Supplementary Material for
Dynamics of the 2001 UK Foot and Mouth Epidemic - dispersal in a heterogeneous
landscape.
Keeling, Woolhouse, Shaw, Matthews, Chase-Topping, Haydon, Cornell, Kappey,
Wilesmith & Grenfell
This document provides a more comprehensive development
of the model; we describe in some detail its formulation, including
initialisation, parameter estimation, within-farm dynamics, and the
assumptions underlying the transmission process. We then present a map
illustrating the spatial risk of infection, in terms of the value of
the basic reproductive ratio of infection, R0, for each
farm. Two animated images are also included, to illustrate the
observed spatio-temporal pattern of the epidemic. Finally, we explain
the different control options used in the main paper and focus in more
detail on the effectiveness of vaccination.
Contents.
Details of the model formulation.
Initialisation.
The model is initialised using the data on all livestock holdings in
the UK from the June 2000 census. Each farm (or holding) is identified
by a unique numerical code - the CPH (county, parish, holding) number.
For each holding the following information is also utilised;
-
location of the farm house
-
area of land occupied
-
number of cattle, sheep, pigs, goats and deer during the summer 2000 survey.
Initially all these farms are considered to be susceptible, with the level
of susceptibility dependent on the number and type of livestock. Almost
all simulations are started from 23
rd February, once the movement
restrictions were firmly in place. Those farms that have just caught the
disease on 23rd February will not be reported until about 9 days later,
hence data on the reported cases until 3rd March are needed to determine
the initial state of all farms. We note that the state of farms prior to
culling is frequently indeterminable; although this does not pose a significant
problem when simulating from near the start of the epidemic, it can cause
difficulties attempting to simulate forward from the latest available data.
Within-farm behaviour.
Following standard epidemic models, farms are classified as susceptible,
exposed, infectious or culled. A susceptible farm can catch
foot and mouth disease (FMD) from any infectious source, after which it
is classified as exposed. During this 4 day exposed period the animals
are incubating the disease but are not yet infectious. The farm next passes
into the infectious class; it can now spread the disease to susceptible
farms. In general, symptoms of the disease appear about 9 days after infection,
after which it may take a further 1 to 4 days to slaughter the entire herd
(this time decreased during the epidemic). Therefore the infectious
period can last between 5 and 9 days after which the holding is classified
as culled.
In practice the times from infection to infectivity, reporting and
slaughter will be distributed about some time varying mean.
For computational convenience we have constrained these time lags to take
their mean value. More detailed simulations have shown that ignoring the
distributed nature of the true lags has very little effect on either the
spatial or temporal pattern of the epidemic. Theoretical considerations
indicate that tighter distributions may lead to a slighter higher level
of secondary cases, but such differences will be absorbed into the parameterisation
of the model.
There has been some speculation about the role and existence of a within-farm
epidemic. Clearly if initially just one animal was infected, then there
should be a buildup of the within-farm epidemic over time and hence an
increase in the farm's infectivity. However, there is no evidence for such
a buildup from the data - the rate at which secondary cases are generated
is approximately constant throughout the infectious period. This may be
due to the aggregated nature of infection, such that many livestock on
a farm get infected at any one time.
Infection processes.
As stated in the main paper, the risk of infection from an infectious
farm to a susceptible one is dependent upon the distance between the farms,
as well as the number and type of livestock in each farm, with these two
factors acting multiplicatively. The probability that a susceptible farm
is infected on any given day is,
Here
Ni is the vector number of livestock (sheep and
cattle) on farm i, and
S and
T are the vectors of susceptibility
(risk of catching the disease) and transmissibility (rate of spreading
the disease) respectively. See below, and note 14 in the paper, for a discussion
of parameter significance
| Species |
Susceptibility (per animal) |
Transmissibility (per animal) |
| Sheep |
1.0
|
2.36 x 10-7
|
| Cattle |
15.2
|
4.30 x 10-7
|
Species level disease parameters for the spread
of disease between farms estimated by a mixture of maximum likelihood methods
and repeated stochastic simulation. These parameters reflect the average
values across the UK and may subsume other factors such as the greater
human contact with cattle. The values are normalised such that the kernel
K is one at the origin.
The effect of distance between farms is captured by the dispersal
kernel K (Figure 1B of main paper) which has been estimated from
the contact tracing performed by MAFF/DEFRA. Again this represents
a national average, and encompasses a variety of transmission routes such
as animal movement, close contact, movement of vehicles and wind-borne
spread.
Slaughter and culling
An important element in the modelling process is to capture the observed
rate of culls and slaughters. The slaughtering of infectious herds is an
effective control measure as it reduces the infectious period of the farm
and therefore reduces the transmission of the disease. The culls of at-risk
farms surrounding an infected premise have two main epidemiological roles,
they remove the farms that are most likely to be incubating the disease
and they reduce the local density of susceptible farms in areas of high
infectivity.
Three types of cull have been classified in the UK control program,
-
CP culls remove Contiguous Premises - those farms that border an IP (infected
premise).
-
DC culls remove Dangerous Contacts - those farms that have been visited
by people, vehicles or livestock from an IP and therefore are potentially
infected by the disease.
-
3km and welfare culls - a systematic removal of livestock either for welfare
reasons or in an attempt to slow the spread of the disease in the worst
areas.
.
The number of daily cases reported, the proportion
of neighbouring premises that are culled, the number of dangerous contacts
that are culled per infected premise, the number of longer range culls
which are mainly concentrated in Cumbria, Devon, Gwynedd, and Dumfries
and Galloway, and the average delay between report and slaughter of an infected
farm. The blue line gives the smoothed values used for the simulations
Note that the length of time from reporting to cull varied through
time, in parallel with the culling effort. Initially there was up to a
week's delay, but later in the epidemic a determined attempt was made to
slaughter the infected premise within 24 hours and to perform all other
associated CP and DC culls within 48 hours.
Contiguous premises are identified within the model by tessellation
about each farm location, using the area of each farm as a weighting. On
average each farm has 6.5 CPs neighbouring it, though this value varies
from region to region and has a large amount of variation between farms.
This will of course differ from the true number of contiguous farms (we
only know the location of the farmhouse, which is not sufficiently detailed
information) and this accounts for some of the discrepancy with the DEFRA
classification. The model is formulated such that premises which are both
CPs and DCs are classified as CPs. Dangerous contacts are assumed to share
the same dispersal kernel as the infection processes. Model results would
indicate that at least1 in 2 dangerous contacts results in the transmission
of infection.
Important events in the spread and control of
FMD
The modelling of the foot and mouth outbreak is complicated both due
to the spatial heterogeneities present,and also because of the temporal
variations in control measures. The table below summarises the approximate timing of the main events
during the early part of the epidemic, chronicling the progress of the
disease and the attempts to control it.
| Early February |
Disease likely to have entered the UK. |
| 19th February |
Foot-and-mouth disease first suspected |
| 20th February |
Foot-and-mouth disease confirmed |
| 23rd February |
Culling initiated of Infected Premises (IP) and Dangerous Contacts
(DC). Movement restrictions are brought into force. |
| 15th March |
Sheep, goats and pigs within 3km of an IP in Lockerbie, Carlisle and
Solway are targeted for culling |
| 23rd March |
Contiguous Premises (CPs) are included in the cull |
| 26th March |
Epidemic reaches its maximum with 54 cases in one day |
| 27th March |
3km cull begins in the Penrith valley, Cumbria |
| 29th March |
24/48 hour policy begins, in which IPs are slaughtered within 24 hours,
and DCs and CPs are culled within 48 hours. |
| 14th April |
3km cull in Cumbria reaches its height. |
| 26th April |
Sheep, pigs and especially cattle from farms with high biosecurity
may be exempt from culls |
| 10th May |
First case reported in the Settle area. |
| 20th June |
First day with no reported cases. |
Parameter Estimation.
We have already noted the difference in transmission and susceptibility
between sheep and cattle. Although these and other parameters are in agreement
with previously-published qualitative measurements, the reported epidemic
remain the only reliable source of data with which to parameterise the
model. The shape of the transmission kernel can be estimated from tracing
of infection, but the other parameters (relative susceptibility
of cattle compared to sheep, transmissibility of cattle, transmissibility
of sheep and ratio of dangerous contacts to infectious contacts) need to
be estimated by comparing the model and data. In a Bayesian approach, approximations
for the parameters were made and these were adjusted using a hybrid mixture
of maximum likelihood calculations and repeated simulation of the stochastic
model. The maximum likelihood approach calculates the probability that
the model would give an accurate prediction one day ahead, (ie predicting
tomorrow's cases from all the previous data). The daily probability is
therefore calculated as the product of the probability of not been infected
for all those farms that remain susceptible multiplied by the product of
being infected for all those farms that are infected on that day.
where
Pi gives the
probability calculated by the model that farm
i is infected on
a given day (as given above). The daily probabilities are multiplied
together to achieve a single likelihood value to be optimised. While
this provides a convenient and deterministic means of optimising the
parameters (such that the probability is maximised), regional biases
in the data mean that repeated simulations of the stochastic model are
necessary for the final fine-tuning of parameters (see note 14 in the
paper). For these stochastic simulations we match both the temporal
pattern of case reports as well as the regional patterns of spatial
cases, by obtaining the least-squares fit at the county level. We note
that the relative transmission rates and susceptibility rates of
cattle and sheep may not be a true reflection of the actual rates, but
may partly reflect the known biases in the data and other
uncertainties.
Impact of transmission heterogeneities on the quality
of fit.
The detailed records of the number and composition of livestock
on all farms in the UK, shows that there are large heterogeneities present
in the system. We illustrate this by plotting the distribution of species
in all farms, those farms that reported infection and those that were culled,
The species composition in all farms, those farms
reporting infection and those farms that are culled. Contours represent the smoothed
distribution of farm-types in terms of numbers of sheep and cattle. Blue contours indicate
minimum values, red contours show the most likely farm-types.
Four key elements are clear from these graphs. First, there is a
very wide range of variability in the number of livestock in each farm,
ranging from a few animals to tens of thousands. Second, we can classify
farms into three basic groups, those with cattle only, those with sheep
only and mixed farms. Third, mixed farms are more likely to be infected
and slaughtered than their basic distribution would suggest. Finally, we
notice that there is a distinct trend toward culling those farms with very
low numbers of sheep - this is particularly pertinent given that cattle
and large farms appear to be responsible for the majority of the disease
transmission.
Having discussed in detail the mechanism and parameterisation of the
model, we now consider the model formulation itself - in particular the
form of the interaction between farms. Throughout the paper, we have assumed
that both the transmission rate and the susceptibility of a farm were proportional
to the number of livestock of each type, and that cattle were much more
susceptible than sheep. The figure below shows the results of models which
ignore these heterogeneities.
The temporal and spatial location from models
which ignore farm level heterogeneities. The left-hand graph ignores species
differences, while the right-hand graph also ignores differences in farm
sizes. For both of these graphs the hybrid fitting procedure was used to
determine the choice of parameters.
When the species parameters are assumed identical, the left-hand
graph shows the results from the best-fit model (Susceptibility
= 1.0 per animal, Transmissibility=5.18 x 10-7 per animal).
We note that the epidemic is somewhat delayed, and the main focus of infection
is Wales. This is as expected; Wales has a proportionately much higher
concentration of sheep, which due to their greater number act as the driving
force in this model epidemic.
If all farms are assumed to be identical (right-hand graph), such
that the farm-size and type of livestock are ignored, then the
best-fit model (Susceptibility = 1.0 per farm,
Transmissibility=0.204 per farm) is in much closer agreement
with the data - although not as good as the full model. The
distribution of cases is far more diverse, with Cumbria experiences a
much smaller epidemic than observed, whereas central England and Devon
fare much worse.
It is therefore clear to us that the heterogeneities between farms
in both the number and type of livestock are very important if we are to
capture the spatial spread of the epidemic. Although the model with identical
farms is able to produce a good match to the temporal dynamics, it is the
spatial dynamics that determine the behaviour of the epidemic in the latter
stages - only a full model can accurately capture these. We note
that our assumption of linear increase in transmission and susceptibility
rates with the number of animals is a simplification of the true behaviour
which is probably non-linear in nature. It remains to be seen if this epidemic
will provide sufficient information to allow the detailed effects of farm
size to be quantified.
The spatio-temporal pattern of epidemics.
As stressed throughout the paper, the spatial aspect of the disease is
very important. Local spread leads to a local depletion of the susceptible
farms and therefore a more rapid decline than mean-field (non-spatial)
models would predict. The localised nature of the dispersal kernel also
means that wave-like spread of the disease can be observed in many regions
(for example a clear spread south-west up the Penrith valley in Cumbria
is observed in the movies below). Our models must capture both the spatial
and temporal aspects of the dynamics if they are to be useful as a management
tool over the full course of the epidemic.
The
movie of the epidemic in the
UK (865K) [readers not using Netscape may wish to download the
image and play it through Real
Player] clearly shows how the disease is aggregated in particular
regions. At a more local scale the movie of the epidemic in
Cumbria (990K) [readers not using Netscape may wish to download
the image and play it through
Real Player] allows us to see some evidence of wave-like spread of
the epidemic and detect the possible role of roads in the transmission
of the disease. (These movies are colour coded, unaffected farms
(green), the observed cases (yellow for infected, red for infectious)
and culled farms (black). Relating the detailed pattern of the
epidemic to road maps and topography is a fruitful area for future
work.
The spatial distribution of high risk areas can be captured by considering
the average value of
R0 in a given region. Given that
R0
is defined as the average number of secondary cases produced by an infectious
farm in a totally susceptible populations, values above one
show regions in which the disease could deterministically increase.
The R0 map for the UK is calculated
from the probability Pi
, by considering how many secondary
cases could be produced by each of the 144000 farms in the DEFRA database.
The results are then aggregated into 10x10km squares. The map gives the
average value of R0 in each square, with the colour-coding highlighting
those areas where R0 > 1 where we expect the number of cases
to initially increase if there was no intervention.
The
R0 risk map captures the observed concentration of
foot and mouth disease in the regions of Cumbria, Wales and Devon.
The effects of initial spread on the subsequent epidemic
It has been suggested that modern farming practices, which involve the
rapid and long-distance movement of livestock may have exacerbated the spread
of FMD. We consider this proposal by varying the initial seeding of the
epidemic before movement restrictions were enforced. This is achieved by
varying the timing of the response to the arrival of FMD in Britain; a more
prompt response is equivalent to reduced early movements or a smaller initial
seed. The average distance between an infectious premise and the secondary
cases it generated was much larger before the movement restrictions (see figure below) - this also leads to a very high early reproductive ratio,
R. More
immediate action (or a smaller initial seed) causes a substantial decrease in
both the total number of cases and the spatial extent of the epidemic (see right-hand figure below). By contrast, the dissemination of a large initial 'dose' of infection
might have sparked major epidemics in areas other than Cumbria - the main
focus of the current outbreak.
Left-hand graph: for reported cases that have been traced, the probability that the distance between primary and secondary cases was less than a given distance. Right-hand graph: the effects of more prompt or slower action, by varying the timing of the movement controls and culling levels. The lower graph shows the total number of cases against the delay. The inset graphs are for taking action six days early, as observed, or six days late. The red dots indicate the seeding of the epidemic - farms that have been infected by the time the movement restrictions were initiated.
A comparison between alternative control strategies.
One of the most useful contributions made by modelling approaches to the
control of this FMD epidemic has been to advise on the potential effectiveness
of various control strategies. The results from six different culling approaches were reported
in table 1 of the main paper, the differences between these approaches is outlined in more detail below,
| Observed control policy |
Infected premises and at risk farms are culled at levels mimicking the observed rates. The delays between reporting and slaughter/culling also follow the observed pattern. The extended 3km and welfare culls of sheep are also included. This provides the model results against which all other control policies are tested. |
| Standard |
Slaughter of infected farms and neighbourhood culls follow the observed levels and delays. The extended 3km and welfare culls are ignored. |
| IP only |
Only the infected premise is removed by slaughter, with the delay following the observed pattern. |
| Prompt |
Although the level of neighbourhood culls follows the observed pattern, the delays follow the 24/48 hou rule such that infected premises are slaughtered within 24 hours and neighbourhood culls are performed within 48 hours. The extended culls are again absent. |
| Intensive |
Throughout the simulations (starting from 23rd February) the level of neighbourhood culls remains constant at the levels achieved in the tail of the epidemic. However, the delays still follow the observed pattern. The extended culls are again absent. |
| 3km ring |
All farms within 3 km of an infected premise are culled. The delay from reporting to slaughter / cull follows the observed pattern. The extended culls are irrelevant. |
Which set of control measures are optimal depends on
what characteristics of the epidemic we wish to minimise. The 3km cull
produces the largest decline in both the number of reported cases and
the duration of the epidemic, but this success is bought at the cost
of many more culled premises. Whether more prompt culling, higher
level of culling or a mixture of the two is the most appropriate
action depends critically upon the available resources and manpower.
We now consider the effects of vaccination on the disease
spread and the number of premises vaccinated for two different
vaccination strategies applied in addition to the Standard cull
policy. Although prophylactic mass vaccination is a successful method
of controlling many human diseases, the rapid spread of FMD means that
the delay from vaccination to protection coupled with the delay from
infection to reporting compromises the efficiency of
vaccination during the outbreak (Ferguson, Donnelly, Anderson 2001 reference
(4); N Ferguson pers. commn.).
Two main strategies are considered. The left-hand
graph shows the effects of vaccinating a fraction of cattle herds within
a 3km ring of any infected premise. The right-hand graph attempts to block
the spatial progress of the disease by vaccinating cattle herds in a 90 degree barrier
between 5 and 10 km in front of the disease, the barrier is erected in
the direction that most susceptible farms lie. In both of these scenarios,
vaccination started on 23rd February and CPs and DCs are culled at the
observed levels.
Vaccination leads to a reduction in both reported cases, the
number of culled herds and epidemic duration. However, the decline
needs to be evaluated relative to the number of animals that would
have to be vaccinated and the resources and manpower this would
require. We also note that these simulations assumed that vaccination
began close to the start of the epidemic (see main text). Evaluating
other ring and barrier vaccination strategies, as well as prophylactic
mass vaccination, is an important area for future modelling work.
Uncertainties in the epidemiology and transmission of FMD
Many uncertainties exist in the epidemiological and farm-level data. These could all have a potential impact on the dynamics and progress of the model, although extensive computer simulations support our arguments that the qualitative nature of our results will still hold. Detailed below are some of the main uncertainties, together with how these would affect the model dynamics.
-
Relative infectivity and susceptibility of sheep and cattle. Experimental results agree with the pattern of species differences used within the model. Quantitative changes to the species parameters will modify the predicted spatio-temporal distribution of outbreaks; our parameters have been chosen to give the best match to the location of high risk areas. However this choice of parameters is contingent on the accuracy of the census distribution of animals on farms (see below).
- Ascertainment of infection in sheep. FMD is much harder to diagnose clinically in sheep, which may reflect the lower contact rates between humans and sheep, particularly hill sheep, compared to cattle and it is possible that some sheep have milder infections which last longer. Such hidden cases in sheep are a potential problem for any model - although they would not affect the fit of the model to the main epidemic, a large number of such infections may significantly increase the tail. In our formulation, hidden cases would be interpreted as more long range (delayed) sparks of infection than we would predict, although to some extent parameterisation the movement kernel would compensate for this. In fact, the current serological evidence from contiguous premises indicates that hidden infections in sheep may not be a widespread phenomenon. However instances like the Brecon Beacons outbreak in July and Hexham in August/September indicate that hidden infections can be !
an important regional issue during
the tail of the epidemic.
- Over-reporting of cases. Subsequent serological testing of those farms diagnosed (and recorded) as having foot-and-mouth disease, has shown that a fraction of these farms may in fact have been disease free. The over-reporting of the true number of cases was largest (up to 25%) shortly after the peak of the epidemic; this would lead us to over-estimate the reproductive ratio, R, of the disease and hence over-estimate the effectiveness of the control measures necessary to limit it.
- Within farm epidemics. The farm has been taken as the basic unit of our model; although in reality there will also be epidemic dynamics within the farm. It has therefore been postulated that the infectivity of a farm should increase through time. Contact tracing does not support this assumption; any effect may be negated by aggregation in the transmission process or rapid reduction in transmission after reporting.
- Estimated spatial distribution of animals of different species. The presence of biases in the livestock data is well accepted, and probably leads to a significant over-estimate of the number of sheep in Wales at the start of the epidemic and smaller variation in other areas. This effects our ability to model the exact spatial distribution of cases, which may explain why we slightly overestimate the number of cases in Wales and Yorkshire. Such a bias will also alter the proportion of mixed and single-species farms recorded in the database, although not sufficient to change our quantitative conclusions. Some of this bias will be absorbed into the parameters by the fitting procedure.
- Seasonal variations in movements of animals. A related factor is the considerable movement of animals, particularly sheep, over the year . This may have some impact on the model dynamics, in particular ignoring the turn-out of cattle may explain why we slightly underestimate the persistence of the disease after May. The effect of movements will be more important if the disease persists until spring 2002, when the large-scale movement of animals at that time of year could lead to recrudescence of cases.
- Contact tracing. Our estimation of the transmission kernel and the culling of dangerous contacts relies on the contract tracing performed by MAFF/DEFRA. Although this is extremely valuable, the exact movement of the disease is complex and sources of infection may not be easily identified. The contact tracing is probably biased toward short-distance infection, which may cause a similar bias in the transmission kernel.
