The potential spread of severe footrot in Norway if no elimination programme had been initiated: a simulation model
A stochastic compartmental model can be used to simulate spread of disease within
a population 11]. In a SILI-compartmental model, the susceptible(S), infected(I) and low susceptible(L) compartments and the transmission of flocks between these compartments describe
the infection dynamics of the population. The susceptible flocks are not, and have
not been, infected with the agent causing the disease. The infected flocks have at
least one sheep infected with the agent causing disease and could infect susceptible
or low susceptible flocks. The low susceptible flocks do not have any animals carrying
the infection, and have a smaller contact network than the susceptible flocks, hence are less at
risk of acquiring a disease than the susceptible flocks. The low susceptible flocks
comprise of flocks with natural barriers towards other sheep flocks (called isolated
flocks) and flocks that have recovered from the disease and by this increased their
biosecurity measures (called recovered flocks). The latent period is assumed to be
zero, and the immunity period for a flock is negligible.
In the model, flocks are transferred from one compartment to another at different
rates. The infection rate (?) is the rate at which susceptible flocks become infected.
This is dependent on the number of contacts, and the risk of transmission of disease
per contact. The recovery rate (?) is the rate of recovery of infected flocks, and
which are accordingly assigned to the low susceptible compartment. The reversion rate
(?) is the rate at which low susceptible flocks become infected, and by this transferred
to the infected compartment.
Spread within subpopulations
Subpopulations can be defined if the spread of disease is not uniform in the population
12], but highly reduced from one geographical area to another. Each subpopulation is
then modelled with their own SILI-compartmental model with their own infection, recovery
and reversion rate. These rates are based on specific values for each subpopulation
that influence the spread of the disease in question.
Spread between subpopulations
Spread of disease is expected to be faster within the subpopulations than between
two subpopulations as flocks within each subpopulation are expected to have more contact
than flocks from two different subpopulations. Different types of contact between
flocks in separate subpopulations may occur, leading to different transmission routes.
Each transmission route between subpopulations are specified and quantified separately.
Only susceptible flocks in the subpopulations are expected to be infected by other
subpopulations as most of the low risk flocks have increased biosecurity and thereby
will not be infected through the between subpopulation transmission route.
Model
Equations 1–3 and Figure 1 show the differential equations of the SILI-compartmental model for one subpopulation
with the possible introduction of infection from other subpopulations. The equations
give the number of flocks in the susceptible (1), infected (2), and low susceptible
(3) compartments in a subpopulation for each year.
(1)
(2)
(3)
Figure 1. Susceptible – Infected – Low susceptible – Infected (SILI) model of severe footrot
among sheep flocks. The simulation model was developed to estimate the potential spread of severe footrot
in Norway if no elimination programme was initiated and to estimate the relative importance
of the different spreading routes. The figure shows the transmission dynamics of severe
footrot within one county (i) for one time step with possible introduction from other counties (j) through sheep movement, cattle movement and sharing of common pastures. The model
was used to calculate all the 19 counties separately.
where i is the subpopulation receiving the infection, j is the subpopulation transmitting the infection and ? is the time interval in years,
S is the number of susceptible flocks, I is the number of infected flocks, L is the number of low susceptible flocks, ? is the rate at which susceptible flocks
become infected, ? is the rate at which infected flocks recover and hence become low
susceptible flocks, ? is the rate at which low susceptible flocks become infected,
?, ? and ? is three possible ways of introduction of infection between subpopulations.
As the starting point for the simulations, the flocks infected with the disease was
assigned to the infected compartment, the isolated flocks were assigned to the low
susceptible compartment, and the remaining flocks were assigned to the susceptible
compartment. Year was the time step and the model was run for the number of years
desired.
Adaptation of the model to footrot in Norwegian sheep flocks
A SILI-compartmental model was developed for estimating the spread of severe footrot
in Norwegian sheep flocks without an elimination program. The isolated flocks in the
low susceptible compartment were defined as sheep farms more than 3 km away from any
other sheep farm. This was based on a study by Grøneng et al. 13] which showed that a geographic distance of more than 3 km between the main buildings
of different sheep farms was not a significant risk factor in the univariable analysis.
We interpret this as sheep farms with more than 3 km distance to the nearest sheep
farm have a lower risk of contracting footrot.
The infection rate of footrot was calculated based on the rate of spread from the
introduction of footrot in Norway in 2005 until the initiation of the elimination
programme in 2009. At this time, severe footrot had only been detected in the county
of Rogaland, but since different regions within the county possessed highly different
rate of spread, the infection rate was expressed by a Pert distribution. Rogaland
County excluding Rennesøy, Rogaland County with Rennesøy and the municipality of Rennesøy
was the regions used to calculate the minimum (min), mode (mod) and maximum (max)
infection rate respectively. The rates were then used in the Pert distribution.
To estimate the infection rate of the regions, the total number of sheep flocks and
the number of flocks assumed to be infected with severe footrot in the region, from
the introduction of the disease (2005 in Rogaland and 2006 in Rennesøy) until the
initiation of the elimination programme in 2009 was used (Table 1). The infection rate was simulated based on Equations 1–3, with a constant annual recovery rate (?) of 5.3% and a reversion rate (?) of 1/3
of the infection rate (see below for descriptions of recovery and reversion rate).
The assumed number of infected flocks was the number detected in the footrot outbreak
in Norway, and the predicted median number of infected flocks was as close to this
number as possible. The appurtenant infection rate was used in the model. The min,
mod and max predicted median number of infected flocks and the appurtenant infection
rates in parentheses were 26 (1.13), 48 (1.31) and 13 (1.36), respectively (Table 1). The Pert distribution for the infection rate for Rogaland was then (?Rog?~?Pert (1.13, 1.31, 1.36)).
The recovery rate was based on spread of severe footrot without an elimination programme
and hence no compensation for sanitation or other measures to eliminate the disease.
The recovered flocks have therefore either undergone sanitation procedure at their
own cost or recovered from the disease spontaneously. Two of 38 flocks completed a
successful sanitation procedure at the farmers own expense in 2008 (Vatn S, Healthy
Feet project, personal communication), corresponding to a recovery rate of 5.3% per
year. Some of the flocks might also recover from the disease with no intervention.
Since it takes a long time for sheep in a flock to recover without human intervention
14], the percentage of these flocks is thought to be small and was not included. The
recovery rate was assumed to be constant for all years.
Since none of the flocks which completed a successful sanitation procedure at the
farmers own expense in 2008 was re-infected, the reversion rate could not be calculated
based on data. The reversion rate was therefore sat based on knowledge of the infection
dynamics. The susceptible flocks were assumed to have a three times higher infection
rate than the low susceptible flocks, hence a reversion rate of (??=??/3) was used.
Spread within subpopulations in Norway
Because of national maedi and scrapie legislation, sheep and goats are not allowed
to be moved from one county to another without derogation. This gives a reduced spread
from one county to another hence each of the 19 counties in Norway was assigned as
a subpopulation. A SILI-compartmental model was constructed for each county. The number
of sheep flocks, cattle herds and combined sheep and cattle flocks was allocated to
each county from the Register of Production Subsidies of 31.07.2012 (Table 2). The register contains all holdings receiving production subsidies in Norway, hence
includes 92% of the total number of sheep flocks; the ones missing are farms with
very few sheep. The number was kept constant for all years.
Table 2. Overview of demographic data, climatic rate and infection rate for modelling spread
of severe footrot
The infection rate calculated earlier was only based on the rate of spread within
the county of Rogaland. To calculate the rate of spread within each of the other counties,
values which would interfere with the spread of footrot are quantified and used to
adjust the minimum, mode and maximum within county infection rate for Rogaland.
One of the values expected to interfere with the infection rate of footrot was the
climate within each county as this is an important factor for the survival of D. nodosus and for the initiation and development of ovine footrot 1],15]. In particular the precipitation and temperature are considered important for the
spread of footrot 15],16]. The geo-coordinates of all sheep farm buildings in Norway (f) (the Agricultural
Property Register, 2011) were linked to a mean value of precipitation and temperature ( from May until October (Norwegian Meteorological Institute, data from 1971 till 2000).
By summarising the mean daily precipitation in mm and mean daily temperature in degrees
Celsius of the individual sheep farms in a county, and dividing by the number of sheep
farms in that county (Nf,i), a climatic value (called Cli) was calculated for each of the 19 counties (Equation 4).
(4)
The fraction between the climatic factors in county i and the climatic factor in Rogaland was incorporated in Equation 5 to adjust the infection rate within each county.
Another value expected to interfere with the infection rate of footrot was the density
of sheep farms within each county. Grøneng et al. 13] showed that a risk factor for contracting the disease is a sheep farm located less
than 1 km from a sheep farm positive for severe footrot. The distances between farms
were calculated based on the locations of the main building on each farm. Hence, for
each sheep farm, the number of other sheep farms within 1 km (neighbour farms) was
obtained. Based on this, the mean number of neighbour farms to the sheep farms within
each county () was calculated (Table 2). The fraction between the mean number of sheep farms within 1 km in county i and the county Rogaland was used to adjust the infection rate in county i (Equation 5). By using the knowledge of the spread of disease in the county of Aust-Agder, the
effect of the fraction between counties was adjusted. In 2013, 14 flocks in the county
of Aust-Agder were diagnosed with severe footrot, and epidemiological investigations
indicate that sheep moved from the county of Rogaland in 2006 were the source. The
spread from the introduction in 2006 to 2013 was simulated based on Equations 1–3, and a value k, adjusting the effect of the density factor between Aust-Agder and
Rogaland, was chosen so that the median value of 2000 replicates matched the number
of infected flocks in Aust-Agder in 2013. A median of 14 (range 1–26) infected flocks
was predicted for k?=?2.3 (Equation 5).
For all counties except Rogaland, a county specific min, mod and max infection rate
was estimated by adjusting the min, mod and max infection rate for Rogaland with the
constant k, the mean number of sheep farms within 1 km and climatic value for the
respective counties. These values were then used in a Pert distribution, where a new
value was estimated for each county and each replicate (Equation 5).
(5)
where i is the county, and Rog is Rogaland County.
The recovery rate was not expected to differ between counties and was expected to
be constant for every year. The reversion rate for each county was defined as one
third of the infection rate (?i?=??i/3).
Spread between subpopulations in Norway
The spread of footrot between counties in Norway was modelled taking three potential
transmission routes into consideration: 1) movement of sheep between counties, 2)
movement of cattle between counties, and 3) introduction by sharing of common pastures
(Figure 1).
Introduction from other counties through sheep movement (?)
Although there is a general ban on movement of sheep from one county to another because
of maedi and scrapie, derogations from the legislation can be authorised by the Norwegian
Food Safety Authority. Two movements of sheep between counties were recorded in 2013
(MATS, the supervision system of the Norwegian Food Safety Authority). There may have
been movements of sheep that have not been reported to the central Food Safety Authority,
but these are believed to be minimal. We therefore assumed that some of the sheep
in 0.05% of the flocks in a county would be moved to each of the neighbouring counties
each year. In addition, some of the sheep in 0.025% of the flocks in a county would
move sheep to each of the counties bordering on neighbouring counties each year. Thus
the number of movements from county j to county i was estimated (MShj,i), and used to calculate the introduction of severe footrot to other counties (Equation
7). For Norway as a whole, this is equivalent to approximately 44 between county movements
of sheep each year. As this is more than reported to the Norwegian Food Safety Authority,
we believe the effect of moving sheep across county borders has been overestimated
rather than underestimated.
Only movement of sheep that is infected with footrot can transmit the disease to other
sheep flocks. This depends on the probability that sheep from an infected flock are
moved , and also on the probability that at least one of the sheep moved is infected (ProbMove). The ProbMove is based on the number of sheep moved and the prevalence of infected sheep within
the flock. The minimum ProbMove value was based on movement of one sheep from a flock with an infection prevalence
of 0.01. The maximum ProbMove value was based on movement of five sheep from a flock with a prevalence of 0.65.
The values were calculated to be 0.01 and 0.995 as shown in Equation 6. Consequently, a uniform distribution with a minimum and maximum value of the ProbMove was used (ProbMove?~?Unif (0.01,?0.995)) in Equation 7.
(6)
The number of sheep moved was based on the knowledge that mostly rams are purchased,
and since the sheep flocks are small, rarely more than two rams are acquired at the
same time. The lowest prevalence was based on one infected sheep in a flock of 100
sheep. The highest prevalence was based on PCR examination of all sheep in three flocks
infected with severe footrot, and the median of these values was used. This was chosen
since only sheep from flocks with a veterinary health certificate may be moved across
county borders. We therefore believe that flocks with a prevalence above 0.65 would
not be allowed to move sheep because they would show pronounced clinical signs of
footrot.
The introduction of severe footrot from other counties by sheep movement is shown
in Equation 7, where the percentage of susceptible sheep flocks in county i. was included in order to calculate the probability of an infected sheep arriving
at a susceptible sheep flock. We expect that a sheep which is infected with footrot
would infect a flock of susceptible animals.
(7)
where i is the county receiving infectn, and j is the county transmitting the infection, I is the number of infected sheep flocks, nSh is the total number of sheep flocks, MSh is the number of flocks that have moved sheep, and S is the number of susceptible sheep flocks.
Introduction from other counties through cattle movement (?)
Cattle that have been in contact with infected sheep may be carriers of virulent D. nodosus and transmit the infection to sheep 17],18]. Hence, virulent D. nodosus might be introduced to a new county by movement of cattle. The number of moved cattle
aged 1 year (MCa) in 2007 was retrieved from the Norwegian National Cattle Register (Norwegian Food
Safety Authority) (Table 2). Cattle aged 1 year were not included as calves are usually not in contact with
sheep, and the probability of a calf being infected by its mother and remain infected
until moved to another flock was expected to be minimal. Only information from 2007
was available. In cases where there was no registered movement between neighbouring
counties in 2007, movement of one head of cattle was imputed. The register also included
records of movement of cattle without information about which county they were moved
from. These were included by giving the unknown county the mean number of cattle flocks,
the mean number of combined flocks and the mean number infected for all the counties.
The number of infected sheep flocks in a county that transmit the disease (Ij), the number of sheep flocks in the county j (nShj), the number of cattle flocks in counties i and j (nCai, nCaj), and the number of combined cattle and sheep flocks in counties i and j (nShCaj, nShCai) (Table 2) are used to calculate the probability of severe footrot being introduced from other
counties by movement of cattle. The probability of a sheep infecting cattle (Sh2Ca)
and vice versa (Ca2Sh) was also needed for the calculation. On the basis of a study
by Knappe-Poindecker et al. 18], the value was found to be 0.1 (gelatin gel test showed five of fifty cattle to be
positive after co-grazing with sheep), while a study by Rogdo et al. 19] found this probability to be 0.3 (18 of 58 cattle were PCR-positive for footrot with
serogroup A). The probability of sheep infecting cattle and vice versa was given by
a uniform distribution (Ca2Sh?~?Unif (0.1,?0.3), Sh2Ca?~?Unif (0.1,?0.3)), a new value was generated for each movement. The percentage of susceptible
sheep flocks in county i was included to enable calculation of the probability of infected cattle entering
a susceptible sheep flock. Equation 8 expresses the introduction of severe footrot
from other counties through movement of cattle:
(8)
where i is the county receiving infection, and j is the county transmitting the infection.
Introduction from other counties through sharing of common pasture ()
In Norway, many sheep flocks are transported to common pastures during the summer.
This is mainly pastures situated in mountain areas. This is an old tradition, and
it is important both for reducing the farmer’s feed expenses and for conserving the
countryside. There are nearly 1000 common pasture groups in Norway, each with several
members and a designated area for their sheep to graze (Norwegian Forest And Landscape
Institute). The organisation of the pasture groups is quite complex, with some common
pasture areas crossing county borders. Some pasture groups also have members from
several counties. Information about common pastures that share borders with common
pastures in other counties and common pastures that have members from different counties
are included in the estimation of cross-county transmission on pasture. In these pastures
there are no fences or other barriers, with the result that sheep from different counties
can mix and transfer infection. The spread of severe footrot on common pasture was
calculated in a same way as the within county infection rates (Equation 5) by adjusting the infection rate for Rogaland for differences in sheep flock density
and climate. Since sheep flocks are free ranging on common pasture, it is difficult
to estimate the mean number of flocks within 1 km, as was done when calculating the
within county infection rates. But sheep flocks are often put on common pasture at
different times and in different areas, and 1–2 ewes with their lambs tend to keep
together within a small area and rarely be in contact with other sheep. Assuming maximum
dispersion of flocks on common pasture, we calculated the mean number of flocks per
1 km2 for each of the common pastures and used this as a proxy for the number for flocks
within 1 km of each other. The higher the density of sheep flocks on common pasture,
the higher the infection rate then will be. The mean number of flocks within 1 km2 on the common pastures where flocks from county j and i can be in contact with each other () was calculated as shown in Equation 9 by adding the density of common pastures in
county j and i which have a common border (Bpast) to the density of common pasture which have members from both counties j and i (Mpast). This was used in Equation 10 to calculate the common pasture infection rates.
(9)
where Nf is the number of sheep flocks on pasture, Apast is the geographic area of the pasture in km2 and n is the number of pastures.
The climate of the common pastures (Clpast) could not be calculated in the same way as the within county climate because we
did not have specific geographical points, but rather large areas across which the
sheep flocks were spread. The common pastures are often situated at a higher altitude
than the general location of sheep farms, and the climate is often colder and dryer.
Given this knowledge, we believe that the climate on common pasture has a lower value
than the climate in any of the counties, so the climatic rate of common pasture (Clpast/ClRog) was assumed to be 0.3, lower than the lowest climate rate (Table 2). The climatic rate was constant for all years, and was used in the calculation of
the common pasture infection rates as shown in Equation 10.
The common pasture infection rate was calculated in the same way as the within county
infection rates (Equation 5) with a Pert distribution for each county and each iteration (Equation 10).
(10)
The introduction of severe footrot from other counties through sharing of common pasture
was calculated on the basis of the common pasture infection rate (?past,j,i), the percentage of infected flocks in county j, and the percentage of susceptible flocks in county i. The number of flocks from county i (npast,i) and county j (npast,j) on common pasture was also included to calculate the number of flocks in county
i that were newly infected by sharing common pasture with county j (Equation 11).
(11)
where i is the county receiving infection, and j is the county transmitting the infection.
Model for Norway
As the starting point for the simulations, one flock in the county of Rogaland was
assigned to the infected compartment, the isolated flocks in each county were assigned
to the low susceptible compartment in the respective counties, and the remaining flocks
in each county were assigned to the susceptible compartments. When the probability
of transferring flocks between compartments resulted in decimal number of flocks,
the decimal number was converted to an integer by performing a Bernoulli trial with
the decimal fraction as the probability. The county results were aggregated to give
the results for Norway.
Scenarios
Basic scenario
The basic scenario was simulation of the spread of severe footrot without any elimination
or control with input values as presented in Tables 1 and 2. For all the scenarios where input parameters were changed, the basic scenario was
used as the reference.
Scenarios with different control measures
The disease can be controlled by reducing the within county or between county transmissions
compared to the basic scenario. Scenarios with a 20%, 40%, 60% and 80% lower infection
rate within the counties were modelled. Scenarios with 20%, 40%, 60%, 80% and 100%
less movement of sheep between counties were modelled. Scenarios with 20%, 40%, 60%,
80% and 100% less movement of cattle between counties were modelled. Scenarios with
20%, 40%, 60%, 80% and 100% fewer flocks sharing common pasture were modelled.
Scenarios with increased between county transmission
Increased between county movement of both sheep and cattle, and an increased number
of flocks on common pastures are scenarios that we might see in the future. Hence
the importance of this factor is highlighted. A five-fold and ten-fold increase was
modelled.
Sensitivity analyses
By increasing and decreasing the basic scenario parameters one by one, an indication
of the robustness of the model and the sensitivity of the model parameters is found.
The sensitivity analysis was performed by stepwise increasing and decreasing of the
parameters, starting with 80%, then 60%, 40% and 20%. The analysis was continued until
the number of infected flocks did not deviate by more than 5% compared to the basic
model. Thus, only the 80% increase and decrease was performed for the parameters which
showed little variance in the results compared to the basic scenario. The parameters
included in the sensitivity analysis were the infection rate, recovery rate, reversion
rate, climatic value, climatic rate on common pasture, number of farms within 1 km
(neighbouring flocks) and number of farms within 3 km (isolated flocks).
Model simulations
The model was run from 2005 and 30 years onward. In addition, the basic scenario where
the time interval was extended to the year 2100 was made. The intention was to capture
the percentage of flocks in each of the compartments when the equilibrium state was
reached. The model was run using R v2.15.1 20] and the additional package deSolve 21]. For each simulation of a scenario, 2000 replicates were made.