1
Title:
2
Estimating animal movement contacts between holdings of different
3
production types.
4
Authors: Tom Lindströma, Scott A. Sissonb, Susanna Stenberg Lewerinc and Uno Wennergrena
5
a
IFM Theory and Modelling, Linköping University, 581 83 Linköping, Sweden
6
b
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia
7
c
Department of Disease Control and Epidemiology, SVA, National Veterinary Institute, 751 89 Uppsala,
8
Sweden
9
Corresponding Author:
10
Uno Wennergren
11
Tel: +46 13 28 16 66
12
Fax: +46 13 28 13 99
13
Email: unwen@ifm.liu.se
14
Correspondence address: See above.
15
16
Key words
17
Markov Chain Monte Carlo; Animal databases; Animal movements; Production types
18
1
1
Abstract
2
Animal movement poses a great risk for disease transmission between holdings. Heterogeneous contact
3
patterns are known to influence the dynamics of disease transmission and should be included in modeling.
4
Using pig movement data from Sweden as an example, we present a method for quantification of between
5
holding contact probabilities based on different production types. The data contained seven production
6
types: Sow pool center, Sow pool satellite, Farrow-to-finish, Nucleus herd, Piglet producer, Multiplying
7
herd and Fattening herd. The method also estimates how much different production types will determine
8
the contact pattern of holdings that have more than one type. The method is based on Bayesian analysis
9
and uses data from central databases of animal movement. Holdings with different production types are
10
estimated to vary in the amount of contacts as well as in with what type of holding they have contact with,
11
and the direction of the contacts. Movements from Multiplying herds to Sow pool centers, Nucleus herds
12
to other Nucleus herds, Sow pool centers to Sow pool satellites, Sow pool satellites to Sow pool centers
13
and Nucleus herds to Multiplying herds were estimated to be most common relative to the abundance of
14
the production types. We show with a simulation study that these contact patterns may also be expected to
15
result in substantial differences in disease transmission via animal movements, depending on the index
16
holding. Simulating transmission for a one year period showed that the median number of infected
17
holdings was 1 (i.e. only the index holding infected) if the infection started at a Fattening herd and 2161 if
18
the infection started on a Nucleus herd. We conclude that it is valuable to include production types in
19
models of disease transmission and the method presented in this paper may be used for such models when
20
appropriate data is available. We also argue that keeping records of production types is of great value
21
since it may be helpful in risk assessments.
22
23
1. Introduction
2
1
In the last decade several major outbreaks with contagious livestock diseases, such as foot and mouth
2
disease (FMD) and classical swine fever (CSF) have occurred in Europe. In the effort to minimize the
3
risks and extent of outbreaks, researchers are increasingly recognizing the impact of the contact structures
4
that mediate transmission of infectious agents between holdings (e.g. Velthuis and Mourits 2007, Dubé et
5
al. 2009, Nöremark et al. 2009, Vernon and Keeling 2009). Depending on the disease, the risk of different
6
contact types may vary but between holding movements of animals are often regarded to be the main risk
7
factor for transmission of livestock disease (Févre et al. 2006, Ortiz-Pelaez et al. 2006, Rweyemamu et al.
8
2008).
9
Large scale experiments of contagious animal diseases are impossible for both financial and practical
10
reasons. Instead risk assessment and contingency plans must be based on modeling scenarios which can be
11
based on previous outbreaks or even outbreaks from other countries. However, models and model
12
parameterizations are not directly applicable from one country to another due to differences in animal
13
population structure and density, production systems, contact networks etc. For instance, Robinson and
14
Christley (2007) showed that livestock markets are expected to have great influence on the transmission of
15
diseases in the UK, while Nöremark et al. (2009) state that such markets are rare in Sweden. Therefore
16
animal movement patterns may be very different in different countries or regions. In addition, even
17
outbreaks of the same disease in the same country may be very different, which is clearly illustrated by the
18
fact that the more recent outbreak of FMD in the United Kingdom in 2007 followed a totally different
19
course (DEFRA 2009) than the 2001 outbreak. Drawing conclusions from just one or a few outbreaks may
20
therefore be problematic. The livestock industry is not static and the structure as well as the movement
21
patterns may change, in particular after major outbreaks (Velthuis and Mourits 2007), which makes
22
forecasting based on historic outbreak data difficult. Modeling based on observed contact structure is a
23
good approach as it can be based on specific and (ideally) updated data of each country. Member states of
24
the EU must keep databases on all livestock holdings and register movements of cattle and pigs. Therefore
3
1
such data is often available for analysis and model parameterization. However, not all national databases
2
contain the same details and this may limit the use for some applications in some countries.
3
Modeling may assume that all holdings have equal probability of contacts but heterogeneity in both the
4
number of contacts and between which holdings these occur have been recorded (Bigras-Poulin et al.
5
2006, Lindström et al. 2009, Nöremark et al. 2009) and such differences are known to affect the
6
transmission dynamics (Mollison 1993, Keeling 2005, Moslonka-Lefebvre et al. 2009). One factor that may
7
lead to non random contact patterns is the production types of the holdings. By production type we mean
8
the different types of specialized production systems, e.g. the separation of breeding animals and animals
9
for slaughter on different holdings, which are usually applied in modern animal production. We may
10
assume that animal movements are more common between some production types than others, and also
11
that these may differ in the direction of the contact. For example, EU requirements do not include details
12
on production type but the pig holding databases in some countries, e.g. Sweden, contains this
13
information.
14
The scope of this paper is twofold. First, we present a method for quantification of the contact structure
15
between holdings with different production types. Previous studies have shown differences in the contact
16
pattern between holdings with different production types (Dickey et al. 2008, Ribbens et al. 2009) and our
17
aim is to provide estimates that can be used in the modeling of such contacts. Commonly,
18
parameterizations of models are based on expert opinions and questionnaires (Christensen et al. 2008.
19
Dickey et al. 2008, Ribbens et al. 2009, Ward et al. 2009). Our method, based on Bayesian analysis,
20
utilizes data found in central databases where animal movements and holding information are stored in
21
such databases at the national level. It also handles the fact that holdings may have more than one
22
production type by analyzing whether some production types may be dominant in determining the
23
contacts between holdings. Bayesian analysis is a flexible tool that allows for parameterization of complex
24
models. Rather than just estimating the maximum likelihood of the model, the data is considered with
25
parameter uncertainty included. This uncertainty may then be incorporated when the model is used for
4
1
simulation or inference based on the parameters. We apply the method to data of pig movements in
2
Sweden from one year.
3
Secondly, we also perform a simplistic simulation of disease transmission via the estimated contact
4
pattern. The intention is not to investigate the dynamics of any specific disease. Rather we investigate how
5
model predictions generally may change with inclusion of production types as estimated with the method
6
presented. More specifically we investigate how the expected number of infected holdings differs
7
depending on the production type of the first infected holding, based on direct transmission via animal
8
movements only. Previous studies have shown that the type of index holding may have large impact on
9
outbreak dynamics. For instance, Boklund et al. (2008) focused on the difference between herds with
10
domesticated and wild boar populations and showed in a simulation study of CSF that the probability of
11
an epidemic changed depending on the type of the index herd. Also, Ward et al. (2009) and Pineda-Krsh et
12
al. (2010) simulated transmission of FMD between cattle and pig herds, respectively, and showed that the
13
index herd affected both the size and duration of an outbreak.
14
15
2. Material and method
16
2.1 Data
17
Data were supplied by the Swedish Board of Agriculture. It contained pig holdings with their reported
18
production types and movements reported from July 2005 until June 2006. Holdings are supposed to be
19
removed from the data-base if they no longer have any pigs but it is known that this is not always the case.
20
Inactive holdings where therefore removed as described in Nöremark et al. (2009) and we use the same
21
data as in that study for spatial analysis of pig holdings. This data has previously been analyzed for spatial
22
(Lindström et al. 2009, Nöremark et al. 2009) and temporal (Nöremark et al. 2009) patterns. Little
23
seasonal variation was observed for pig movements.
5
1
Movements of cattle and pigs, as well as information related to livestock holdings, are registered by the
2
Swedish Board of Agriculture and stored in different databases. The database is designed for multiple
3
purposes, including the basis of agricultural subsidies, contact with individual farmers and disease tracing.
4
EU legislations require that all states keep databases including all holdings and register movements of
5
cattle and pigs. The holdings with pigs are registered with a unique number, the production-place number
6
(PPN), and the database contains information on postal address, species kept and approximate number of
7
animals kept on the holding. The type of production is also included. Holdings that are geographically
8
separated from each other should have different PPNs. Thus one farmer can have several holdings. The
9
PPN number system is also used for transport vehicles transporting cattle or pigs.
10
Movements of pigs are recorded at the group level, not for each pig. The reports include the date, the
11
number of pigs, and the start and end holdings of the movements. The movements are reported by the
12
farmer at the PPN sending the animals. Farmers should report within seven days after the event, either
13
electronically or using a form sent by ordinary mail. Due to single reporting on group level, the movement
14
reports could not be matched or checked with location, but earlier work on cattle data has shown that
15
errors may be common in these data (Nöremark et al. 2009). A total number of 3084 holdings and 20231
16
movements were included in the analysis. Data on production type are recorded at the first registration of
17
an animal holding but there is no requirement for updating this information.
18
The Swedish pig industry has a pyramidal structure with transports predominantly going downward in the
19
system. Permanent trade agreements between pig farmers are quite common. The different production
20
types are defined as Nucleus herds (that produce breeding animals), Multiplying herds (that receive
21
breeding animals and breed gilts for sale to piglet producers), Piglet producers, Fattening herds and
22
Farrow-to-finish herds. Moreover, there is a special type of production system involving several holdings,
23
namely Sow pools. These consist of a Sow pool center (where sows are covered or inseminated) and Sow
24
pool satellites (where the sows farrow). Thus, the sows are regularly moved back and forth between the
25
central unit and the satellites and they don’t always farrow in the same satellite herd every time.
6
1
We included seven production types which may be selected by the farmer on the report form. Farmers also
2
have the possibility to select a box for “other” and leave free text information, but this was excluded from
3
the analysis. Instructions state that a maximum number of two production types may be reported for each
4
PPN but the maximum number of production types reported was five. 20% of the holdings had reported
5
more than one type and 7.6% of the holdings had no information on production type or had only given
6
free text information. The number of holdings that had reported each production type is presented in Table
7
1.
8
9
2.2 Model and parameter specification
10
The available data includes reported production types, which we denote 𝑹. We indicate 𝑅𝑓𝑘 = 1 if holding
11
𝑓 (𝑓 = 1,2, … 𝑛 where 𝑛 is the number of holdings) has reported production type 𝑘 (𝑘 = 1,2, … 𝐾 where 𝐾
12
is the number of production types including the additional type “Missing information”, which is referred
13
to as type 𝑚) and 𝑅𝑓𝑘 = 0 if holding 𝑓 has not reported production type 𝑘. Also, data include start, 𝒔, and
14
end, 𝒆, holdings of all 𝑇 movements and we use the notation that 𝑒𝑡 and 𝑠𝑡 are the end and start holdings,
15
respectively, of movement 𝑡. We say that 𝑹 is fixed and given the observed data we search the posterior
16
distribution of two parameters of interest, 𝒉 and 𝒗. The parameter 𝒉 models how common movements
17
between production types are (see below for more details). The parameter vector 𝒗 has 𝐾 − 1 elements
18
(production type 𝑚 is excluded, see below) and indicates whether some production types are dominant in
19
determining the contacts of a holding with more than one production type. If there is no difference
20
between the types, then 𝑣𝑘 = 1⁄(𝐾 − 1) for all production types 𝑘 = 1,2, … 𝐾 − 1.
21
Parameter matrix 𝒉 has 𝐾 × 𝐾 elements and indicates how common movements between the different
22
production types are and ℎ𝐼𝐽 = 1⁄𝐾 2 if there is no difference in probability of contacts between holdings.
23
A higher value indicates that movements between the types are more common than expected by random.
24
Hence, 𝒉 can be seen as a measurement of the commonness of movements between holdings with
7
1
different production types but accounting for how frequent production types are, and we refer to this as the
2
commonness index. Both 𝒗 and 𝒉 are defined such that the sum of the elements equals to one.
3
To include the fact that many holdings have more than one production type we model movements between
4
holdings assuming that these act as a mixture of the types. The proportions are given through 𝒗 and 𝑹 and
5
each holding 𝑓 is assumed to be of type 𝑘 with the proportion
𝑣̂𝑓𝑘 =
𝑅𝑓𝑘 𝑣𝑘
for 𝑘 ≠ 𝑚
∑𝐾 𝑅𝑓𝐾 𝑣𝐾
} if 𝑅𝑓𝑚 ≠ 1
𝑣̂𝑓𝑚 = 0
(1)
𝑣̂𝑓𝑘 = 0 for 𝑘 ≠ 𝑚
} if 𝑅𝑓𝑚 = 1
𝑣̂𝑓𝑚 = 1 if 𝑘 = 𝑚
6
where 𝑘 = 1,2, … 𝐾 − 1 and type 𝑚 refers to “Missing information” in the data analyzed. The reason why
7
this has to be treated separately is that this enforced type never is shared with any other type for any
8
holding (as it by definition then would not be of type 𝑚). Hence, 𝑣̂𝑓𝑚 = 1 independent of 𝒗. In the case
9
where 𝑘 ≠ 𝑚, equation 1 is interpreted as 𝒗 normalized over the reported production types of each
10
holding 𝑓.
11
The basic outline of the model is that we search the posterior distribution
𝑃(𝒗, 𝒉|𝒆, 𝒔, 𝑹) ∝ 𝐿(𝒆, 𝒔|𝒗, 𝒉, 𝑹)𝑝(𝒗, 𝒉)
(2)
12
where 𝐿(𝒆, 𝒔|𝒗, 𝒉, 𝑹) is the likelihood of all movements and 𝑝(𝒗, 𝒉) is the prior. The likelihood of one
13
movement, 𝐿(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝒉, 𝑹), may be rewritten by 𝐿(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝒉, 𝑹) = 𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 )𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹),
14
hence formulating the probability of 𝑒𝑡 , 𝑠𝑡 conditional on the production types of the start ( 𝐼𝑡 ) and end
15
holding (𝐽𝑡 ). We use the functions
𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) =
8
̂𝐼 𝑁
̂̇𝐽𝐼
ℎ𝐼𝐽 𝑁
̂𝐼 𝑁
̂̇𝐽𝐼
∑𝐼 ∑𝐽 ℎ𝐼𝐽 𝑁
(3)
1
and
𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 ) =
𝑣̂𝑠𝐼 𝑣̂𝑒𝐽
,
̂𝑁
̂̈
𝑁
(4)
𝐼 𝐽𝑠
2
̂𝐼 = ∑𝑓 𝑣̂𝑓𝐼 , 𝑁
̂̇𝐽𝐼 = ∑𝑓 𝑣̂𝑓𝐽 (1 − 𝑣̂𝑓𝐼 ⁄𝑁
̂𝐼 ) and 𝑁
̂̈𝐽𝑠 = ∑𝑔≠𝑠 𝑣̂𝑔𝐽 may be interpreted as the amount of
where 𝑁
3
holdings of each production type, taking into account that a holding may be a proportion of each type, and
4
̂̇𝐽𝐼 and 𝑁
̂̈𝐽𝑠 are adjusted so that the normalizations of equation 3 and 4 account for the exclusion of 𝑒𝑡 =
𝑁
5
𝑠𝑡 . A movement will not end up at the same holding as it is sent from. The likelihood of all observed start
6
and end holdings is written
𝑇
𝐿(𝒆, 𝒔|𝒗, 𝒉, 𝑹) = ∏(𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 )𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) )
(5)
𝑡=1
7
Appendix A gives a more detailed description of the model.
8
We use a Markov Chain Monte Carlo (MCMC) for parameter estimation. This is described in Appendix
9
B.
10
11
2.3 Simulation
12
We set up a simple simulation model where contacts via animal movements between holdings are
13
modeled with probabilities given by section 2.2. Programs were written in MatLab 7.8. In each simulation
14
the model is parameterized with random draws from the posterior distribution, as provided by the MCMC
15
output. We simulated transmission between the same holdings and their reported production types as was
16
used in the analysis of 𝒉 and 𝒗 and the holdings are used as infective units. Each holding will be in either
17
state S (susceptible) or I (infected). The simulation assumes instant infection through directed contacts, i.e.
18
if a movement occurs from infected holding A to B, B will instantly become infected and all subsequent
9
1
movements from B to any uninfected holding C will lead to infection. If however there is a movement
2
from uninfected holding D to infected holding E, D will not become infected.
3
We investigated how the expected number of infected holdings varies depending on the production type of
4
the initially infected holding. Therefore we varied the type of the index holding and selected randomly
5
from holdings that had reported the current production type. We ran the simulations 500 times for each
6
production type and recorded the number of infected holdings after 388, 1552 and 20231 movements,
7
which corresponds to the expected number of movements for 7, 28 and 365 days, respectively. Note that
8
these numbers also include movements between non infected holdings. We analyzed the simulation results
9
with Kruskal-Wallis tests and box plots.
10
11
3. Results
12
Estimates of 𝒉 are shown in Table 2. The five highest commonness indices were found for, in decreasing
13
order, movements from Multiplying herds to Sow pool centers, Nucleus herds to other Nucleus herds,
14
Sow pool centers to Sow pool satellites, Sow pool satellites to Sow pool centers and Nucleus herds to
15
Multiplying herds. Movements involving fattening herds were estimated to be relatively rare, and in
16
particular movements from such holdings. Movements to and from Farrow-to-finish type holdings also
17
had low commonness indices. Sow pool satellites only had high commonness indices for movements from
18
Sow pool centers and except for the high commonness index of movements back to Sow pool centers,
19
Sow pool satellites tend to send pigs to Fattening herds. Movements from Nucleus herds had generally
20
high commonness index but estimates were low for all incoming movements except from other Nucleus
21
herds.
10
1
Estimates for 𝒗 are shown in Table 3. Sow pool centers were estimated to be dominant in determining the
2
contacts with other holdings. Fattening herds were estimated to predict very little about the contacts of a
3
herd when this production type was reported together with other types.
4
The predicted numbers of infected holdings depending on the production type where the infection was
5
initialized are shown in Figure 1. The highest median was found for simulations where the initial infection
6
started in a Nucleus herd followed by Multiplying herds and Sow pool centers. Simulations with the initial
7
infection on holdings with Fattening herds, Farrow-to-finish and Missing information showed low median
8
infection rate but many outliers with large number of infections. The trend was most apparent in the one
9
year simulation, where the lowest median, including the index holding, was found for Fattening herds
10
(median=1) and the largest for Nucleus herds (median=2161). The Kruskal-Wallis test showed that for all
11
tested time periods 𝑝 ≪ 0.01 and the null hypothesis that the number of infected holdings is independent
12
of the production type of the initial holding is rejected.
13
14
4. Discussion
15
Largely, our results on general movement patterns between production types reflect what is already known
16
about the Swedish pig industry. The analysis of 𝒉 shows that the top five estimates are found for
17
movements between types that are known to move many animals between them. Also, the commonness
18
index for movement between Sow pool centers is estimated to be high, indicating that trade of animals
19
between different holdings of this type would be common. However, trade between Sow pool centers is
20
not allowed in the system of Sow pools (the idea being that each Sow pool should function as one unit
21
without contact with herds outside the pool except for some sourcing herds for gilts). This finding is thus
22
contrary to what is known about the actual practice of these units and indicates that there may be
23
misclassification of the production type of some herds in this category. Also, the analysis estimates that
24
there is a high commonness index for movements from Nucleus herds to Sow pool centers. Hence, while
11
1
there is (as expected) many movements from Nucleus herds to Multiplying herds and from Multiplying
2
herds to Sow pool centers, movements directly from Nucleus herds to Sow pool centers are also estimated
3
to be common. We cannot exclude the possibility that these unexpected results are due to erroneous
4
reports on production types. The production types involved are quite rare (see Table 1) and thereby more
5
sensitive to data quality.
6
From Table 2 we observe that Fattening herds have higher commonness indices for incoming than
7
outgoing movements in contacts with all other types. This result is expected since Fattening herds only
8
produce pigs for slaughter and slaughterhouses were not included in this study. By comparing the
9
estimated values of ℎ𝐼𝐽 to ℎ𝐽𝐼 in Table 2 (i.e. movements from type 𝐼 to 𝐽 compared to type 𝐽 to 𝐼) we may
10
further conclude that the commonness index of movements generally differs depending on the direction.
11
The credibility intervals of ℎ𝐼𝐽 and ℎ𝐽𝐼 , 𝐽 ≠ 𝐼, only overlap in 4 out of 28 cases. One important exception
12
is the commonness index of movements between Sow pool centers and Sow pool satellites which are very
13
similar in estimates independent of direction. This similarity is expected given that the system is based on
14
the same number of sows moving through the entire system, back and forth between the centre and the
15
satellite holdings.
16
The fact that many holdings have more than one production type needs to be incorporated when modeling
17
contacts. One may assume that a holding that for instance has two types will have a contact structure that
18
is 50% of each type. The analysis of 𝒗 however show that such supposition is incorrect, at least for the
19
data analyzed in this study. There is great difference in how much the production types determine the
20
contacts of a holding. The type Sow pool center is very dominant while Fattening herd determines very
21
little about the contacts of a holding when this type is reported concurrently with others. So rather than
22
assuming 50% of each type, a holding 𝑓 that has reported two production types is expected to have a
23
contact structure that is determined by 𝑣̂𝑓𝑘 (as defined by equation 1) of each type 𝑘 = 1,2. Hence, a
24
holding that has reported e.g. both Sow pool center and Fattening herd is expected to interact with other
25
holdings as 99.7% (using mean estimates of 𝒗 as shown in Table 3, 0.83⁄(0.83 + 0.0023)) Sow pool
12
1
center and only 0.3% Fattening herd. The expected contact pattern of such a holding would be very
2
different if it was assumed to be determined equally by the two types as the estimates for 𝒉 are very
3
different between movements involving Sow pool centers and Fattening herds (see Table 2).
4
The assumptions of the simulation model are too crude to capture the dynamics of any real disease as
5
there is no intra-herd dynamics and all other contacts are neglected. Simplistic models with holdings as
6
infective units do however, as pointed out by Vernon and Keeling (2009), allow for investigation of the
7
effects of the contact structure. In the simulation study, disease transmission was estimated to be highest if
8
the index holding was a Nucleus herd, and also high if the index holding was of types Sow pool center,
9
Sow pool satellite or Multiplying herd. The median number of cases was very low if the index holding
10
was of type Farrow-to-finish, Fattening herd or had Missing information. Also, the Kruskal-Wallis test
11
showed that the number of infected holdings differed depending on the index holding, which is expected
12
both from the analyzed contact structure as well as previous studies where production types have been
13
included in outbreak simulations (Boklund et al. 2008, Ward et al. 2009, Pineda-Krsh et al. 2010). This is
14
also according to what is expected and planned for in contingency plans and disease surveillance
15
programs.
16
We conclude that the observed contact heterogeneities are also expected to influence the dynamics of
17
disease transmission. However, a more realistic model needs to include other factors, such as intra-herd
18
dynamics, incubation time, mode of spread and other disease specific aspects. Moreover, there are other
19
factors influencing the probability of contacts between holdings via animal movements. Distance is known
20
to be an important factor (Lindström et al. 2009, Ribbens et al. 2009) and we may expect holdings with
21
larger herd sizes to have more contacts (Ribbens et al. 2009). We do however argue that these factors, if
22
included, should be analyzed as being dependent on the production type. For instance, more short distance
23
contacts may occur when more permanent agreements between farmers are present, such as for the actors
24
in Sow pools. And a large Farrow-to-finish holding might not necessarily have many incoming contacts as
25
the whole production chain is integrated on the holding.
13
1
Our results however clearly show that production types influence the contact pattern and this in turn is
2
expected to have implications for disease transmission. Hence, there is great value in including this
3
information in animal databases and it has the potential to improve risk assessment. Ideally, animal
4
databases should include exact geographic location, type of production, type of housing, animal species
5
kept on the premises and the number of animals of each species, for each animal holding. Moreover,
6
yearly updating of the data would be optimal. In reality, this may not be feasible for existing databases but
7
for countries setting up new systems, these aspects should be considered.
8
We believe that production types should be included in disease spread models when data is available and
9
urge other researchers to include this in their studies. However, one must remember that the contact
10
patterns may change quickly, due to structural changes in pig production. Larger, more specialized units
11
may lead to less frequent reporting of more than one production system. Some types of production may,
12
however, become more common in smaller units or mixed with others. Therefore, analysis of the contact
13
patterns needs to be updated to avoid erroneous assumptions about the structure. Also, reliable data is
14
essential if analysis of the contact pattern is to be used in risk assessment of between holding disease
15
transmission. Clear guidelines to the farmers may improve the quality of the data and thereby the
16
possibility to utilize this for risk assessments of disease spread.
17
18
5. Conclusion
19
We have presented a model for analysis of animal movements between holding of varying production
20
types and applied it to pig movement data from Sweden. We have shown that there is great difference
21
between which production types influence the contact pattern when a holding has more than one type as
22
well as heterogeneity in the contact pattern between the types. The results also demonstrate that there
23
generally is a difference in direction of contacts. We have further shown that the contact heterogeneity is
24
expected to influence the dynamics of disease spread via the considered contacts. Hence we believe that
14
1
models based on contact patterns between holdings may be improved by inclusion of production types and
2
we argue that the method presented in this paper may be used for parameterization if data is available. It is
3
therefore valuable to include such information in central databases of livestock holdings.
4
5
Conflict of interest
6
We have no conflict of interest.
7
8
Acknowledgement
9
We thank the Swedish Emergency Management Agency (KBM) for funding and the Swedish Board of
10
Agriculture for supplying the data used.
11
12
Appendix A. Model details
13
Here follow some clarifications on the model used. In determining the distribution of 𝐿(𝒆, 𝒔|𝒗, 𝒉, 𝑹), we
14
start with the hypothetical case of no difference between production types and no difference in the
15
probability of movements between them. In such a setup, the probability that a movement 𝑡 originates at
16
holding 𝑠 and ends up at holding 𝑒 is given by
𝑃(𝑒𝑡 , 𝑠𝑡 |𝑛) = 𝑃(𝑠𝑡 |𝑛)𝑃(𝑒𝑡 |𝑛, 𝑠𝑡 ) =
1
1 1
𝑛𝑛 −1
1
(A.1)
17
where 𝑛 is the total number of holdings. 𝑃(𝑒𝑡 |𝑛, 𝑠) = 𝑛−1 rather than 𝑛 since a movement may not end up
18
at the same holding as it originates.
15
1
If however there are different production types, each holding has exactly one type (𝑅𝑓𝑘 = 1 for only one
2
type 𝑘) and the probability of contacts between these vary, then
𝑃(𝑒𝑡 , 𝑠𝑡 |𝒉, 𝑹) =
ℎ𝑖𝑗
∑𝐼 ∑𝐽 ℎ𝐼𝐽 𝑁𝐼 𝑁̇𝐽
(A.2)
3
where 𝐼, 𝐽 are production types of 𝑠 and 𝑒 respectively. Conditionality on 𝑹 is expressed through 𝑵 which
4
is a vector of size 𝐾and 𝑁𝑘 is the number of holdings with production type 𝑘, i.e. ∑𝑘 𝑅𝑓𝑘 . Since
5
movements starting and ending at the same holding are excluded, 𝑁̇𝐽 = 𝑁𝐽 if 𝐼 ≠ 𝐽 and 𝑁̇𝐽 = 𝑁𝐽 − 1 if 𝐼 =
6
𝐽. The parameter 𝒉 determines whether movements are more or less likely between types 𝐼, 𝐽 and equation
7
A.2 can be rewritten as equation A.1 if ℎ𝐼𝐽 = 1⁄𝐾 2 for all 𝐼, 𝐽.
8
If the interest is not probability of individual holdings, but rather the probability of contact between
9
production types, the probability of a movement from type 𝐼 to type 𝐽 is given by
𝑃(𝐼, 𝐽|𝒉, 𝑹) =
ℎ𝐼𝐽 𝑁𝐼 𝑁̇𝐽
∑𝐼 ∑𝐽 ℎ𝐼𝐽 𝑁𝐼 𝑁̇𝐽
(A.3)
10
This is obtained by summation of the probability of movements between holdings of types 𝐼, 𝐽 divided by
11
the probability of all possible movements. If we include the fact that many holdings have more than one
12
production type, equation A.3 may be rewritten as equation 3. Hence, equations 3 and A.3 are identical if
13
̂𝐼 , i.e. the
all holdings have exactly one production type. The probability of the start holding is then 𝑣̂𝑠𝐼 ⁄𝑁
14
̂𝐼 . Similarly, the probability of
proportion of 𝑠 being of type 𝐼 divided by the total amount of this type, 𝑁
15
̂̈𝐽𝑠 and the probability of start and end holding of movement 𝑡 conditional on
the end holding is 𝑣̂𝑒𝐽 ⁄𝑁
16
production types 𝐼𝑡 , 𝐽𝑡 is given by equation 4.
17
18
Appendix B. Parameter estimation
16
1
To facilitate computation of the full conditional distribution, we introduce an indicator variable (Gelman
2
et al. 2004), 𝑼, of size 𝐾 × 𝐾 × 𝑇, and rewrite equation 5 as
𝑇
𝐾
𝐾
𝐿(𝒆, 𝒔, 𝑼|𝒗, 𝒉, 𝑹) = ∏ ∏ ∏(𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 )𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) )𝑈𝐼𝐽𝑡
𝑡=1 𝐼=1 𝐽=1
(B.1)
3
where 𝑈𝐼𝐽𝑡 = 1, probabilistically, for exactly one combination of 𝐼, 𝐽 for each movement 𝑡 with (full
4
conditional) probability
𝑃𝑟(𝑈𝐼𝐽𝑡 = 1|𝒗, 𝒉, 𝑹) =
𝑣̂𝑠𝐼 𝑣̂𝑒𝐽 ℎ𝐼𝐽
.
∑𝐼 ∑𝐽 𝑣̂𝑠𝐼 𝑣̂𝑒𝐽 ℎ𝐼𝐽
(B.2)
5
Note that unlike the standard formulation of mixture models in Gelman et al. 2004, the mixture
6
components in equation B.1 include the mixing distribution 𝒗. This is because 𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 ) and
7
𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) are normalized over the proportion of holdings belonging to each production type (see
8
equations 3 and 4, respectively). The full model distribution is then written as
𝑃(𝒗, 𝒉, 𝑼|𝒆, 𝒔, 𝑹) ∝ 𝐿(𝒆, 𝒔|𝒗, 𝒉, 𝑼, 𝑹)𝑝(𝒗, 𝒉, 𝑼).
(B.3)
9
Parameters 𝒉, 𝒗 and 𝑼 are estimated with MCMC. Parameter 𝑼 is updated with Gibbs sampling by
10
drawing one random number for each 𝑡 from a multinomial distribution with probabilities given by
11
equation B.2. The full conditional distribution of 𝒉 and 𝒗 is of non standard form and Metropolis-Hastings
12
updates have to be used (see below). The conditional distribution of 𝒉 is based on equation 3 and with
13
inclusion of the indicator variable 𝑼 the conditional distribution of 𝒉 is given by
𝑇
𝐾
𝐾
𝑃(𝒉|𝒗, 𝑼, 𝑹) ∝ ∏ ∏ ∏(𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) )𝑈𝐼𝐽𝑡 𝑃(𝒉)
𝑡=1 𝐼=1 𝐽=1
(B.4)
14
̂ and 𝑵
̂̇ , and 𝑃(𝒉) is the prior distribution of 𝒉.
̂, 𝑵
where conditionality on 𝒗 and 𝑹 is expressed through 𝒗
15
We use an uninformative prior for 𝒉 (𝐷𝑖𝑟𝑖𝑐ℎ𝑙𝑒𝑡(1,1, … 1)). For each update and 𝑡, (𝑃(𝐼, 𝐽|𝒗, 𝒉, 𝑹) )𝑈𝐼𝐽𝑡
16
will deviate from one for only one combination of 𝐼, 𝐽 and the probability distribution of 𝒉 can be given by
17
𝐿1 = 𝑀𝑢𝑙𝑡𝑖𝑛𝑜𝑚𝑖𝑎𝑙(𝑀1,1 , 𝑀1,2 , … 𝑀2,1 , 𝑀2,2 … 𝑀𝑘,𝑘−1 , 𝑀𝑘,𝑘 |𝑝1,1 , 𝑝1,2 , … 𝑝2,1 , 𝑝2,2 … 𝑝𝑘,𝑘−1 , 𝑝𝑘,𝑘 )
1
where 𝑝𝐼,𝐽 = 𝑃(𝐼, 𝐽|𝒗, 𝒉, 𝑹) as given in equation 3 and 𝑀𝐼,𝐽 = ∑𝑡 𝑈𝐼𝐽𝑡 .
2
The parameter 𝒗 is included in both 𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 ) and 𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) in equation B.1 and so the
3
conditional distribution of 𝒗 is given by
(B.5)
𝑃(𝒗|𝒉, 𝑼, 𝑹, 𝒆, 𝒔)
𝑇
𝐾
𝐾
∝ ∏ ∏ ∏(𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 )𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) )𝑈𝐼𝐽𝑡 𝑃(𝒗)
𝑡=1 𝐼=1 𝐽=1
𝑇
𝐾
𝐾
𝑇
𝐾
(B.6)
𝐾
= ∏ ∏ ∏ 𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 )𝑈𝐼𝐽𝑡 ∏ ∏ ∏ 𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) 𝑈𝐼𝐽𝑡 𝑃(𝒗)
𝑡=1 𝐼=1 𝐽=1
𝑡=1 𝐼=1 𝐽=1
4
where 𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) is given as in equation B.5 and 𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 ) for each 𝑡 is given by equation
5
12. For further notation we write
𝑇
𝐾
𝑈𝐼𝐽𝑡
𝐾
𝑣̂𝑠𝐼 𝑣̂𝑒𝐽
𝐿2 = ∏ ∏ ∏ (
)
̂𝑁
̂̇
𝑁
𝑡
𝐼=1 𝐽=1
𝑈𝐼𝐽𝑡
𝐼 𝐽𝑠
(B.7)
6
̂𝐼 𝑁
̂̇𝐽𝑠 ))
and note that for each update ((𝑣̂𝑠𝐼 𝑣̂𝑒𝐽 )⁄(𝑁
7
For 𝒉 or 𝒗 we use Metropolis-Hastings updates which involves proposing new parameter values and then
8
accepting or rejecting this proposal. Since there are many similarities in the updates of 𝒉 or 𝒗, we use the
9
indication 𝜽 (consisting of 𝜃1 , 𝜃2 , … 𝜃𝑙 ) when referring to either 𝒉 or 𝒗. Both 𝒉 and 𝒗 are defined as 𝜃1 +
10
𝜃2 + ⋯ +𝜃𝑙 = 1 and 0 < 𝜃𝑚 < 1 for all 𝑚. Therefore the elements are dependent and cannot be updated
11
separately, and the proposed values need to follow that definition. Hence, proposals are performed using a
12
Dirichlet distribution. If the current position of an iteration is 𝜽̇ then 𝜽̈ is proposed from 𝑞 (𝜽̈|𝜽̇, 𝐴(𝜽̇)) =
13
𝐷𝑖𝑟𝑖𝑐ℎ𝑙𝑒𝑡(𝜽̈|𝑩) where 𝑩 = 𝜽̇𝐴(𝜽̇) which is centered at 𝜽̇ and 𝐴(𝜽̇) controls the width of the proposal
deviates from one for only combination of 𝐼, 𝐽.
18
1
distribution. It is possible to use a fixed value 𝐴̈ such that 𝐴(𝜽̇) = 𝐴̈ for all 𝜽𝑡 but computational problems
2
may arise for very small 𝜃𝑚 . Generation of Dirichlet random numbers are based on random numbers from
3
either the Gamma or the Beta distribution (Gelman et al. 2004). For numerical reasons, small values of the
4
𝛼 parameter of either Gamma or Beta distribution generates values equal to exactly zero, which also gives
5
𝜃𝑚 = 0 and all subsequent proposal will be 𝜃𝑚 = 0 for any value of 𝐴̈. We therefore use 𝐴(𝜽) =
6
𝐶𝑙𝑖𝑚 /𝜃𝑚𝑖𝑛 where 𝐶𝑙𝑖𝑚 is the critical value for where numerical problems occur in generation of Dirichlet
7
random numbers. The acceptance ratio is then given by
𝑃(𝜽̈| ∙)𝑃(𝜽̈)𝑞 (𝜽̇|𝜽̈, 𝐴(𝜽̈))
𝑚𝑖𝑛 (1,
)
𝑃(𝜽̇| ∙)𝑃(𝜽̇)𝑞 (𝜽̈|𝜽̇, 𝐴(𝜽̇))
(B.8)
8
where 𝑃(𝜽| ∙) is the posterior distribution of the parameter vector 𝜽 and 𝑃(𝜽) is the prior distribution of 𝜽.
9
The acceptance ratio for 𝒉, is given by
𝑚𝑖𝑛 (1,
𝐿̈1 𝑃(𝒉̈)𝑞 (𝒉̇|𝒉̈, 𝐴(𝒉̈))
)
𝐿̇1 𝑃(𝒉̇)𝑞 (𝒉̈|𝒉̇, 𝐴(𝒉̇))
(B.9)
10
where 𝐿̇1 and 𝐿̈1 are the distributions given by equation B.5 for the current and proposed values of 𝒉,
11
respectively. Good mixing is hard to obtain for 𝒉 since it is based on a random walk in 63 (i.e. 𝐾 2 − 1)
12
̂ is a vector with 𝐾
dimensions. We therefore make partial updates by rewriting ℎ𝐼𝐽 = ℎ̂𝐼 ℎ̌𝐼𝐽 where 𝒉
13
𝐾 ̌
̌ is a matrix with 𝐾 × 𝐾 elements and ∑𝐾
̂
elements and 𝒉
𝐼=1 ℎ𝐼 = 1 and ∑𝐽=1 ℎ𝐼𝐽 = 1. For every iteration we
14
̂ and 𝐾 updates on 𝒉
̌ , in the latter case proposing values (from a Dirichlet
perform one update on 𝒉
15
distribution as described above) of ℎ̌𝑖1 , ℎ̌𝑖2 , … ℎ̌𝑖𝐾 for the 𝑖th partial update. The formulation of equation
16
B.9 is however still valid since the acceptance ratio is based on the probability density of 𝒉 but 𝑞 will be
17
different for each partial update.
18
The acceptance ratio for 𝒗, is given by
19
𝑚𝑖𝑛 (1,
𝐿̈1 𝐿̈2 𝑃(𝒗̈ )𝑞(𝒗̇ |𝒗̈ , 𝐴(𝒗̈ ))
)
𝐿̇1 𝐿̇2 𝑃(𝒗̇ )𝑞(𝒗̈ |𝒗̇ , 𝐴(𝒗̇ ))
(B.10)
1
where 𝐿̇1 and 𝐿̈1 are the probability distributions given by equation B.5 for the current and proposed
2
values of 𝒗, respectively, and 𝐿̇2 and 𝐿̈2 are given analogously from equation B.7.
3
4
References
5
Bigras-Poulin, M., Thompson, R.A., Chriel, M., Mortensen, S., Greiner, M., 2006. Network analysis of
6
Danish cattle industry trade patterns as an evaluation of risk potential for disease spread. Prev. Vet. Med.
7
76, 11-39.
8
Boklund, A., Goldbach, S.G., Uttenthal, A., Alban, L., 2008. Simulating the spread of classical swine
9
fever virus between a hypothetical wild-boar population and domestic pig herds in Denmark. Prev. Vet.
10
Med. 85, 187-206.
11
Christensen, J., McNab, B., Stryhn, H. Dohoo, I., Hurnik, D., Kellar, J., 2008. Description of empirical
12
movement data from Canadian swine herds with an application to a disease spread simulation model.
13
Prev. Vet. Med. 83, 170-185.
14
DEFRA, 2009. FMD: 2007 outbreak.
15
http://www.defra.gov.uk/foodfarm/farmanimal/diseases/atoz/fmd/2007/index.htm
16
Dickey, B.F., Carpenter, T.E., Bartell, S.M., 2008. Use of heterogeneous operation-specific contact
17
parameters changes predictions for foot-and-mouth disease outbreaks in complex simulation models. Prev.
18
Vet. Med. 87, 272–287.
20
1
Dubé, C., Ribble, C., Kelton, D., McNab, B., 2009. A review of network analysis terminology and its
2
application to foot-and-mouth disease modelling and policy development. Transbound. Emerg. Dis. 56,
3
73–85.
4
Févre, E.M., Bronsvoort, B.M.de C., Hamilton, K.A., Cleaveland, S., 2006. Animal movements and the
5
spread of infectious diseases. TRENDS Microbiol. 14, 125-131.
6
Gelman A., Carlin, J. B., Stern, H. S., Rubin, D. B., 2004. Bayesian Data Analysis (2nd Edition). Chapman
7
& Hall/CRC.
8
Keeling, M., 2005., The implications of network structure for epidemic dynamics. Theo. Pop. Bio. 67. 1-8.
9
Lindström, T., Sisson, S.A., Nöremark, M., Jonsson A., Wennergren, U., 2009. Estimation of distance
10
related probability of animal movements between holdings and implications for disease spread modeling.
11
Prev. Vet. Med. 91,85-94.
12
Mollison, D., Isham, V., Grenfell, B., 1993. Epidemics: Models and Data. J. R. Statist. Soc. B. 157, 115-
13
149.
14
Moslonka-Lefebvre, M., Pautasso, M., Jeger, M., 2009. Disease spread in small-size directed networks:
15
Epidemic threshold, correlation between links to and from nodes, and clustering. Theo. Pop. Bio. 260,
16
402-411.
17
Nöremark, M., Håkansson, N., Lindström, T., Wennergren, U., Sternberg Lewerin, S., 2009. Spatial and
18
temporal investigations of reported movements, births and deaths of cattle and pigs in Sweden. Acta Vet.
19
Scand. 51:37.
20
Ortiz-Pelaez A, Pfeiffer D. U., Soares-Magalhães R.J. Guitian F.J., 2006. Use of social network analysis
21
to characterize the pattern of animal movements in the initial phases of the 2001 foot and mouth disease
22
(FMD) epidemic in the UK. Prev. Vet. Med. 75, 40-55.
21
1
Pineda-Krch, M., O'Brien, J.M., Thunes, C., Carpenter, T.E., 2010. Potential impact of introduction of
2
foot-and-mouth disease from wild pigs into commercial livestock premises in California. Am. J. Vet. Res.
3
71, 82-88.
4
Rweyemamu, M., Roeder, P., Mackay, D., Sumption, K., Brownlie, J., Leforban, Y., Valarcher, J.F.,
5
Knowles, N.J., Saraiva, V., 2008. Epidemiological patterns of foot-and-mouth disease worldwide.
6
Transbound. Emerg. Dis. 55, 57-72.
7
Ribbens, S., Dewulf, J., Koenen, F., Mintiens, K., de Kruif, A., Maes, D., 2009. Type and frequency of
8
contacts between Belgian pig herds. Prev. Vet. Med. 88, 57–66.
9
Robinson, S.E., Christley, R.M., 2007. Exploring the role of auction markets in cattle movements within
10
Great Britain. Prev. Vet. Med. 81, 21-37.
11
Velthuis, A.G., Mourits, M.C., 2007. Effectiveness of movement-prevention regulations to reduce the
12
spread of foot-and-mouth disease in The Netherlands. Prev. Vet. Med. 82, 262-281.
13
Vernon, M.C., Keeling, M.J., 2009. Representing the UK’s cattle herd as static and dynamic networks.
14
Proc. Roy. Soc. London B. 276, 469-476.
15
Ward, M.P., Highfield, L.D., Vongseng, P., Graeme Garner, M., 2009. Simulation of foot-and-mouth
16
disease spread within an integrated livestock system in Texas, USA. Prev. Vet. Med. 88, 286-297.
17
18
Table 1.
19
Table shows the production types and the number and percentages of holdings that have reported having
20
them. Note that the percentages sum to more than 100% as holdings may have more than one type.
Production type
Nr holdings
Percent of holdings
22
Sow pool center
36
1.2%
Sow pool satellite
245
7.9%
Farrow-to-finish
720
23.3%
Nucleus herd
63
2.0%
Piglet producer
1249
40.5%
Multiplying herd
88
2.9%
Fattening herd
1147
37.2%
Information missing
233
7.6%
1
2
23
1
Table 2.
2
Posterior mean estimated values (underlined) of commonness indices, 𝒉, for movements between production types given such that the estimate for
3
movements from type 𝐼 to type 𝐽, ℎ𝐼𝐽 , is found in row 𝐼, column 𝐽. Estimates are given as ℎ𝐼𝐽 x103, and 95% credibility intervals are given in
4
brackets. A high value of ℎ𝐼𝐽 means that movements from type 𝐼 to type 𝐽 are estimated to be common relative to a homogeneous contact pattern.
5
Estimated mean values larger than average commonness index (i.e. 1/64=0.016) are shown in bold.
FROM
TO
Sow pool
center
Sow pool
satellite
Farrow-tofinish
Nucleus
herd
Piglet
producer
Multiplying
herd
Fattening
herd
Missing
information
Sow pool
center
77
(63,94)
120
(110,140)
0.79
(0.41,1.3)
0.59
(0.014,2.2)
4.0
(3.3,4.7)
6.1
(3.3,9.9)
10
(9.2,12)
13
(11,16)
Sow pool
satellite
120
(110,130)
1.6
(1.1,2.1)
0.033
(0.001,0.095)
0.11
(0.003,0.43)
0.015
(0.002,0.038)
0.11
(0.002,0.34)
9.3
(8.6,10)
0.51
(0.24,0.84)
Farrow-tofinish
2.4
(1.7,3.1)
0.047
(0.002,0.13)
0.35
(0.28,0.43)
0.037
(0.001,0.14)
0.12
(0.087,0.15)
0.42
(0.22,0.67)
1.8
(1.6,2.0)
2.3
(2.0,2.5)
Nucleus
herd
69
(56,82)
0.51
(0.11,1.2)
12
(10,13)
130
(120,150)
12
(11,13)
120
(100,130)
3.6
(3.0,4.3)
16
(13,18)
Piglet
producer
5.3
(4.5,6.2)
0.5
(0.35,0.65)
0.45
(0.39,0.52)
0.13
(0.031,0.26)
0.29
(0.25,0.33)
0.097
(0.008,0.22)
9.5
(9.0,10)
3.2
(2.9,3.5)
Multiplying
herd
150
(140,170)
1.3
(0.58,2.3)
20
(18,22)
0.73
(0.079,2.1)
25
(23,26)
15
(11,20)
11
(10,13)
8.8
(7.2,11)
Fattening
herd
0.95
(0.61,1.3)
0.019
(0.001,0.049)
0.015
(0.005,0.030)
0.076
(0.015,0.17)
0.019
(0.010,0.031)
0.39
(0.24,0.58)
0.18
(0.15,0.22)
0.17
(0.11,0.24)
Missing
information
2.1
(1.3,3.2)
0.11
(0.018,0.22)
0.16
(0.095,0.24)
0.53
(0.19,1.0)
0.066
(0.034,0.10)
0.14
(0.007,0.38)
0.97
(0.82,1.1)
0.84
(0.60,1.1)
6
24
1
Table 3.
2
Posterior mean values (underlined) of elements in parameter vector 𝒗 used to model how much production types will determine the contacts of
3
holdings with more than one type. 95% credibility intervals are shown in brackets.
Sow pool center
0.83
Sow pool satellite
0.037
Farrow-to-finish
0.012
Nucleus herd
0.047
Piglet producer
0.019
Multiplying herd
0.052
Fattening herd
0.0023
(0.68,0.93)
(0.016,0.064)
(0.0045,0.022)
(0.018,0.095)
(0.0074,0.036)
(0.020,0.10)
(0.00086,0.0044)
25
1
Figure captions
2
Figure1. Box plots with number of infected holdings after simulation of disease transmission via pig
3
transports for 7 days (top), 28 days (middle) and 365 days (bottom). Each box shows the distribution of
4
simulations initialized with index holding with different production type.
26