Download Is a sampled network a good enough descriptor for epidemic

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Is a sampled network a good enough descriptor for epidemic
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predictions? Missing links and appropriate choice of representation
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Jenny Lennartssona, Annie Jonssona, Nina Håkanssona,b and Uno Wennergrenb,*
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Systems Biology Research Centre, Skövde University, Box 408, 541 28, Skövde, Sweden
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b
IFM Theory and Modeling, Linköping University, 581 83 Linköping, Sweden
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* Corresponding author. IFM Theory and Modeling, Linköping University, 581 83
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Linköping, Sweden. Phone: +46 13 281666. Fax: +46 13 281399
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E-mail address: [email protected]
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ABSTRACT
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Lack of complete data sets can be a limitation in network analysis. Here, we studied how
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link density affects the properties of disease transmission networks. Networks with
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weighted links were used to run scenarios assuming distance dependent probabilities of
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disease transmission, which were subsequently compared with scenarios where
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probabilities of disease transmission were randomly drawn (i.e. non-distance dependence).
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In both types of scenarios, two link sampling methods were tested, one based on distance
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dependence and the other on a random approach. This allowed us to study how link density
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influences the spread of disease in networks generated using different link sampling
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methods and transmission scenarios. We conclude that, under the assumption of distance
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dependence of both link sampling and disease transmission, predictions about the extent of
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an epidemic can be drawn from a network, even at a link density that is low, albeit higher
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than in most empirical studies. In reality, neither sampling procedures nor disease
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transmissions fit distance dependence perfectly. Our results show how this enforces an even
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higher level of link density in sampled networks to achieve reasonable predictions for
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disease transmission.
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Keywords: epidemic modelling, link density, link sampling, network analysis, disease
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transmission
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1. INTRODUCTION
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During recent years, there has been growing interest in and use of network analysis in
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epidemiology. A network consists of interacting units, here denoted nodes, and these units
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are connected through relationships we call links. Examples of nodes are individual animals
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or animal holdings, and links can be visits or animal transports. The pattern of links
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between the nodes gives rise to networks with contact structures that differ depending on
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both the amount of links and how the links are organized. Here, we classify networks into
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three categories: (i) the complete network (Wasserman and Faust, 1994), which includes all
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theoretically possible links (figure 1a); (ii) the real-world network, which comprises all
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realizations of links during a specified time period (figure 1b); (iii) the sampled network,
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which encompasses the links measured during the sample period (figure 1c). In addition, to
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estimate the link structure of networks, probabilities of occurrence or disease transmission
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can be measured per time unit of individual links, in which case the network is referred to
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as being weighted (Barrat et al., 2004). In contrast to classical epidemiological models such
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as SI, SIR, and SEIR, network models relax the assumption of homogeneous mixing (mass-
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action type of assumptions), because all nodes are not linked to all other nodes, or the links
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are weighted, for example depending on distance.
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The sampled network can be estimated through sample surveys, literature studies, or
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contact tracing, or by use of databases (e.g. national databases for animal movements). The
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estimation is cumbersome, and it might be expected that estimated networks will lack some
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links and even some nodes (Clauset et al., 2008). It is the real-world network that is of
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interest to register, but we have to use the sampled network to represent it. Hence, there is a
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need to evaluate the effect of missing links in order to assess or possibly reduce errors when
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networks are applied. The current study focused on how the number of links in sampled
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networks affects predictions of the size of epidemics. We simulated spread of disease in
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networks with different link densities and different scenarios that mimicked sampling
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procedures.
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A real-world weighted contact network consists of all contacts that occur during a specific
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time period, where the link weights are estimated, as probabilities, from the contact
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frequencies. Another time period constitutes yet another event along with its specific
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contacts, which may very well lead to another network with a different set of links. Thus,
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the question arises as to whether the properties of the two real networks will differ.
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Furthermore, it must be asked whether the properties of the sampled networks for those two
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events will differ. Is it possible that a property of the first sampled network (e.g. the spread
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of disease) can be valid as an approximation of the spread of disease during a second event?
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Obviously, a time period that is too short will result in a poor approximation of any of the
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two events. On the other hand, a measured period with a very large time frame will yield a
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nearly complete network that is almost a perfect approximation. Beyond reality is the
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infinite sampling procedure that results in a complete network in which all links exist with
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specified probabilities. Somewhere in between the short time period and the excessive
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sampling, there is a sampled network that has enough links and sufficiently estimated
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probabilities to generate an adequate approximation.
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In the present study, we concentrated on disease transmission networks with weighted
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links, because it might be expected that the probability of contact can be higher for some
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links than for others. We ran scenarios with the assumptions of distance dependent
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probabilities and compared the results with scenarios based on randomly drawn
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probabilities. The distance dependence was tested for disease transmission and link
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sampling, both separately and in combination. In a worst-case scenario, there would be a
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mismatch when using distance dependent transmission probabilities together with random
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link sampling. Hence, in addition we studied how the necessary amount of measured links
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also depended on the mismatch between a real-world network and the sampling procedure.
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We chose to focus our investigation on networks in veterinary medicine, specifically the
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spread of infectious diseases between animal holdings, because the use of network analysis
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is increasing in that field (Barthélemy et al., 2005; Ortiz-Pelaez et al., 2006).
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In veterinary medicine, network analysis and modelling can be employed to predict the
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spread of disease and epidemic size, and also to examine the effects of various intervention
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methods, such as vaccination, stand still, and stamping out. An example of this is a study
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performed by Corner et al. (2003), which was aimed at examining a network of wild
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brushtail possums with regard to transmission of the pathogen Mycobacterium bovis and
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social contacts between the animals. In another investigation, Kiss et al. (2006) analysed
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networks of sheep movements within Great Britain and found that, during an epidemic, the
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most efficient strategy is to concentrate control interventions on highly connected nodes.
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Despite the increased use of network analysis in epidemiology, there are shortcomings
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related to missing links and how to represent a structure when only a single sample is
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available. Collected network data are often incomplete (Christley et al., 2005; Clauset et al.,
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2008; Eames et al., 2009; Guimerà and Sales-Pardo, 2009; Heath et al., 2008; Ortiz-Pelaez
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et al., 2006), for instance, there can be missing animal movements or unknown locations of
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herds in databases. Accordingly, Perkins et al. (2009) demonstrated that network structures
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are only approximations of contacts and that it is almost impossible to identify all contacts
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when collecting data.
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Properties such as spread of disease can vary depending on the structure of the network
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(Keeling, 2005; Kiss et al., 2006, Newman et al., 2001; Shirley and Rushton, 2005). Thus,
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due to the relationship between disease transmission and network structure, results based on
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networks with missing links may be misleading. In practice, this means that there will be
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problems with missing data, which will lead to lost links in the representation of a network.
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The links are lost due to errors that occur during the sampling period or as a consequence of
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the finite length of the sampling period. Guimerà and Sales-Pardo (2009) introduced a
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method to use a single measure of a network, called a sampled network, to generate a more
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correct representation, i.e. an approximation of the real-world network. Those researchers
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focused on networks that were reduced during sampling, and by measuring and classifying
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the structure of the sampled network, they were able to identify either missing or spurious
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links. By comparison, our study was more general in nature and handled the relationship
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between link density and estimates of properties such as spread of disease and specific
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network measures. Our results can be combined with the findings of Guimerà and Sales-
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Pardo (2009) by stressing when to expect missing links.
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When conducting a survey to achieve a sampled network, it is important to consider the
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time window of the sampling period. For example, Kao et al. (2007) studied the
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relationship between the network of UK livestock movement and disease dynamics on
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different time scales. This was achieved by simulating transmission of two diseases, scrapie
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and foot-and-mouth disease, which differ greatly regarding the time scales of the incubation
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and infectious periods. Kao and colleagues concluded that, in order for network analysis to
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be a valuable tool in epidemiological modelling, it is important to consider the time scale as
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well as the potentially infectious contacts. In another study, Robinson et al. (2007)
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investigated animal movement networks evolving over time in Great Britain, and their
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findings point out the importance of temporal scale. With increasing length of the time
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period under consideration, the networks became progressively more connected, and in that
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way fuelled the spread of disease. Those authors also found a seasonal pattern with a peak
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in spring and August. Thus, depending on the question to be examined or when comparing
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different networks, it is important to choose the appropriate temporal scale (Vernon and
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Keeling, 2009). In the current study, we discuss this time window problem in relation to the
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difficulty of achieving a sufficient number of links during a selected period.
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2. METHODS
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2.1 The model
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Figure 2 illustrates the process of network generation and simulation in our study. The first
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step involved placement of animal holdings in the landscape, and the second the link-
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forming procedures, which in this case were related to empirical sampling. The third step
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comprised simulation runs of disease transmission. The network-generating algorithm,
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simulations, and calculations were implemented and run in MATLAB (version R2009a).
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2.1.1 Landscape of animal holdings
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The number of animal holdings was set to 500, and these entities were randomly placed in a
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landscape of size 34 x 34 (see figure 2). The holding density was chosen according to
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actual farm density in southern Sweden. Each animal holding was considered to be a node,
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which implies that each animal was not modelled individually.
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2.1.2 Link sampling and link density
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The animal holdings were connected by distance dependence (eq. 1; Håkansson et al.,
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2010; Lindström et al., 2008) between those entities (Dl) or completely at random (Rl).
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P(lij )  K exp  dij ab
(1)
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In the equation, P(lij) is the probability that a link is formed between nodes i and j, and di,j is
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the Euclidian distance between holdings i and j. Parameters a and b are set by the
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parameters kurtosis, к, and standard deviation, σ (see Lindström et al., 2008). The constant
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K normalized the distribution such that the probabilities of all possible links summed to
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one. For distance dependent link sampling, Dl, we used a kurtosis value of 10/3, meaning
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an exponential distribution and a standard deviation of one. The links were sampled
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randomly and successively from this probability distribution (eq. 1), until the desired link
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density was achieved. Since stochasticity was included in the method, it was also possible
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to sample links between holdings that were more distant from each other, even at a low link
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density. For random distribution of links, Rl, the links were sampled one at time, with the
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same probability for all links. To avoid edge effects, periodic boundaries were used
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(Lindström et al., 2008) along the edges of the 34 x 34 landscape.
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Link density, D, represents the actual connections, L, in a network as a proportion of all
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theoretical possible links in that network (Wasserman and Faust, 1994, eq. 2).
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D  2L nn 1
(2)
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In our study, n represented the number of animal holdings in the network. In the
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simulations, we varied link density between 0.001 and 1.0, and a density of 1.0 indicated a
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complete network (figure 1a) that included all theoretical connections. Inasmuch as the link
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density of the networks was set when generating the networks, the mean link degree was
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also given from the start (table 1).
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2.1.3 Disease transmission
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As in the link sampling process, we assumed two different processes for the transmission
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probabilities of a disease, one distance dependent, Dt, and the other completely random, Rt.
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The two processes could represent two diseases with different behaviours. Transmission
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rates were determined using the same processes as applied in the link sampling (see §2.1.2).
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Hence, Dt was set by equation 1 and the same parameter values as Dl, and the transmission
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probabilities of Rt were arbitrarily set to 0.01.
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2.1.4 Model scenarios
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Combining two link sampling processes and two disease transmission processes yielded
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four different scenarios that we designated DlDt, DlRt, RlDt and RlRt (figure 2), and these
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can be described as follows. The RlRt scenario is an example of a mass action mixing
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model (Keeling, 2005) that assumes that all links have the same probability of transmitting
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disease combined with a matching random procedure for link sampling. Matching in this
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context is considered in the sense of process but not necessarily with respect to the
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occurrence of events, i.e. two different realizations of the randomization from the same
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process. The DlDt, which comprises linking and transmission probabilities for each link, is
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a distance dependent scenario that involves matching between the process of probability of
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measure and probability of transmission. Considering the combinations in the remaining
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two scenarios, DlRt and RlDt, the link sampling procedure does not match the actual process
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that generates probability of transmission. For example, in RlDt, transmission is distance
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dependent and yet the link sampling procedure is random and hence expected to be
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ineffective. Accordingly, in this case, link sampling is random, which, regardless of
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distance, implies that some of the first connections detected will have low probabilities,
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while some that have high probability of transmission will not be detected within the
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sampling time frame.
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2.1.5 Simulation model
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To simulate disease transmission in the sampled networks, we used a general and very
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simple epidemiological model, where the holdings could be in either of two phases:
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susceptible (S) or infectious (I) (eq. 3).
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dS dt   SI
dI dt  SI
(3)
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Parameter λ in the equation is the probability of disease transmission from an infective
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holding through a link to a susceptible holding, and the variables S(t) and I(t) are the
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number of holdings in the susceptible and the infected phase, respectively, at time t. We did
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not incorporate incubation time, and hence animal holdings in contact with an infected
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holding were already able to infect other holdings during the next time step. Furthermore, a
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recovery phase was not included in the model, and thus an infected holding remained in the
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infectious phase during the remaining simulation time. Undirected links were used, and the
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disease could thus be transmitted in both directions along the links. Disease transmission
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could occur only between animal holdings that were connected by a link. It should be noted
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that the probability of a link in the sampled network was according to Rl or Dl, whereas the
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probability of transmission was according to Rt or Dt.
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2.2 Simulation runs
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For each link density presented in table 1, simulations were run separately for all four
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scenarios illustrated in figure 2. One hundred different networks were generated for each of
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the two link sampling processes, Dl and Rl, and each link density (figure 2). Also, for each
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density and link sampling procedure, 10 replicates of randomly distributed holdings were
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created, and, for each of these landscapes of holdings, 10 replicates of networks were made
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by using one of the two sampling processes (see §2.1.2). For each of these sampled
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networks, 10 simulations were performed per transmission process, Dt and Rt, by initiating
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the spread from a randomly chosen animal holding. In all, 1000 simulations were run per
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scenario and link density. Simulation period was set to 300 time steps, and numbers of
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infected animal holdings were calculated for each time step.
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2.3 Analysis
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To compare the different scenarios and prediction powers determined by link density, we
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analysed the extent of the spread of disease as the mean number of infected holdings per
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time step, and also, the mean number of time steps elapsed until a specified proportion of
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holdings was infected (here 10%, 50% and 90%). To characterize the networks and to
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ascertain how a change in link density would affect the structure and function of the
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networks, we used the following network measures: degree assortativity, clustering
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coefficient and fragmentation index.
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Degree assortativity (Newman, 2002) is a measure of to what extent nodes with equal
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respectively unequal degree are connected. Values range from minus one to one. A value
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near one indicates that a larger proportion of holdings with equal degree are linked to each
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other. Assortativity near minus one corresponds to a network where holdings with a
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different degree have a higher probability of being connected. A value of zero implies that
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the connections between holdings are not dependent on node degree.
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The clustering coefficient (Watts and Strogatz, 1998) for a holding is the number of links
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that exist between neighbours of that holding divided by all possible links between the
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neighbours. Here, we used the average clustering coefficient for the whole network; this
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measure ranges between zero and one, where one indicates that the network is highly
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clustered.
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The fragmentation index (Borgatti, 2003; Webb, 2005) measures to what extent a network
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is disconnected. This index ranges from zero to one; a low value indicates that the network
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is highly connected, and a high value means that the networks are very fragmented.
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3. RESULTS
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The results show that, for the scenario with distance dependent link sampling and disease
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transmission (DlDt), a link density of around 0.04 gave the same number of infected animal
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holdings as it did for networks with a larger proportion of connections (figure 3). Under the
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assumptions of our model, these findings suggest that such low proportions of links in the
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network were sufficient to examine the extent of the disease transmission. The scenario
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comprising random link sampling and distance dependent disease transmission (RlDt)
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required a higher link density until a limit was reached where additional links had no
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influence. For the scenarios involving random transmission (DlRt and RlRt), the number of
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infected animal holdings increased with increasing link density, and no limit was reached.
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Since the spread of disease is stochastic, we also studied the variation in different
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realizations. We found variation between realizations in both cases, that is, incomplete
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networks compared with complete networks with link density of 1.0. Hence, it is important
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to assess both the expected and the measured variation. Figure 4 shows the median values
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of the 1000 replicates of the simulations of the DlDt scenario plotted with the first and third
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quartile on each side on the median curve. For a link density as low as 0.001 (figure 4a), the
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median was one for the whole time period, because only in some cases was the disease
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transmitted to other holdings. When link density increased to 0.01 (figure 4b), the
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difference between the first and third quartiles also increased. The difference was small at
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the beginning of the simulation time when few holdings were infected but increased over
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the time period. Moreover, if link density increases further, up to 0.02 (figure 4c) and 0.03
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(figure 4d), the difference between the first and third quartiles decreases. For link densities
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from 0.04 and higher (figures 4e and 4f), the shape of the curves and the distances between
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them are almost the same, which implies that measured and expected stochastic processes
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generate equal variation between realizations. Of course, in the last part of the simulation
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time, when almost all holdings are infected, the variation between realizations decreases
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towards zero.
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The time until a given proportion of the holdings were infected differed depending on the
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link sampling scenario and the disease transmission scenario (figure 5). The random disease
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transmission scenarios (DlRt and RlRt) required almost the same length of time to reach a
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given proportion of infected animal holdings. In addition, they occurred at a much faster
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rate compared to the distance dependent disease transmission scenarios (DlDt and RlDt)
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(figure 3 and 5). Considering all the scenarios, the slowest transmission rate was found for
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RlDt, i.e. random link sampling and distance dependent transmission.
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Figure 6 shows a comparison of the four scenarios with regard to the number of infected
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holdings at a given link density. At low link densities, all methods gave different results.
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When link density increases, the two distance dependent disease transmission scenarios
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(DlDt and RlDt) approach each other. As well as the two random disease transmission
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scenarios (DlRt and RlRt) did. It can be seen that the higher the link density, the greater the
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similarity between the results for the different distance dependent disease transmission
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scenarios. As mentioned above, disease spread was much faster with random transmission
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than with distance dependent transmission.
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The average assortativity for the networks depended on the link creation method that was
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used (figure 7a). Distance dependent link creation led to higher values of assortativity
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compared to random link creation. As expected, the networks produced by random linking
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had assortativity close to zero at all link densities.
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The average clustering coefficient for all networks increased with increasing link density
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(figure 7b). The clustering coefficients for the networks generated by distance dependent
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link sampling were higher than the values for the networks made by random link creation.
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When link density was increased, the random link sampling approached the distance
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dependent link sampling. The networks generated by the random link sampling gave
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clustering coefficients that were equal to the link density in question. Of course, the
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clustering coefficient was one for all networks when the link density was one, and all
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animal holdings were connected to each other.
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For both link sampling scenarios, the fragmentation index for the networks was close to one
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when link density was 0.001 (table 2). When we increased link density to 0.01, the
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fragmentation index decreased dramatically. In both link sampling scenarios, the index
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reached zero when link density was 0.03 or higher.
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4. DISCUSSION
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Our aim was to study the effects of using a disease transmission network with missing links
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to predict the spread of disease. We investigated whether it is possible to predict anything
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about the size of such dissemination using only a proportion of all theoretically possible
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links. According to the results, a link density of 0.04 gave the same mean number of
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infected animal holdings as a higher link density when spread of disease was simulated in a
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scenario in which both the probability of identifying a link, and disease transmission, was
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distance dependent (the DlDt scenario). Also, the variation between different realizations of
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disease spread converged to expected variation at link density 0.04. When considering
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distance dependent disease transmission and random link sampling (RlDt), as expected, the
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numbers of infected animal holdings reached the same level as in the DlDt scenario.
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Although most of the links were needed to attain that rate since the less probable (longer
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distance) links will be included when using random sampling than with distance dependent
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sampling. For random disease transmission (scenario DlRt and RlRt), the number of infected
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holdings increased with increased link density, which implies that a much higher link
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density is required to reach relevant approximations of spread of disease. The discussion
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below addresses the implications of our results in relation to sampling procedures and the
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effects of using networks with missing links.
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Studies using empirical data have shown that only a small fraction of all possible
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connections in a network actually occurs (Webb, 2006; Eames et al., 2009). When sampling
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data, it is almost impossible to trace all connections between nodes, even if the number is
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small, and this often leads to incomplete data sets. Therefore, it is important to consider link
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density, or mean link degree, when modelling networks. If simulations in a scenario DlDt
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network with a link density of 0.04 or higher are compared with simulations in a complete
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network, both will result in the same mean number of infected holdings and the same
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variation in that mean. This implies that a link density of 0.04 is sufficient and further
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sampling is unnecessary. Another important issue to consider when using empirically
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sampled networks is the time window for the sampling period. Using an “incorrect” time
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window can lead to missing links or unnecessary sampling. The period chosen has an
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impact on how complete the network will be: a longer time window can result in a more
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connected network compared to one that is based on a very short time window. The lengths
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of the time windows used in different studies have varied. For example, Kiss et al. (2006)
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chose a four-week time scale in their investigation of sheep movements in Great Britain,
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and Robinson and Christley (2007) used periods of 10 weeks to analyse animal transports.
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Such studies may indeed provide a very good description of a network during the actual
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time period under consideration, but that information may not be suitable for making
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predictions.
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Combining our results with the time frame of a study can help emphasize the problem and
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focus on number of links that are actually measured. By definition, a shorter period will
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result in fewer measured links, but the question is whether such a sample can suffice to test
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for spread of disease over a period that is longer than the one that is actually measured.
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Obviously, in the DlDt scenario, a link density of 0.04 is a guarantee for correct
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measurement of disease transmission during any given time period. However, at a link
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density below 0.02, our results indicate that measuring disease dissemination will be
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erroneous even during short periods. By comparison, a density of 0.03 may hold until a
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period comprising 50–100 time steps is reached, i.e. when the 0.03 curve diverges from the
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curves for higher densities (see figure 3) and there is a large overlap in variation between
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realizations (figure 4). These conclusions are true only when considering a perfect link
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sampling procedure such as in the DlDt scenario. On the other hand, if the sample
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procedure is not that perfect, even more links must be included. Our RlDt scenario
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represents the opposite extreme, i.e. a complete random link sampling procedure that is in
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no way related to the probability of contacts. In such a case, almost all links have to be
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sampled, even to make estimations during short time periods. In real life, a link sampling
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procedure fall somewhere in between these two extremes. Considering any time period or
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sampling procedure, it is not recommendable to base analysis on link densities below 0.02.
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Of course, this conclusion applies to our setup: how we modelled the spread of disease,
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what distance dependence we used (eq. 1), the number and spatial configuration of the
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holdings. Still our results show that to achieve reliable measurements, it may be necessary
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to include a higher link density than expected. Link density is a measure that very much
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depends on the node density; the more nodes the less link density will suffices. Yet, link
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density is a relevant measure when determining how large proportion of all links is needed
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to get a fair estimate of disease spread. On the other hand, mean degree is a more general
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measure and our study show that at least a mean degree of 10 links is required. Our method
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includes periodic boundaries which expel boundary effects and hence our results applies for
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larger, to infinite, sets of nodes given that the configuration of nodes/animal holdings are
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realized by our set up of 500 randomly distributed nodes. Hence, a mean degree of 10 links
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should hold for larger sets of randomly distributed animal holdings although link density
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consequently will decrease. Furthermore, our methodology can be applied to any specific
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system, other spatial configurations, latency periods etc, to assess the necessary level of
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link density. The link density can be achieved by making a single measurement over a
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sufficiently long period of time or by conducting repetitive sampling over shorter time.
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Empirical investigations of networks have shown that link densities is often very small, i.e.
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merely a parts per thousand or a few per cent of the total number of theoretical connections
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in the networks. An example of this is a study by Ortiz-Pelaez et al. (2006), which was
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conducted to analyse animal movements during the initial phase of an epidemic of foot-
422
and-mouth disease that occurred in Great Britain in 2001. The network in that investigation
423
had a mean link degree of 1.22, which corresponds to a link density that is as low as about
424
0.0019. A low link density was also found in the Swedish animal transport network
425
(Nöremark et al., submitted). It is important to remember that the measured contacts in an
426
empirical network are simply subsets of realizations of all possible contacts, which means
427
that the number of links in such a network is in fact a subset of the links that have been
428
realized at the time of data collection. Actually, there are probabilities for a huge number of
429
additional connections, but they are not even realized during the chosen time period of a
430
network study. For instance, when a link density of 0.01 was used in our investigation, all
19
431
theoretical connections were possible but only 1% of them were realized, and those may
432
have differed between the replicates. When modelling virtual networks, it is also important
433
to consider the link density and mean degree. Kiss et al. (2005) performed epidemiological
434
modelling using virtual networks with mean degrees varying between 5 and 20. In their
435
results, there is an indication that a mean degree of 15-20 was enough to estimate final
436
epidemic size. Besides the investigation by Kiss et al. (2005) few other network studies
437
have focused on link densities and missing links. When comparing either different
438
theoretical studies or theoretical results with empirical investigations one has to use
439
relevant measures. In general, our study indicates that a mean link degree of at least 10
440
links is required and in empirical studies do have too few links in their estimates. The
441
results of Kiss et al. (2005) also support our findings.
442
443
Diseases will be spread faster by a network with randomly distributed links than by
444
clustered networks (Kiss et al., 2005; Watts and Strogatz, 1998). We generated such
445
random networks in our study when we applied random transmission probabilities (RlRt
446
and DlRt), and RlRt represents the full random scenario with rapid spread of disease. For a
447
scenario such as DlRt, with random transmissions and distance dependent link sampling, the
448
transmission rate is slightly slower at any given link density. The link sampling procedure
449
of DlRt erroneously assumes density-dependent contact, and yet the contact structure is
450
random. In a case like this, the rate of the real network, i.e. with random transmission
451
probabilities, is higher than the rate of the sampled network, since the link sampling
452
procedure will miss some important long distance links. Hence, in this mismatch between
453
sampling and transmission probabilities, even higher link density is necessary when
20
454
sampling to reach the correct levels of spread of disease (compare RlRt and DlRt).
455
Lindström et al. (2009) have shown that the spatial kernel explaining the distance
456
dependence of contacts between holdings due to transport is a mix of distance
457
independence, mass action mixing, and distance dependence. In our setup, the mass action
458
mixing is represented by Rt, and, once again, the reality is somewhere in between these two
459
extremes, the DlRt and the DlDt scenario. Consequently, our results for the RlRt and DlRt
460
scenarios imply that the link density levels of 0.03 and 0.04 that were found can be
461
expected to be too low, since the mass action component in contact structures creates even
462
higher demands on link density.
463
464
It is recognized that random networks have a low level of clustering compared to other
465
kinds of networks, such as small-world networks (Shirley and Rushton, 2005; Watts and
466
Strogatz, 1998). We measured the clustering coefficient for each of our networks, and, as
467
expected, found lower values for those generated by random sampling than for those
468
generated by distance dependent link sampling. The degree of fragmentation of a network
469
influenced the extent to which diseases could spread between the holdings. Fragmentation
470
index is a measure of the extent of disconnection of networks, and, in our study, only the
471
networks with link density below 0.03 resulted in disconnection. Link densities of 0.03 or
472
higher gave rise to connected graphs, indicating that it is possible for a disease to spread
473
between all animal holdings in these networks. Since we know that a link density of 0.03
474
corresponds to a mean link degree of almost 7.5, the values of the fragmentation index
475
seem reasonable. It is plausible that a disconnection in a network would reduce the spread
476
of the disease immensely, and hence any disconnection that is apparent after a link
21
477
sampling procedure should be scrutinized. If a disconnection is the result of a specific
478
realization and thus is not necessarily the same in any other realizations (i.e. new time
479
period), this will jeopardize any conclusions drawn from the study. This is evident
480
considering the observed variation, in our results, in the rate of spread for different link
481
densities (figure 4), which emphasizes the difference between a network that represents one
482
specific time period with all its measures and a network that can be used to predict and
483
estimate rate for any given time.
484
485
We were interested in determining how many animal holdings that could become infected
486
and the rate of disease transmission, and thus incubation time was not included in our
487
model. This is a simplification, because diseases differ with respect to incubation time,
488
which can vary from only a few days to as long as a number of years. However, our model
489
can easily be extended to encompass a more complex disease context by including a
490
recovery phase and incubation time. We calculated the number of infected animal holdings
491
as a measure of the spread of disease. In practice, this might not be particularly relevant,
492
because it is not desirable to allow disease transmission to proceed for such a long time.
493
Obviously, it would be preferable to adopt control strategies as soon as possible after
494
identifying an infection. Notwithstanding, the findings of our study do have implications
495
regarding what link density ought to be achieved when testing different strategies.
496
497
4.1 Conclusions
498
Our results indicate that to estimate network properties such as spread of disease, it might
499
be necessary to construct link sampling procedures that yield high link densities. More
22
500
specifically, our scenarios based on Swedish farms show that, if the sampling procedure is
501
ideal a density of 0.02 (mean degree of 5) can suffice to estimate disease transmission over
502
shorter time periods, whereas 0.04 (mean degree of 10) is required for longer periods.
503
Nevertheless, in reality, link sampling procedures are not perfect, and some mass-action
504
mixing component can be expected in the contacts between holdings. Our results
505
demonstrate that these two components of reality enforce an even higher level of link
506
density and thereby represent a relevant measure of spread of disease.
507
508
509
ACKNOWLEDGEMENTS
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We would like to thank the Swedish Civil Contingencies Agency (MSB) for funding this
512
project. We also like to thank Patricia Ödman for revising the English.
513
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TABLE CAPTIONS
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Table 1. Link densities used in simulations and the corresponding mean link degree for the
628
networks.
629
630
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Table 2. Fragmentation index according to link density and the link sampling method used.
632
29
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FIGURE CAPTIONS
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Figure 1. Network categories: (a) complete network, (b) real-world network, (c) sampled network.
636
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Figure 2. Model flow chart. Flow chart showing relationships between the different components of
638
the model.
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Figure 3. Mean number of infected holdings per time step in the four linking and disease
641
transmission scenarios. Disease transmission was distance dependent in scenarios DlDt (a) and RlDt
642
(b) but random in DlRt (c) and RlRt (d). Also, distance dependent link creation was applied in DlDt
643
(a) and DlRt (c), whereas links were generated randomly in RlDt (b) and RlRt (d). The link densities
644
were as follows: 0.001 (---), 0.005 (…), 0.01 (--.--), 0.02 (__), 0.03 (-○-), 0.04 (-*-), 0.05 (-□-), 0.1
645
(-♦-), 0.25 (-◦-), 0.5 (-▼-), 0.75 (-x-) and 1.0 (-+-). Corresponding mean link degrees can be found
646
in table 1.
647
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Figure 4. The median values of the 1000 replicates of the simulations of the DlDt scenario plotted
649
with the first and third quartile on each side on the median curve. The solid line shows the median
650
number of infected holdings per time step and the dashed lines represent the first and third quartiles
651
of the replicates. Link densities: (a) 0.001, (b) 0.01, (c) 0.02, (d) 0.03, (e) 0.04, (f) 1.0. Note that the
652
scales of the y-axes differ in (a) and (b). Corresponding mean link degrees can be found in table 1.
653
654
Figure 5. Number of time steps passed before 10% (a), 50% (b) and 90% (c) of all holdings in the
655
network were infected. The time depended on which of the four scenarios was used. The scenarios
656
are designated as follows: dashed line, DlDt ; dotted line, RlDt; solid line, DlRt; dash-dot line, RlRt.
30
657
For scenario RlDt, the number of infected holdings did not reach any of the given proportions during
658
the simulation time.
659
660
Figure 6. Mean number of infected holdings per time step for a given link density and the four
661
scenarios, designated as follows: dashed line, DlDt ; dotted line, RlDt; solid line, DlRt; dashed-dotted
662
line, RlRt. Link densities: (a) 0.001, (b) 0.01, (c) 0.03, (d) 0.05, (e) 0.07, (f) 0.1, (g) 0.5, (h) 1.0.
663
Note that the scales of the y-axes differ in (a). Corresponding mean link degrees can be found in
664
table 1.
665
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Figure 7. Average assortativity (a) and clustering coefficient (b) illustrated for the networks
667
according to the connections of the holdings. Distance dependent linking is indicated by a dashed
668
line and random linking by a solid line.
31