1
Influence on disease spread dynamics of herd characteristics in a structured livestock
2
industry.
3
Tom Lindström1, Susanna Sternberg Lewerin2 and Uno Wennergren1
4
5
1
IFM Theory and Modelling, Linköping University, 581 83 Linköping, Sweden
6
2
Department of Disease Control and Epidemiology, SVA, National Veterinary Institute, 751
7
89 Uppsala, Sweden
8
9
Keywords: Disease spread modelling, Heterogeneous contact structure, Animal movement
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11
Summary: Studies of between herd contacts may provide important insight to disease
12
transmission dynamics. By comparing the result from models with different level of details in
13
the description of animal movement we studied how factors influence the final epidemic size
14
as well as the stochastic behaviour of an outbreak. We investigated the effect of contact
15
heterogeneity of pig herds in Sweden due to herd size, between herd distance and production
16
type. Our comparative study suggests that the production type structure is the most influential
17
factor. Hence, our results imply that production type is the most important factor to obtain
18
valid data for and include when modelling and analyzing this system. The study also revealed
19
that all included factors reduce the final epidemic size and also have yet more diverse effects
20
on initial rate of disease spread. This implies that a large set of factors ought to be included to
21
assess relevant predictions when modelling disease spread between herds. Furthermore our
1
1
results show that a more detailed model predicts more stochastic outbreaks and conclude that
2
this is an important factor to consider in risk assessment.
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4
1. Introduction
5
With the aim of providing deeper understanding of the dynamics of livestock disease,
6
researchers have recently focused much attention on the contact pattern between herds [1-6].
7
Such analysis may be used for predictions about disease transmission [7] as well as estimation
8
of the effect of a changed contact pattern [8]. This study focuses on animal movements.
9
Different diseases spread with different vectors, but movements of animals between herds are
10
generally considered to be the main risk factor for livestock disease transmission [9].
11
Theoretical considerations may assume a homogeneous contact pattern with equal
12
probabilities of infections between all herds, known as mass action mixing (MAM), but more
13
realistic models need to include mixing heterogeneities. Perhaps most obvious, the number of
14
contacts may be expected to differ between herds [5,10]. Infected herds with many contacts
15
may rapidly infect a large number of other herds, and thereby function as “super spreaders”
16
[11]. Outbreaks in systems with super spreaders are characterized by stochastic transmission
17
dynamics [12]. In other systems where contacts may be considered undirected, such as for e.g.
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sexually transmitted diseases, it may further be expected that the presence of actors with many
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contacts will result in a rapid increase in number of infections [13]. The contact structure of
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animal movements may however be considered highly directed and a herd that sends many
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animals to other herds may not necessarily receive many [4].
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Also, the contact structure may show heterogeneities by some sort of clustering which is
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known to influence the dynamic of disease transmission [14]. If infectious contacts occur
24
more frequently within a group of herds, the reproductive ratio generally decreases more
2
1
rapidly with decreasing number of susceptibles. Assuming that a herd can only be infected
2
once, reoccurring contacts between two herds will not generate additional disease
3
transmissions. Also, the effect of clustering may be illustrated by how it is usually quantified
4
in network studies, i.e. clustering coefficient, which measures the frequency of “triplets” in
5
the contact network [14].
6
In this study we focus on three factors at herd and between herd levels that may cause
7
heterogeneous contact patterns. First, clustering is typically the effect of a strong spatial
8
component in the contact pattern. Consequently, if contacts occur more likely at shorter
9
distance we may expect a more rapid decrease in the number of new infections due to
10
depletion of local susceptible herds. Secondly, a large herd may be expected to have more
11
contacts and may thereby be more susceptible to infections as well as potentially infecting
12
many other herds. Thirdly, the production type of a herd is expected to influence both the
13
number of contacts as well as with which other herds the contacts occur. Heterogeneity due to
14
different production types is therefore expected to result in stochastic disease transmission as
15
well as clustering patterns in the contact structure. In addition, the number of movements
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from herds of one type to another may be very different from the number of reversed
17
movements [4,5].
18
Lindström et al. [5] presented a model that estimated the probability of movements of pigs
19
between herds based on herd sizes, production types and between herd distances. In this study
20
we test how inclusion of these aspects are expected to change the dynamics of disease
21
transmission via animal movements. We expect that important characteristics may be missed
22
if these aspects are disregarded in modelling of between-herd contacts. Our aim is to
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investigate the effect of including detailed herd data on the expected dynamics of disease
24
transmission via animal movements. We thereby address the importance of collecting such
25
data.
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1
2
2. Transmission and mass action mixing
3
In this study we study infections at the herd level. We follow the assumptions of SIR models,
4
where a herd is considered to be either susceptible (S), infected (I) or recovered (R). We
5
further assume that recovered herds remain in state R for the time of the outbreak (hence
6
become immune) and no new herds are added during the time of an outbreak. If all herds are
7
equal and has equal probability of contacts, disease transmission could be modeled as a MAM
8
process. In this study we are not interested in the dynamics of any specific disease, but rather
9
the influence of the observed contact structure, and we therefore want to make as few
10
assumptions as possible about the disease. In order to observe relevant dynamics we do
11
however need to make assumptions about the recovery of a herd. We therefore assume that an
12
infected herd has a constant recovery rate 𝑟. Further we assume that there is a constant rate of
13
movements (which is generally a fair assumption on a seasonal time scale for the considered
14
pig movements [2]). We thereby formulate a discrete time model where every time step
15
consists of first a probability of recovery for every infected herd and one movement which
16
results in a new infection if the source herd is infected and the destination herd is susceptible.
17
The expected number of herds remaining infected is (1 − 𝑟)𝐼𝑡 and with the assumption of
18
MAM, the probability of the source farm being infected is ((1 − 𝑟)𝐼𝑡 )⁄𝑁, where 𝑁 is the
19
total number of herds and 𝐼𝑡 is the current number of infected herds, and the probability of the
20
destination herd being susceptible is given by 𝑆𝑡 ⁄(𝑁 − 1). Hence, the expected number of
21
susceptible, infected and recovered herds at time 𝑡 + 1 is given by
4
𝑆𝑡+1 = 𝑆𝑡 −
(1 − 𝑟)𝐼𝑡 𝑆𝑡
𝑁
𝑁−1
𝐼𝑡+1 = (1 − 𝑟)𝐼𝑡 +
(1 − 𝑟)𝐼𝑡 𝑆𝑡
𝑁
𝑁−1
(1)
𝑅𝑡+1 = 𝑅𝑡 + 𝑟𝐼𝑡
1
If a disease is initiated with one infected herd at 𝑡 = 0, then 𝐼0 = 1, 𝑆0 = 𝑁 − 1 and an
2
outbreak is expected if 𝐼1 = (1 − 𝑟𝑐 )(1 + 1⁄𝑁) > 1, or rewritten
𝑟𝑐 < 1 −
1
1
𝑁+1
(2)
3
where 𝑟𝑐 is the critical value for when an initial increase in the number of infected herds is
4
expected. In the analysis of pig movements in Lindström et al. [5] 20231 movements between
5
3084 herds were analyzed. With 𝑁 = 3084 in equation 2, 𝑟𝑐 = 3.24 × 10−4 . The expected
6
time that a herd remains infected, 𝑔, is given as 𝑔 = 1⁄𝜆, where 𝜆 = − log(1 − 𝑟). In order
7
to calculate 𝑔𝑐 , the critical expected time a herd must remain infected for an expected initial
8
increase in the number of infected herds, we set 𝑟 = 𝑟𝑐 and with an average of
9
20231/365=55.4 daily movements we estimate 𝑔𝑐 at 55.7 days.
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3. The pig farming industry of Sweden
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The pig industry has a pyramidal structure with transports predominantly going downward in
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the system; from nucleus (breeder) herds to multiplying (gilt producing) herds, from
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multiplying herds to piglet producing herds and farrow-to-finish herds, and finally from piglet
15
producing herds to fattening herds. Farrow-to-finish herds and fattening herds mainly send
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animals to slaughterhouses. An exception from the general movement pattern is the sow pool
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system, where the sows are covered in the central unit and they farrow in the satellite herds.
5
1
Satellite herds can be either piglet producing or farrow-to-finish herds. The sows are regularly
2
moved back and forth between the central unit and the satellite herds. In Lindström et al. [5] it
3
was shown that sow pool centers, nucleus herds and multiplying herds are expected to
4
transmit a disease to a large number of herds, based on the contact structure of animal
5
movements. It was also showed that sow pool centers likely generate transmission locally,
6
while many long distance movements are expected from multiplying herds, which may
7
rapidly spread diseases to distant areas.
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4. The contact model
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In Lindström et al. [5], a model was proposed where the probability of a movement between
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two herds depends on production type, between herd distance and herd sizes. Parameters were
12
estimated using a hierarchical Bayesian approach, implementing Markov Chain Monte Carlo
13
(MCMC) (see e.g. [15] for details on MCMC techniques). It is beyond the scope of this paper
14
to give a full description of the model and the parameter estimations involved, and readers are
15
referred to the original paper for further details. The basic concept of the model is that it is
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assumed that the probability of a movement from herd 𝑠 to 𝑑, 𝑃(𝑑, 𝑠|𝜽), where 𝜽 refers to a
17
set of parameters yet to be specified, can be written as
𝑃(𝑑, 𝑠|𝜽) = 𝑃(𝑑|𝑠, 𝜽)𝑃(𝑠|𝜽)
(3)
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meaning that the destination herd is conditional on the source herd. Herds may have more
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than one production type, and the model was formulated as a mixture model. Probability of
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movements between herds were given conditional on the probability of a movement between
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production types 𝐼 and 𝐽, referring to the production types of the source and destination herd,
22
respectively. Schematically, the model is written
6
𝑃(𝑑, 𝑠|𝜽) = ∑ ∑ 𝑃(𝑑|𝑠, 𝐼, 𝐽, 𝜽)𝑃(𝑠|𝐼, 𝜽)𝑃(𝐼, 𝐽|𝜽)
𝐼
(4)
𝐽
1
This should be interpreted such that the probability 𝑃(𝑑, 𝑠|𝜽) is given as the probability of 𝑑
2
conditional on 𝑠 and production types 𝐼, 𝐽 as 𝑃(𝑑|𝑠, 𝐼, 𝐽, 𝜽), and the probability of source herd
3
𝑃(𝑠|𝐼, 𝜽) is conditional on the production type 𝐼. The distribution 𝑃(𝐼, 𝐽|𝜽) is defined such
4
that ∑𝐼 ∑𝐽 𝑃(𝐼, 𝐽|𝜽) = 1 and is given as
̂, 𝒉) =
𝑃(𝐼, 𝐽|𝜽) = 𝑃(𝐼, 𝐽|𝒗
̂𝐼 𝑁
̂̇𝐽𝐼
ℎ𝐼𝐽 𝑁
̂𝐼 𝑁
̂̇𝐽𝐼
∑𝐼 ∑𝐽 ℎ𝐼𝐽 𝑁
(5)
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̂𝐼 = ∑𝑓 𝑣̂𝑓𝐼 is the amount of herds registered with production type 𝐼, taken
where is given as 𝑁
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̂̇𝐽𝐼 = ∑𝑓 𝑣̂𝑓𝐽 (1 −
into account that a herd is a fraction 𝑣̂𝑓𝐼 of production type 𝐼. Similarly, 𝑁
7
𝑣̂𝑓𝐼
)
̂𝐼
𝑁
8
probability of the destination herd being the same as the source herd. For modelling of a
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system with 𝐾 production types, 𝒉 is a matrix of dimensions 𝐾 × 𝐾 and is defined as
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∑𝐼 ∑𝐽 ℎ𝐼𝐽 = 1, and ℎ𝐼𝐽 is a measurement of how common movements between production
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types 𝐼 and 𝐽 are relative to the abundance of the types. If ℎ𝐼𝐽 = 1⁄𝐾 2 for all 𝐼𝐽, then
12
movements between all production types are equally probable.
13
The probability 𝑃(𝑠|𝐼, 𝜽) was assumed to be dependent on the production type 𝐼 and herd
14
size. The latter was given by the maximum capacity, which is reported separately for fattening
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pigs and sows. This was modeled as
is the amount of herds registered with production type 𝐽, but adjusted to exclude the
̂, 𝐼, 𝑺) =
𝑃(𝑠|𝐼, 𝜽) = 𝑃(𝑠|𝒗
𝑣̂𝑠𝐼 𝐺(𝑆1𝑠 , 𝑆2𝑠 , 𝑐̇1𝐼 , 𝑐̇2𝐼 )
∑𝑓 𝑣̂𝑓𝐼 𝐺(𝑆1𝑓 , 𝑆2𝑓 , 𝑐̇1𝐼 , 𝑐̇2𝐼 )
(6)
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where 𝑆1𝑠 and 𝑆2𝑠 are the maximum capacity of fattening pigs and sows, respectively,
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reported for herd 𝑠, and 𝑺 refers to the maximum capacity of both demographic groups on all
7
1
herds. The parameters 𝑐̇1𝐼 and 𝑐̇2𝐼 determine how probability of movements depends on 𝑆1
2
and 𝑆2 , respectively, for production type 𝐼. The probability of movements were assumed to
3
depend on herd size with a power law relationship and 𝐺 were given as
𝐺(𝑆1𝑓 , 𝑆2𝑓 , 𝑐̇1𝐼 , 𝑐̇2𝐼 ) = (𝑆1𝑓 + 1)
𝑐̇1𝐼
(𝑆2𝑓 + 1)
𝑐̇2𝐼
(7)
4
The use of 𝑆𝑢𝑓 + 1 (𝑢 = 1,2) rather than simply 𝑆𝑢𝑓 was used to avoid zero probability of
5
movements from herds with maximum capacity reported as zero for any of the demographic
6
groups. If 𝑐̇𝑢𝐼 = 0, then there is no relationship between maximum capacity of demographic
7
group 𝑢 and the probability of movements starting at a herd for production type 𝐼.
8
The probability of the destination herd 𝑑 of a movement was assumed to depend on both herd
9
size of 𝑑 and the distance between herds 𝑠 and 𝑑, 𝐷𝑠𝑑 , and given as
̂, 𝑠, 𝐼, 𝐽, 𝑺, 𝑫, 𝒄̈ 𝑱 , 𝜅𝐼𝐽 , 𝑉𝐼𝐽 )
𝑃(𝑑|𝒗
=
𝑣̂𝑑𝐽 𝐺(𝑆1𝑑 , 𝑆2𝑑 , 𝑐̈1𝐽 , 𝑐̈2𝐽 )𝐹(𝐷𝑠𝑑 , 𝜅𝐼𝐽 , 𝑉𝐼𝐽 )
∑𝑓 𝑣̂𝑓𝐽 𝐺(𝑆1𝑓 , 𝑆2𝑓 , 𝑐̈1𝐽 , 𝑐̈2𝐽 )𝐹(𝐷𝑠𝑓 , 𝜅𝐼𝐽 , 𝑉𝐼𝐽 )
(8)
,𝑓 ≠ 𝑠
10
Here, 𝑐̈1𝐽 and 𝑐̈2𝐽 are equivalents of 𝑐̇1𝐼 and 𝑐̇2𝐼 , respectively, but used for modelling of the
11
probability of the destination herd depending on herd sizes. Consequently, 𝑐̈𝑢𝐽 = 0 (𝑢 = 1,2)
12
means that there is no relationship between maximum capacity of demographic group 𝑢 and
13
the probability of movements starting at a herd for production type 𝐽. The function
14
𝐹(𝐷𝑠𝑑 , 𝜅𝐼𝐽 , 𝜈𝐼𝐽 ) is a spatial kernel function and describes how probability of movements
15
decreases with distance between herds, where 𝜅𝐼𝐽 and 𝑉𝐼𝐽 are the kernel kurtosis and variance,
16
respectively, as given in [16]. In equation 8, distance dependence is incorporated by
17
normalizing over all possible destination farms and kurtosis are expected to be of less
18
importance compared to the variance, independent of the spatial arrangement of farms [17].
19
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1
5. Reduced models and simulations
2
In order to test the effect of the factors included in the contact model on outbreak dynamics
3
we simulated disease transmission via animal movements with different models. If nothing
4
else is stated, all models were parameterized by random draws (one draw for every
5
simulation) from the full posterior distribution estimated with the MCMC as given in
6
Lindström et al. [5]. The following six models were used:
7
𝑀𝐹𝑢𝑙𝑙 : The full model as described in section 4.
8
𝑀𝑉̂,𝜅̂ : Production type differences in contact probabilities related to distance was removed by
9
setting 𝑉𝐼𝐽 = 𝑉̂ and 𝜅𝐼𝐽 = 𝜅̂ for all 𝐼, 𝐽. The values 𝑉̂ and 𝜅̂ were given, for each simulation,
10
by random draws from the estimated hierarchical priors of the parameters. For details on
11
hierarchical priors, see e.g. [18].
12
𝑀𝑁𝑜𝐷 : By setting 𝐹(𝐷𝑠𝑑 , 𝜅𝐼𝐽 , 𝑉𝐼𝐽 ) = 1, distance dependence is removed. Hence, probabilities
13
of contacts depend only on production types and herd sizes.
14
𝑀ℎ̂ : If ℎ𝐼𝐽 = ℎ̂ = 1⁄𝐾 2 for all 𝐼, 𝐽, then the production type structure of the contact pattern is
15
removed.
16
𝑀𝑐=0 : The probability of contacts depending on herd size was removed by setting 𝑐̇1𝐼 = 𝑐̇2𝐼 =
17
𝑐̈1𝐽 = 𝑐̈2𝐽 = 0 for all 𝐼, 𝐽.
18
𝑀𝑀𝐴𝑀 : All herds have the same probability of sending and receiving movements.
19
By implementing reduced versions of the full model we aim to show what difference the
20
included factors have on the dynamic and thereby may be missed if these factors are excluded
21
in modelling of spread of livestock disease via between herd contacts. The full model is by no
22
means a “true model”, but it’s the most detailed description of the system. 1000 replicates
9
1
were simulated with each model and the herds included in the analysis in [5] were used as
2
infective units in the simulations. A herd was assumed to instantly become infected if a
3
movement was simulated from infected herd A to susceptible herd B, and any subsequent
4
movement from B to a susceptible herd were assumed to result in a new infected herd. The
5
mean expected time a herd remains infected (and infective) was set to 𝑔 = 91. At least for
6
𝑀𝑀𝐴𝑀 , this is expected to result in an outbreak, given the analysis of section 2.
7
To compare the dynamics predicted by the different models we plot the mean number of
8
infected individuals versus time from first infection. We also compare the final epidemic size
9
as well as the number of currently infected and recovered herds at 𝑡 = 1552, corresponding to
10
the expected number of movements of four weeks. The latter gives an indication of the
11
stochastic behaviour of the initial stage of an outbreak. We are however interested in the
12
dynamics also at later stages and therefore require a measure that captures the variability of
13
the change in the number of infected herds. For this, we first define 𝐼̂𝑡,𝑥,𝑚 and 𝐼̂𝑡+𝜏,𝑥,𝑚 as the
14
simulated number of infected herds at time 𝑡 and 𝑡 + 𝜏, respectively, for replicate 𝑥 and
15
̇
model 𝑚. Further, we define 𝐼𝑡+𝜏,𝑥,𝑚
as the number of infected herds at time 𝑡 + 𝜏 expected
16
from a MAM transmission. This is calculated by equation 1, given the simulated number of
17
susceptible, infected and recovered herds at 𝑡. We then define 𝐺̂𝑡,𝑥,𝑚 = 𝐼̂𝑡+𝜏,𝑥,𝑚 ⁄𝐼̂𝑡,𝑥,𝑚 and
18
̇
𝐺̇𝑡,𝑥,𝑚 = 𝐼𝑡+𝜏,𝑥,𝑚
⁄𝐼̂𝑡,𝑥,𝑚 , as the simulated and predicted growth rate, respectively, at time 𝑡,
19
replicate 𝑥 and model 𝑚 and calculate the growth rate ratio 𝐶𝑡,𝑥,𝑚 = 𝐺̂𝑡,𝑥,𝑚 ⁄𝐺̇𝑡,𝑥,𝑚 . As both
20
𝐺̂𝑡,𝑥,𝑚 and 𝐺̇𝑡,𝑥,𝑚 contains 𝐼̂𝑡,𝑥,𝑚 in the numerator, this may however be rewritten as 𝐶𝑡,𝑥,𝑚 =
21
̇
𝐼̂𝑡+𝜏,𝑥,𝑚 ⁄𝐼𝑡+𝜏,𝑥,𝑚
. This was calculated for 𝑡 = 0, 𝜏, 2𝜏 … 𝑍𝑥,𝑚 𝜏 where 𝑍𝑥,𝑚 is defined as the
22
largest integer satisfying 𝑍𝑥,𝑚 𝜏 < 𝐸𝑥,𝑚 . Here 𝐸𝑥,𝑚 is the time of extinction for replicate 𝑥 and
23
model 𝑚 and we chose 𝜏 = 776, corresponding to the expected number of movements of 14
10
1
̅
days. We then define 𝐶𝑥,𝑚
as the mean value of 𝐶 for replicate 𝑥 and calculate the coefficient
2
of variation as
̅ )2 ]
√E [(𝐶𝑡,𝑥,𝑚 − 𝐶𝑥,𝑚
𝜌𝑥,𝑚 =
̅
𝐶𝑥,𝑚
.
(9)
3
This is a measurement of the stochasticity of the growth rate, adjusted to take into account the
4
expected (from MAM assumptions) differences in growth rate depending on the current
5
number of susceptible, infected, and recovered herds. As we use a coefficient of variation
6
measure we also avoid confusing differences in mean growth rate with the stochastic
7
behaviour of the disease transmission. The initial stage of an outbreak (which is analyzed
8
separately) is inherently stochastic due to the risk of early extinction [19], but this is not the
9
effect we want to capture with 𝜌𝑥,𝑚 . We therefore exclude simulations where the disease went
10
extinct prior to reaching at least 10 herds when comparing 𝜌𝑥,𝑚 of the different models.
11
12
6. Results and discussion
13
Figure 1 shows the mean number of infected herds at time after first infection. Removing the
14
production type structure, as is done in 𝑀ℎ̂ , has the highest impact when comparing the
15
curves to the full model, 𝑀𝐹𝑢𝑙𝑙 . However, other factors are also essential as 𝑀ℎ̂ differs from
16
𝑀𝑀𝐴𝑀 , most noticeably in the modes of the curves. The maximum value for 𝑀ℎ̂ occurs
17
earlier, and there is initially a more rapid increase in the number of infected herds. Such
18
dynamic is expected if some actors, in this case (generally) larger herds, are found to have
19
more contacts [13]. Further, the final epidemic size is smaller for 𝑀ℎ̂ than 𝑀𝑀𝐴𝑀 (Table 1)
20
indicating that while there is initially on average a (slightly) higher rate of infections for 𝑀ℎ̂ ,
21
the reversed is found in later stages of the outbreak. Kiss et al. [20] conclude that
11
1
heterogeneity in the number of contacts, which in 𝑀ℎ̂ is introduced as an effect of herd size,
2
may lead to a smaller final epidemic size. Also, 𝑀ℎ̂ includes distance dependence in contact
3
probability which leads to local depletion of susceptibles. This effect is more prominent in
4
later stages of an outbreak, and therefore more so in 𝑀ℎ̂ than 𝑀𝐹𝑢𝑙𝑙 as the epidemic usually is
5
larger.
6
The effect of herd sizes and between herd distances is also shown more explicitly by the
7
difference between 𝑀𝑐=0 and 𝑀𝑁𝑜𝐷 , respectively, and the full model 𝑀𝐹𝑢𝑙𝑙 . Both show a
8
slightly larger mean epidemic size (Table 1) and, as expected, the mode of 𝑀𝑐=0 is shifted
9
right compared to 𝑀𝐹𝑢𝑙𝑙 in Figure 1. The only model showing a lower epidemic size than
10
𝑀𝐹𝑢𝑙𝑙 is 𝑀𝑉̂,𝜅̂ (Table 1). There is a higher degree of variability in movement distances in 𝑀𝐹𝑢𝑙𝑙
11
which allows more extreme long distance movements. Interactions between which production
12
types are commonly in contact with each other and how the contact probability is influenced
13
by distance in the full model may further explain the difference between 𝑀𝐹𝑢𝑙𝑙 and 𝑀𝑉̂,𝜅̂ . This
14
result may therefore be specific for the analyzed pig movements and should not necessarily be
15
expected in other systems. Our results do however show that, in this system, the effect was
16
apparent and the models differ in later stages of the outbreak (Figure 1).
17
The embedded axes in Figure 1 show the mean differences in number of infected herds at the
18
initial stages of the simulated outbreaks, but important features of the outbreak is neglected if
19
only the mean is considered. Figure 2 shows the distribution of initially infected herds
20
generated with each model. Little difference is found for the percentiles 50% and 75%, but
21
more extreme values are found for models deviating from MAM assumptions, particularly
22
models 𝑀𝐹𝑢𝑙𝑙 , 𝑀𝑉̂,𝜅̂ and 𝑀𝑁𝑜𝐷 . Hence, while 𝑀𝑀𝐴𝑀 is a good predictor of the mean number
23
of cases at the initial stage (there is little differences between 𝑀𝑀𝐴𝑀 and 𝑀𝐹𝑢𝑙𝑙 in the
24
embedded axes of Figure 1), it overlooks the possibility of rapid increase in the number of
12
1
infected herds. Some production types, such as Nucleus herds and Multiplying herds, send
2
many animals to other herds and while these types are rare, they may function as super
3
spreaders and rapidly spread the disease to many other herds if they are infected at an early
4
stage of an outbreak. The simulations are initiated with one randomly infected herd and most
5
herds are of types that rarely send animals other than to slaughterhouses.
6
The production type structure also causes stochastic behaviour in the later stages of the
7
outbreak scenario. While the lowest median of 𝜌 was found for 𝑀𝑀𝐴𝑀 (Figure 3), removal of
8
the production type structure, as is done in 𝑀ℎ̂ , reduces the stochasticity more than removal of
9
other factors (as in 𝑀𝑁𝑜𝐷 or 𝑀𝑐=0). However, as for the final epidemic size, both herd sizes
10
and the spatial component affect the stochasticity and the only model that doesn’t show a
11
significantly lower median of 𝜌 compared to 𝑀𝐹𝑢𝑙𝑙 is 𝑀𝑉̂,𝜅̂ . The spatial kernels estimated in
12
Lindström et al. [5] were mainly leptokurtic, i.e. a fat tail describing long distance contacts.
13
These may spark new infections in areas were local depletion of susceptibles has not yet
14
occurred, and such dynamic has been observed for instance in the UK 2001 outbreak of foot
15
and mouth disease [21]. To avoid the false impression that high stochasticity implies low
16
accuracy of the model, it needs to be stressed that the stochastic behaviour, here induced in
17
particular from the production type structure, is a realistic and important feature of disease
18
spread modelling.
19
20
7. Conclusion
21
Of the factors included in the full model, the production type structure is the factor most
22
influential on the dynamics and hence the most important factor to obtain valid data for and
23
include for modelling. Exclusion of this structure overestimates the final epidemic size but
24
underestimates the stochasticity of outbreaks, both in initial and later stages. However, contact
13
1
heterogeneities due to herd sizes and between herd distances also contribute substantially to
2
the dynamics. Exclusion of the former leads to underestimation of the initial growth of the
3
epidemic but slight overestimation of the final epidemic size. Also, if contact probabilities are
4
modelled as independent of between herd distances, the final epidemic size is somewhat
5
overestimated. Exclusion of production type differences in contact probabilities due to
6
distance however underestimated the epidemic size in the analyzed system. While the
7
presented work only takes into account disease transmission via animal movements it still
8
demonstrates that detailed and updated registers on the animal population in which a model is
9
applied is essential. For a structured farming industry, such as the pig farming considered
10
here, production types are particularly important. This may not be the most important factor in
11
less structured industries, but our result show that this needs to be considered in modelling
12
and risk assessment.
13
14
Acknowledgement
15
We thank the Swedish Board of Agriculture for supplying the data and Swedish Civil
16
Contingencies Agency for funding.
17
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19
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1
Figure 1. Mean number of infected herds vs. time from first infection, given from 1000 simulations with each
2
model, where 𝑀𝐹𝑢𝑙𝑙 is the most detailed model and 𝑀𝑀𝐴𝑀 assumes equal probability of infectious contacts.
3
Other models listed in the legend refers to different simplifications of 𝑀𝐹𝑢𝑙𝑙 . Embedded axes show the same as
4
main axes but only the 20 first weeks.
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6
Figure 2. Percentiles of the total number of infected herds (including both currently infected and recovered
7
herds), 4 weeks after initial infection. The figure shows the results from 1000 simulations with each model.
8
9
Figure 3. Boxplot of the stochastic measurement 𝜌 from simulations with each model. Replicates with final
10
epidemic size less than 10 herds were excluded. Notches show 95% confidence interval of the median, the box
11
edges indicate 25th and 75th percentiles, respectively, and whiskers indicate the range of the distribution within
12
2.7 standard deviations. Outliers are indicated by +.
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18