity of cattle versus sheep, β2 as the relative susceptibility of cattle versus sheep, β3
and β4 as the rate of spatial transmission for Infected and Notified premises respectively,
with β5 governing its rate of decay with increasing distance. We allow for a latent period in
which an individual is infected but not yet infectious by multiplying βij by a time-dependent
infectivity function (see Supplementary Information).
Since we make inference using Bayesian methodology, we require prior distributions for each
of the parameters in the model. For this, we turn to previous studies as our source of prior
information. Kypraios 15 provides a Bayesian analysis of the UK Foot-and-Mouth outbreak
in 2001 using a similar model from which we take priors for β1 ,β2 , and ψ. A combination
of the posterior distance kernel provided in Kypraios 15 and the empirical kernel provided
by Savill et al. 20 , which are remarkably consistent, inform our choice of β3 ,β4 , and β5 . For
each parameter, we use a Gamma-distribution with mean and variance chosen to match our
prior information (see Posterior plots in Supplementary Information).
Finally, since the infection times are not observed, we use a m odified Gumbel distribution
to model the time between Infection and Notification:
−bx −1)
FD (d) = e−a(e
with parameters a and b fixed to give a mean Infection - Notification time of 7.5 days 15 .
The posterior distributions are then evaluated for each observational timepoint using the
Adaptive Reversible-Jump Markov chain Monte Carlo algorithm described in the Supplementary Information.
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50
40
30
20
10
0
Number of individuals
Known IPs
Median undetected IPs
IP4
IP5
Sep 14
IP6 IP7
IP8
Sep 24
Oct 04
Date in 2007
50
100
Median
95% credibility interval
0
New cases in (Date,7+Date]
150
(a) The epidemic time line. Solid line shows the number
of detected premises. Dashed line gives the predicted
median number of undetected infections.
Sep 14
Sep 24
Oct 04
Date in 2007
(b) Predicted numbers of new cases within a 7 day time
period after the observation time.
Figure 1: Epidemic time line, occult and new cases predictions.
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Occult prob.
0.005 - 0.010
0.011 - 0.050
0.051 - 0.100
0.101 - 0.350
13th Sept
17th Sept
0.351 - 1.000
Known IP
Susceptibles
21st Sept
24th Sept
30th Sept
5th Oct
Figure 2: Probability of infected-but-undetected farms during Foot-and-Mouth 2007 outbreak
IP
IP5
IP6
IP7
IP8
Method
ba
sp
ba
sp
ba
sp
ba
sp
13th Sept
17 (0.42)
11
50 (0.17)
14
35 (0.23)
6
34 (0.23)
24
17th Sept
21st Sept
24th Sept
12 (0.11)
30
11 (0.11)
14
13 (0.11)
42
13 (0.08)
10
1 (0.34)
2
1 (0.25)
10
Table 1: Occult risk ranking calculated from Baysian Analysis (ba) occult probabilities (in
brackets), and by spatial proximity to currently infected individuals (sp)
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