1
Title:
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Estimating animal movement contacts between holdings of different
3
production types.
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Authors: Tom Lindströma, Scott A. Sissonb, Susanna Stenberg Lewerinc and Uno Wennergrena
5
a
IFM Theory and Modelling, Linköping University, 581 83 Linköping, Sweden
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b
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia
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c
Department of Disease Control and Epidemiology, SVA, National Veterinary Institute, 751 89 Uppsala,
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Sweden
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Corresponding Author:
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Uno Wennergren
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Tel: +46 13 28 16 66
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Fax: +46 13 28 13 99
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Email: unwen@ifm.liu.se
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Correspondence address: See above.
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Key words
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Markov Chain Monte Carlo; Mixture Models; Dirichlet distribution; Commonness index; Animal
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databases
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2
Abstract
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Animal movement poses a great risk for disease transmission between holdings. Heterogeneous contact
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patterns are known to influence the dynamics of disease transmission and should be included in modeling.
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Using pig movement data from Sweden as an example, we present a method for quantification of between
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holding contact probabilities based on different production types. The data contained seven production
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types: Sow pool center, Sow pool satellite, Farrow-to-finish, Nucleus herd, Piglet producer, Multiplying
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herd and Fattening herd. The method also estimates how much different production types will determine
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the contact pattern of holdings that has more than one type. The method is based on Bayesian analysis and
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uses data from central data bases of animal movement. Holdings with different production types are
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estimated to vary in the amount of contacts as well as in with what type of holding they have contact with,
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and the direction of the contacts. We show with a simulation study that these contact patterns may also be
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expected to result in differences in disease transmission via animal movements, depending on the index
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holding. We conclude that models of disease transmission may be improved if production types are
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included in the models when data is available. We also argue that there is great value in keeping records of
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production types since this may be helpful in risk assessments.
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1. Introduction
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In the last decade several major outbreaks with contagious livestock diseases, such as foot and mouth
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disease (FMD) and classical swine fever (CSF) have occurred in Europe. In the effort to minimize the
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risks and extent of outbreaks, researchers are increasingly recognizing the impact of the contact structures
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that mediate the transmission between holdings (e.g. Velthuis and Mourits 2007, Dubé et al. 2009,
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Nöremark et al. 2009, Vernon and Keeling 2009). Depending on the disease, the risk of different contact
1
types may vary but generally between holding movements of animals are regarded to be the main risk
2
factor for transmission of livestock disease (Févre et al. 2006).
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Large scale experiments of contagious animal diseases are impossible for both financial and practical
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reasons. Instead risk assessment and contingency plans must be based on modeling scenarios which can be
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based on previous outbreaks or even outbreaks from other countries. However, models and model
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parameterizations are not directly applicable from one country to another due to differences in animal
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population structure and density, production systems, contact networks etc. For instance, Robinson and
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Christley (2007) showed that livestock markets have great influence on the transmission of diseases in the
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UK, while Nöremark et al. (2009) state that such markets are rare in Sweden. Therefore animal movement
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patterns are very different. In addition, even outbreaks of the same disease in the same country may be
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very different, a fact that is clearly illustrated by the fact that the more recent outbreak of FMD in the
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United Kingdom in 2007 followed a totally different course (DEFRA 2009) than the 2001 outbreak.
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Drawing conclusions from just one or a few outbreaks may therefore be problematic. The livestock
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industry is not static and the structure may change, in particular after major outbreaks (Velthuis and
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Mourits 2007), which makes forecasting based on historic outbreak data difficult. Modeling based on
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observed contact structure is a good approach as it can be based on specific and (ideally) updated data of
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each country. Member states of the EU must keep databases on all livestock holdings and register
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movements of cattle and pigs. Therefore such data is often available for analysis and model
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parameterization.
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Modeling may assume that all holdings have equal probability of contacts but heterogeneity in both the
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number of contacts and between which holdings these occur have been recorded (Bigras-Poulin et al.
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2006, Lindström et al. 2009, Nöremark et al. 2009) and such differences are known to affect the
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transmission dynamics (Mollison 1993, Keeling 2005, Moslonka-Lefebvre et al. 2009). One factor that may
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lead to non random contact patterns is the production types of the holdings. We may assume that animal
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movements are more common between some production types than others, and also that these may differ
1
in the direction of the contact. EU requirements do not include details on production type but the pig
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holding databases in some countries, e.g. Sweden, contains this information.
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The scope of this paper is twofold. First, we present a method for quantification of the contact structure
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between holdings with different production types. Previous studies have shown differences in the contact
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pattern between holdings with different production types (Dickey et al. 2008, Ribbens et al. 2009) and our
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aim is to provide estimates that can be used in the modeling of such contacts. Our method, based on
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Bayesian analysis, utilizes data found in central data-bases where animal movements and holding
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information are stored at national level. It also handles the fact that holdings may have more than one
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production type by analyzing whether some production types may be dominant in determining the
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contacts between holdings. We apply the method to data of pig movements in Sweden from one year.
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Secondly, we also perform a simplistic simulation of disease transmission via the estimated contact
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pattern. The intention is not to investigate the dynamics of any specific disease. Rather we investigate how
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model predictions generally may change with inclusion of production types as estimated with the method
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presented. More specifically we investigate how the expected number of infected holdings differs
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depending on the production type of the first infected holding, based on direct transmission via animal
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movements only.
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2. Material and method
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2.1 Data
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Data was supplied by the Swedish Board of Agriculture. It contained pig holdings with their reported
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production types and the recorded movements between them. Holdings are supposed to be removed from
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the data-base if they no longer have any pigs but it is known that this is not always the case. Inactive
1
holdings where therefore removed as described in Nöremark et al. (2009) and we use the same data as in
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that study for spatial analysis of pig holdings.
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The Swedish pig industry has a pyramidal structure with transports predominantly going downward in the
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system. Permanent trade agreements between pig farmers are quite common. The different production
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types are defined as Nucleus herds (that produce breeding animals), Multiplying herds (that receive
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breeding animals and breed gilts for sale to piglet producers), Piglet producers, Fattening herds and
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Farrow-to-finish herds. Moreover, there is a special type of production system involving several holdings,
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namely Sow pools. These consist of a Sow pool center (where sows are covered or inseminated) and Sow
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pool satellites (where the sows farrow). Thus, the sows are regularly moved back and forth between the
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central unit and the satellites and they don’t always farrow in the same satellite herd every time.
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Movements of cattle and pigs, as well as information related to livestock holdings, are registered by the
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Swedish Board of Agriculture and stored in different databases. The holdings with pigs are registered with
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a unique number, the production-place number (PPN), and the database contains information on postal
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address, species kept and approximate number of animals kept on the holding. The type of production is
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also included. Holdings that are geographically separated from each other should have different PPNs.
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Thus one farmer can have several holdings. The PPN number system is also used for transport vehicles
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transporting cattle or pigs.
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Movements of pigs are reported on group level, not for each pig. The reports include the date, the number
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of pigs, the holding of origin and the holding of destination. The movements are reported by the farmer at
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the PPN of destination. Farmers should report within seven days after the event, either electronically or
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using a form sent by ordinary mail. Due to single reporting on group level, the movement reports could
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not be matched or checked with location, but earlier work on cattle data has shown that errors may be
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common in these data (Nöremark et al. 2009).
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A total number of 3084 holdings and 20231 movements were included in the analysis. We included seven
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production types which may be ticked by the farmer on the report form. Farmers also have the possibility
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to tick a box for “other” and leave free text information, but this was excluded from the analysis.
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Instructions state that a maximum number of two production types may be reported for each PPN but the
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maximum number of production types reported was five. 20% of the holdings had reported more than one
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type and 7.6% of the holdings had no information on production type or had only given free text
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information. The number of holdings that had reported each production type is presented in Table 1.
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2.2 Model specification
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The available data includes reported production types, which we denote 𝑹. We indicate 𝑅𝑓𝑘 = 1 if holding
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𝑓 (𝑓 = 1,2, … 𝑛 where 𝑛 is the number of holdings) has reported production type 𝑘 (𝑘 = 1,2, … 𝐾 where 𝐾
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is the number of production types including the additional type “Missing information”, which is referred
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to as type 𝑚) and 𝑅𝑓𝑘 = 0 if holding 𝑓 has not reported production type 𝑘. Also, data include start, 𝒔, and
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end, 𝒆, holdings of all 𝑇 movements and we use the notation that 𝑒𝑡 and 𝑠𝑡 are the end and start holdings,
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respectively, of movement 𝑡. We say that 𝑹 is fixed and search the posterior distribution of parameters 𝒗
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and 𝒉 given the observed movement data. The parameter vector 𝒗 has 𝐾 − 1 elements (production type 𝑚
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is excluded, see below) and indicates whether some production types are dominant in determining the
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contacts of a holding with more than one production type. If there is no difference between the types, then
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𝑣𝑘 = 1⁄(𝐾 − 1) for all production types 𝑘 = 1,2, … 𝐾 − 1.
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Parameter matrix 𝒉 has 𝐾 × 𝐾 elements and indicates how common movements between the different
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production types are and ℎ𝐼𝐽 = 1⁄𝐾 2 if there is no difference in probability of contacts between holdings.
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A higher value indicates that movements between the types are more common than expected by random.
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Hence, 𝒉 can be seen as a measurement of the commonness of movements between holdings with
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different production types but accounting for how frequent production types are, and we refer to this as the
1
commonness index. Both 𝒗 and 𝒉 are defined such that the sum of the elements equals to one and each
2
element is larger than zero and smaller than one.
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The posterior distribution of the model parameters are
𝑃(𝒗, 𝒉|𝒆, 𝒔, 𝑹) ∝ 𝐿(𝒆, 𝒔|𝒗, 𝒉, 𝑹)𝑝(𝒗, 𝒉)
(1)
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In determining the distribution of 𝐿(𝒆, 𝒔|𝒗, 𝒉, 𝑹), we start with the hypothetical case of no difference
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between production types and no difference in the probability of movements between them. In such a
6
setup, the probability that a movement 𝑡 originates at holding 𝑠 and ends up at holding 𝑒 is given by
𝑃(𝑒𝑡 , 𝑠𝑡 |𝑛) = 𝑃(𝑠𝑡 |𝑛)𝑃(𝑒𝑡 |𝑛, 𝑠𝑡 ) =
1
𝑛−1
1 1
𝑛𝑛 −1
(2)
1
𝑛
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where 𝑛 is the total number of holdings. 𝑃(𝑒𝑡 |𝑛, 𝑠) =
8
at the same holding as it originates.
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If however there are different production types, each holding has exactly one type (𝑅𝑓𝑘 = 1 for only one
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rather than since a movement may not end up
type 𝑘) and the probability of contacts between these vary, then
𝑃(𝑒𝑡 , 𝑠𝑡 |𝒉, 𝑹) =
ℎ𝑖𝑗
∑𝐼 ∑𝐽 ℎ𝐼𝐽 𝑁𝐼 𝑁̇𝐽
(3)
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where 𝑖, 𝑗 are production types of 𝑠 and 𝑒 respectively. Conditionality on 𝑹 is expressed through 𝑵 which
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is a vector of size 𝐾and 𝑁𝑘 is the number of holdings with production type 𝑘, i.e. ∑𝑘 𝑅𝑓𝑘 . Since
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movements starting and ending at the same holding are excluded, 𝑁̇𝐽 = 𝑁𝐽 if 𝐼 ≠ 𝐽 and 𝑁̇𝐽 = 𝑁𝐽 − 1 if 𝐼 =
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𝐽. The parameter 𝒉 determines whether movements are more or less likely between types 𝐼, 𝐽 and equation
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3 can be rewritten as equation 2 if ℎ𝐼𝐽 = 1⁄𝐾 2 for all 𝐼, 𝐽.
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If the interest is not probability of individual holdings, but rather the probability of contact between
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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
(4)
1
This is obtained by summation of the probability of movements between holdings of types 𝐼, 𝐽 divided by
2
the probability of all possible movements.
3
Many holdings however have more than one production type, and 𝑅𝑓𝑘 may be larger than one for more
4
than one 𝑘. We therefore model movements between holdings that act as a mixture of the types. The
5
proportions are given through 𝒗 and 𝑹 and 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
(5)
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
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where 𝑘 ≠ 𝑚, equation 5 is interpreted as 𝒗 normalized over the reported production types of each
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holding 𝑓.
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Equation 3 is then rewritten as
𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝒉, 𝑹) = ∑
𝑖,𝑗
𝑣̂𝑖𝑠 𝑣̂𝑗𝑒 ℎ𝑖𝑗
̂𝐼 𝑁
̂̇𝐽𝑠
∑𝐼 ∑𝐽 ℎ𝐼𝐽 𝑁
(6)
12
̂𝐼
for 𝑒 ≠ 𝑠. The number of holdings of each production type, 𝑁𝐼 and 𝑁̇𝐽 , in equation 4 are replaced by 𝑁
13
̂̇𝐽𝑠 in equation 6 and these are given by 𝒗
̂ through
and 𝑁
̂𝐼 = ∑ 𝑣̂𝑓𝐼
𝑁
(7)
𝑓
1
and
̂̇𝐽𝑠 = ∑ 𝑣̂𝑔𝐽
𝑁
(8)
𝑔≠𝑠
2
respectively. These may be interpreted as the amount of holdings of each production type, taking into
3
account that a holding may be a proportion of each type. Note that in equation 5, 𝑣̂𝑓𝑘 = 1 if 𝑅𝑓𝑘 = 1 for
4
̂𝐼 and
only one type 𝑘 and zero for all other. Consequently, if all holdings have only one production type 𝑁
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̂̇𝐽𝑠 will be the same as 𝑁𝐼 and 𝑁̇𝐽 , respectively.
𝑁
6
To make analysis of parameters easier we seek an expression of 𝑒𝑡 , 𝑠𝑡 of movement 𝑡 conditional on the
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production types 𝐼𝑡 , 𝐽𝑡 . The analog of equation 4 based on proportions of holdings is
𝑃(𝐼, 𝐽|𝒗, 𝒉, 𝑹) =
̂𝐼 𝑁
̂̈𝐽𝐼
ℎ𝐼𝐽 𝑁
̂𝐼 𝑁
̂̈𝐽𝐼
∑𝐼 ∑𝐽 ℎ𝐼𝐽 𝑁
(9)
8
̂̈𝐽𝐼 differs from 𝑁
̂̇𝐽𝑠 in equation 8 in that the start holding 𝑠 is not known when writing
where the use of 𝑁
9
the likelihood of types 𝐼, 𝐽. Also, the start holding might not be exactly of type 𝐼 and exclusion of
10
movements to the same holding as the origin cannot be achieved through 𝑁̇𝐽 as in equation 4 (i.e. 𝑁̇𝐽 = 𝑁𝐽
11
̂̈𝐽𝐼 is adjusted by the probability of 𝑒𝑡 = 𝑠𝑡 given type 𝐼 of the
if 𝐼 ≠ 𝐽 and 𝑁̇𝐽 = 𝑁𝐽 − 1 if 𝐼 = 𝐽). Instead 𝑁
12
̂𝐼 and therefore
start holding, which for holding 𝑔 is 𝑣̂𝑔𝐼 ⁄𝑁
𝑣̂
̂̈𝐽𝐼 = ∑ 𝑣̂𝑔𝐽 (1 − 𝑔𝐼 )
𝑁
̂𝐼
𝑁
(10)
𝑔
13
We may then give probability of 𝑒𝑡 , 𝑠𝑡 conditional on the probability of the production types 𝐼𝑡 , 𝐽𝑡 and
14
rewrite equation 6 as
𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝒉, 𝑹) = 𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 )𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹)
1
(11)
where 𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) is given from equation 9 and
𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 ) =
𝑣̂𝑠𝐼 𝑣̂𝑒𝐽
̂𝑁
̂̇
𝑁
(12)
𝐼 𝐽𝑠
2
This can be explained intuitively such that if types 𝐼𝑡 , 𝐽𝑡 of movement 𝑡 is known, then the probability of
3
that movement going from holding 𝑠 to 𝑒 is simply the proportion of holding 𝑠 being of type 𝐼 and holding
4
𝑒 being of type 𝐽 divided by the total proportions of holdings of production types 𝐼, 𝐽 by the definition of
5
̂𝐼 and 𝑁
̂̇𝐽𝑠 in equation 7 and 8, respectively.
𝑁
6
The likelihood of unknown parameters 𝒗, 𝒉 is then written as
𝑇
𝐿(𝒆, 𝒔|𝒗, 𝒉, 𝑹) = ∏(𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 )𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) )
(13)
𝑡=1
7
8
2.2.1 Indicator variable
9
To facilitate computation of the full conditional distribution, we introduce an indicator variable (Gelman
10
et al. 2004), 𝑼, of size 𝐾 × 𝐾 × 𝑇, and rewrite equation 13 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
(14)
𝑡=1 𝐼=1 𝐽=1
11
where 𝑈𝐼𝐽𝑡 = 1, probabilistically, for exactly one combination of 𝐼, 𝐽 for each movement 𝑡 with
12
probability
𝑃𝑟(𝑈𝑖𝑗𝑡 = 1|𝒗, 𝒉, 𝑹) =
𝑣̂𝑠𝑖 𝑣̂𝑒𝑗 ℎ𝑖𝑗
∑𝑖 ∑𝑗 𝑣̂𝑠𝑖 𝑣̂𝑒𝑗 ℎ𝑖𝑗
(15)
1
Note that unlike the standard formulation of mixture models in Gelman et al. 2004, the mixture
2
components in equation 14 include the mixing distribution 𝒗. This is because 𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 ) and
3
𝑃𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) are normalized over the proportion of holdings belonging to each production type (see
4
equations 9 and 12, 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)
(16)
5
6
2.3 Parameter estimation
7
We use a Markov Chain Monte Carlo (MCMC) to estimate 𝒉, 𝒗 and 𝑼. Programs were written in MatLab
8
7.8. Parameter 𝑼 is updated with Gibbs sampling by drawing one random number for each 𝑡 from a
9
multinomial distribution with probabilities given by equation 15. The full conditional distribution of 𝒉 and
10
𝒗 is of non standard form and Metropolis-Hastings updates have to be used (see below). Conditional
11
distribution of 𝒉 is based on equation 9 and with inclusion of indicator variable 𝑼 the conditional
12
distribution of 𝒉 is given by
𝑇
𝐾
𝐾
𝑃(𝒉|𝒗, 𝑼, 𝑹) ∝ ∏ ∏ ∏(𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) )𝑈𝐼𝐽𝑡 𝑃(𝒉)
(17)
𝑡=1 𝐼=1 𝐽=1
13
where conditionality on 𝒗 and 𝑹 is expressed through 𝑵 by equations 5, 7 and 10 and 𝑃(𝒉) is the prior
14
distribution of 𝒉. We use uninformative prior (𝐷𝑖𝑟𝑖𝑐ℎ𝑙𝑒𝑡(1,1, … 1)). For each update and 𝑡,
15
(𝑃(𝐼, 𝐽|𝒗, 𝒉, 𝑹) )𝑈𝐼𝐽𝑡 will deviate from one for only one combination of 𝐼, 𝐽 and the likelihood of 𝒉 for
16
each 𝑡 can be given by the probability density of
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 )
17
where 𝑝𝐼,𝐽 = 𝑃(𝐼, 𝐽|𝒗, 𝒉, 𝑹) as given in equation 9 and 𝑀𝐼,𝐽 = ∑𝑡 𝑈𝐼𝐽𝑡 .
(18)
1
Parameter 𝒗 is included in both 𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 )and 𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹) in equation 14 and the conditional
2
distribution of 𝒗 is given by
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
(19)
𝑇
𝐾
𝐾
𝑇
𝐾
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
3
𝑈𝐼𝐽𝑡
𝑈𝐼𝐽𝑡
where ∏𝑇𝑡=1 ∏𝐾,𝐾
is given as in equation 18 and ∏𝐾,𝐾
𝐼=1,𝐽=1 𝑃(𝐼𝑡 , 𝐽𝑡 |𝒗, 𝒉, 𝑹)
𝐼=1,𝐽=1 𝑃(𝑒𝑡 , 𝑠𝑡 |𝒗, 𝑹, 𝐼𝑡 , 𝐽𝑡 )
4
for each 𝑡 is given by equation 12. For further notation we write
𝑇
𝐾
𝑈𝐼𝐽𝑡
𝐾
𝑣̂𝑠𝐼 𝑣̂𝑒𝐽
𝐿2 = ∏ ∏ ∏ (
)
̂𝑁
̂̇
𝑁
𝑡
𝐼=1 𝐽=1
(20)
𝐼 𝐽𝑠
5
𝑈𝐼𝐽𝑡
̂𝐼 𝑁
̂̇𝐽𝑠 ))
and note that for each update ((𝑣̂𝑠𝐼 𝑣̂𝑒𝐽 )⁄(𝑁
deviates from one for only combination of 𝐼, 𝐽.
6
For 𝒉 or 𝒗 we use Metropolis-Hastings updates which involves repeatedly proposing parameter values and
7
subsequent acceptation or rejection. Since there are many similarities in the updates of 𝒉 or 𝒗 and we use
8
the indication 𝜽 (consisting of 𝜃1 , 𝜃2 , … 𝜃𝑙 ) when referring to either 𝒉 or 𝒗. Both 𝒉 and 𝒗 are defined as
9
𝜃1 + 𝜃2 + ⋯ +𝜃𝑙 = 1 and all 0 < 𝜃𝑚 < 1. Therefore the elements are dependent and cannot be updated
10
separately, and the proposed values need to follow that definition. Hence, proposals are performed using
11
Dirichlet distribution. If the current position of an iteration is 𝜽̇ then 𝜽̈ is proposed from 𝑞 (𝜽̈|𝜽̇, 𝐴(𝜽̇)) =
12
𝐷𝑖𝑟𝑖𝑐ℎ𝑙𝑒𝑡(𝜽̈|𝑩) where 𝑩 = 𝜽̇𝐴(𝜽̇) which is centered at 𝜽̇ and 𝐴(𝜽̇) controls the width of the proposal
13
distribution. It is possible to use a fixed value 𝐴̈ such that 𝐴(𝜽̇) = 𝐴̈ for all 𝜽𝑡 but problems may arise for
14
very small 𝜃𝑚 . Generation of Dirichlet random numbers are based on random numbers from either the
15
Gamma or the Beta distribution (Gelman et al. 2004). For numerical reasons, small values of the 𝛼
1
parameter of either Gamma or Beta distribution generates values equal to exactly zero, which also gives
2
𝜃𝑚 = 0 and all subsequent proposal will be 𝜃𝑚 = 0 for any value of 𝐴̈. We therefore use 𝐴(𝜽) =
3
𝐶𝑙𝑖𝑚 /𝜃𝑚𝑖𝑛 where 𝐶𝑙𝑖𝑚 is the critical value for where numerical problems occur in generation of Dirichlet
4
random numbers.
5
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))
(21)
6
where 𝑃(𝜽| ∙) is the posterior distribution of the parameter vector 𝜽 and 𝑃(𝜽) is the prior distribution of 𝜽.
7
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))
(22)
8
where 𝐿̇1 and 𝐿̈1 are the likelihoods given by equation 18 for the current and proposed values of 𝒉,
9
respectively. A good mixing is hard to obtain for 𝒉 since it is based on a random walk in 63 (i.e. 𝐾 2 − 1)
10
̂𝑰𝒉
̌ 𝑰𝑱 where ∑𝐾
̂
dimensions. We therefore make partial updates by rewriting 𝒉𝑰𝑱 = 𝒉
𝐼=1 ℎ𝐼 = 1 och
11
̂
̌
̌
∑𝐾
𝐽=1 ℎ𝐼𝐽 = 1 and for every iteration we perform one update on 𝒉𝑰 and 𝐾 updates on 𝒉𝑰𝑱 , in the latter case
12
proposing values of ℎ̌𝑖1 , ℎ̌𝑖2 , … ℎ̌𝑖𝐾 for the 𝑖th partial update. The formulation of equation 22 is however
13
still valid since acceptance ratio is based on 𝒉 but 𝑞 will be different for each partial update.
14
The acceptance ratio for 𝒗, is given
𝑚𝑖𝑛 (1,
𝐿̈1 𝐿̈2 𝑃(𝒗̈ )𝑞(𝒗̇ |𝒗̈ , 𝐴(𝒗̈ ))
)
𝐿̇1 𝐿̇2 𝑃(𝒗̇ )𝑞(𝒗̈ |𝒗̇ , 𝐴(𝒗̇ ))
(23)
15
where 𝐿̇1 and 𝐿̈1 are the likelihoods given by equation 18 for the current and proposed values of 𝒗,
16
respectively, and 𝐿̇2 and 𝐿̈2 are given analogously from equation 20.
1
2
2.4 Simulation
3
We set up a simple simulation model where contacts via animal movements between holdings are
4
modeled with probabilities given by section 2.2. In each simulation the model is parameterized with
5
random draws from the parameter estimates given by the MCMC. We simulate transmission between the
6
same farms and their reported production types as was used in the analysis of 𝒉 and 𝒗 and the holdings are
7
used as infective units. Each holding will be in either state S (susceptible) or I (infected). The simulation
8
assumes instant infection through directed contacts. I.e. if a movement occurs from infected holding A to
9
B, B will instantly become infected and all consecutive movement from B to any uninfected holding C
10
will lead to infection. If however there is a movement from uninfected holding D to infected holding E, D
11
will not become infected.
12
We want to investigate how the expected number of infected holdings varies depending on the production
13
type of the initially infected holding. Therefore we vary the type of the initially infected holding and select
14
randomly from holdings that have reported the current production type. We run the simulations 500 times
15
for each production type and record the number of infected holdings after 388, 1552 and 20231
16
movements, which corresponds to the expected number of movements for 7, 28 and 365 days,
17
respectively. Note that these numbers also include movements between non infected holdings. We analyze
18
the results with Kruskal-Wallis tests and box plots.
19
20
3. Results
21
Estimates of 𝒉 are shown in Table 2. The five highest commonness indices were found for, in decreasing
22
order, movements from Multiplying herds to Sow pool centers, Nucleus herds to other Nucleus herds,
23
Sow pool centers to Sow pool satellites, Sow pool satellites to Sow pool centers and Nucleus herds to
1
Multiplying herds. Movements involving fattening herds were estimated to be relatively rare, and in
2
particular movements from such holdings. Movements to and from Farrow-to-finish type holdings also
3
had low commonness indices. Sow pool satellites only had high commonness indices for movements from
4
Sow pool centers and except for the high commonness index of movements back to Sow pool centers,
5
Sow pool satellites tend to send pigs to Fattening herds. Movements from Nucleus herds had generally
6
high commonness index but estimates were low for all incoming movements except from other Nucleus
7
herds.
8
Estimates for 𝒗 are shown in Table 3. Sow pool centers were estimated to be dominant in determining the
9
contacts with other holdings. Fattening herds were estimated to predict very little about the contacts of a
10
herd when this production type was reported together with other types.
11
The predicted numbers of infected holdings depending on the production type where the infection was
12
initialized are shown in Figure 1. The highest median was found for simulations where the initial infection
13
started in a Nucleus herd followed by Multiplying herds and Sow pool centers. Simulations with the initial
14
infection on holdings with Fattening herds, Farrow-to-finish and Missing information showed low median
15
infection rate but many outliers with large number of infections. The Kruskal-Wallis test showed that for
16
all tested time periods 𝑝 ≪ 0.01 and the null hypothesis that the number of infected holdings is
17
independent of the production type of the initial holding is rejected.
18
19
4. Discussion
20
Largely, our results on general movement patterns between production types reflect what is already known
21
about the Swedish pig industry. The analysis of 𝒉 shows that the top five estimates are found for
22
movements between types that are known to move many animals between them. However, commonness
23
index for movements between Sow pool centers is estimated to be high, indicating that trade of animals
1
between different holdings of this type would be common. However, trade between Sow pool centers is
2
not allowed in the system of Sow pools (the idea being that each Sow pool should function as one unit
3
without contact with herds outside the pool except for some sourcing herds for gilts). This finding is thus
4
contrary to what is known about the actual practice of these units and indicates that there may be
5
misclassification of the production type of some herds in this category. Also, the analysis estimate that
6
there is a high commonness index for movements from Nucleus herds to Sow pool centers. Hence, while
7
there is (as expected) many movements from Nucleus herds to Multiplying herds and from Multiplying
8
herds to Sow pool centers, movements directly from Nucleus herds to Sow pool centers are also estimated
9
to be common. We can not exclude the possibility that these unexpected results are due to erroneous
10
reports on production types. The production types involved are quite rare (see Table 1) and thereby more
11
sensitive to data quality.
12
From Table 2 we observe that Fattening herds have higher commonness indices for incoming than
13
outgoing movements in contacts with all other types. This result is expected since Fattening herds only
14
produce pigs for slaughter and slaughterhouses were not included in this study. By comparing the
15
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
16
further conclude that the commonness index of movements generally differs depending on the direction.
17
The credibility intervals of ℎ𝐼𝐽 and ℎ𝐽𝐼 , 𝐽 ≠ 𝐼, only overlap in 4 out of 28 cases. One important exception
18
is the commonness index of movements between Sow pool centers and Sow pool satellites which are very
19
similar in estimates independent of direction. This similarity is expected given that the system is based on
20
the same number of sows moving through the entire system, back and forth between the centre and the
21
satellite holdings.
22
The fact that many holdings have more than one production type needs to be incorporated when modeling
23
contacts. One may assume that a holding that for instance has two types will have a contact structure that
24
is 50% of each type. The analysis of 𝒗 however show that such supposition is incorrect, at least for the
25
data analyzed in this study. There is great difference in how much the production types determine the
1
contacts of a holding. The type Sow pool center is very dominant while Fattening herd determines very
2
little about the contacts of a holding when this type is reported concurrently with others. So rather than
3
assuming 50% of each type, a holding 𝑓 that has reported two production types is expected to have a
4
contact structure that is determined 𝑣̂𝑓𝑘 (as defined by equation 5) of each type 𝑘 = 1,2. Hence, a holding
5
that has reported e.g. both Sow pool center and Fattening herd is expected to interact with other holdings
6
as 99.7% (using mean estimates of 𝒗 as shown in Table 3, 0.83⁄(0.83 + 0.0023)) Sow pool center and
7
only 0.3% Fattening herd. The expected contact pattern of such a farm would be very different if it was
8
assumed be determined equally by the two types as the estimates for 𝒉 are very different between
9
movements involving Sow pool centers and Fattening herds (see Table 2).
10
The assumptions of the simulation model are too crude to capture the dynamics of any real disease as
11
there is no intra-herd dynamics and all other contacts are neglected. Simplistic models with holdings as
12
infective units do however, as pointed out by Vernon and Keeling (2009), allow for investigation of the
13
effects of the contact structure. In the simulation study, disease transmission was estimated to be highest if
14
the index holding was a Nucleus herd, and also high if the index holding was of types Sow pool center,
15
Sow pool satellite or Multiplying herd. The median number of cases was very low if the index holding
16
was of type Farrow-to-finish, Fattening herd or had Missing information. Also, the Kruskal-Wallis test
17
showed that the number of infected holdings differed depending on the index holding. This is also
18
according to what is expected and planned for in contingency plans and disease surveillance programs.
19
We conclude that the observed contact heterogeneities are also expected to influence the dynamics of
20
disease transmission. However, a more realistic model needs to include other factors, such as intra-herd
21
dynamics, incubation time, mode of spread and other disease specific aspects. Moreover, there are other
22
factors influencing the probability of contacts between holdings via animal movements. Distance is known
23
to be an important factor (Lindström et al. 2009, Ribbens et al. 2009) and we may expect holdings with
24
larger herd sizes to have more contacts (Ribbens et al. 2009). We do however argue that these factors, if
25
included, should be analyzed as being dependent on the production type. For instance, more short distance
1
contacts may occur when more permanent agreement between farmers are present, such as for the actors
2
in Sow pools. And a large Farrow-to-finish holding might not necessarily have many incoming contacts as
3
the whole production chain is integrated on the farm.
4
Our results however clearly show that production types influence the contact pattern and this in turn is
5
expected to have implications for disease transmission. Hence, there is great value in including this
6
information in animal databases and it has the potential to improve risk assessment. We believe that
7
production types should be included in disease spread models when data is available and urge other
8
researchers to include this in their studies. However, one must remember that the contact patterns may
9
change quickly, due to structural changes in pig production. Larger, more specialized units may lead to
10
less frequent reporting of more than one production system. Some types of production may, however,
11
become more common in smaller units or mixed with others. Therefore, analysis of the contact patterns
12
needs to be updated to avoid erroneous assumptions about the structure. Also, reliable data is essential if
13
analysis of the contact patter is to be used in risk assessment of between holding disease transmission.
14
Clear guidelines to the farmers may improve the quality of the data and thereby the possibility to utilize
15
this for risk assessments of disease spread.
16
17
5. Conclusion
18
We have presented a model for analysis of animal movements between holding of varying production
19
types and applied it to pig movement data from Sweden. We have shown that there is great difference
20
between which production types influence the contact pattern when a holding has more than one type as
21
well as heterogeneity in the contact pattern between the types. The results also demonstrate that there
22
generally is a difference in direction of contacts. We have further shown that the contact heterogeneity is
23
expected to influence the dynamics of disease spread via the considered contacts. Hence we believe that
1
models based on contact patterns between holdings may be improved by inclusion of production types. It
2
is therefore valuable to include such information in central data bases of livestock holdings.
3
4
Conflict of interest
5
We have no conflict of interest.
6
7
Acknowledgement
8
We thank the Swedish Emergency Management Agency (KBM) for funding and the Swedish Board of
9
Agriculture for supplying the data used.
10
11
References
12
Bigras-Poulin, M., Thompson, R.A., Chriel, M., Mortensen, S., Greiner, M., 2006. Network analysis of
13
Danish cattle industry trade patterns as an evaluation of risk potential for disease spread. Prev. Vet. Med.
14
76, 11-39.
15
DEFRA, 2009. FMD: 2007 outbreak.
16
http://www.defra.gov.uk/foodfarm/farmanimal/diseases/atoz/fmd/2007/index.htm
17
Dickey, B.F., Carpenter, T.E., Bartell, S.M., 2008. Use of heterogeneous operation-specific contact
18
parameters changes predictions for foot-and-mouth disease outbreaks in complex simulation models. Prev.
19
Vet. Med. 87, 272–287.
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
Ribbens, S., Dewulf, J., Koenen, F., Mintiens, K., de Kruif, A., Maes, D., 2009. Type and frequency of
21
contacts between Belgian pig herds. Prev. Vet. Med. 88, 57–66.
1
Robinson, S.E., Christley, R.M., 2007. Exploring the role of auction markets in cattle movements within
2
Great Britain. Prev. Vet. Med. 81, 21-37.
3
Velthuis, A.G., Mourits, M.C., 2007. Effectiveness of movement-prevention regulations to reduce the
4
spread of foot-and-mouth disease in The Netherlands. Prev. Vet. Med. 82, 262-281.
5
Vernon, M.C., Keeling, M.J., 2009. Representing the UK’s cattle herd as static and dynamic networks.
6
Proc. Roy. Soc. London B. 276, 469-476.
7
Table 1.
8
The production types and the number of farms that has reported having them
9
10
Production type
Nr farms
Sow pool center
36
Sow pool satellite
245
Farrow-to-finish
720
Nucleus herd
63
Piglet producer
1249
Multiplying herd
88
Fattening herd
1147
Information missing
233
1
Table 2.
2
Mean estimated values (underlined) of commonness indices, 𝒉, for movements between production types given such that the estimates 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 interval are given in
4
brackets. A high value of ℎ𝐼𝐽 means that movements from type 𝐼 to type 𝐽 is 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)
1
1
Table 3.
2
Mean estimated 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)
1
Figure captions
2
Figure1. Box plots with number of infected farms after simulation of disease transmission via pig
3
transports for 7 days (top), 28 days (middle) and 365 days (bottom). The simulations were initialized with
4
index holding with different production type.