Dynamical patterns of cattle trade movements

Abstract

Despite their importance for the spread of zoonotic diseases, our understanding of the dynamical aspects characterizing the movements of farmed animal populations remains limited as these systems are traditionally studied as static objects and through simplified approximations. By leveraging on the network science approach, here we are able for the first time to fully analyze the longitudinal dataset of Italian cattle movements that reports the mobility of individual animals among farms on a daily basis. The complexity and inter-relations between topology, function and dynamical nature of the system are characterized at different spatial and time resolutions, in order to uncover patterns and vulnerabilities fundamental for the definition of targeted prevention and control measures for zoonotic diseases. Results show how the stationarity of statistical distributions coexists with a strong and non-trivial evolutionary dynamics at the node and link levels, on all timescales. Traditional static views of the displacement network hide important patterns of structural changes affecting nodes' centrality and farms' spreading potential, thus limiting the efficiency of interventions based on partial longitudinal information. By fully taking into account the longitudinal dimension, we propose a novel definition of dynamical motifs that is able to uncover the presence of a temporal arrow describing the evolution of the system and the causality patterns of its displacements, shedding light on mechanisms that may play a crucial role in the definition of preventive actions.

Figure 1. Degree distributions for networks aggregated on different timescales .

Since a single value of (for days) yields multiple snapshots, each panel shows one distribution obtained for a given snapshot (circles) superimposed to a subset of the distributions obtained for the other snapshots at the same value of (grey lines). Panels A to D report the distributions of the in-degree , that show very large fluctuations and a power-law like behavior with exponent close to in all cases. Panels E to H present the distributions of the out-degree , characterized by a cut-off that strongly depends on the length of the aggregating time window.

Figure 2. Strength distributions for networks aggregated on different timescales .

Panels A to D report the distributions of the in-strength . Interestingly, the definition used to weight the links does not affect the distribution of the incoming traffic: the distributions and are very close. Panels E to H present the distributions of the out-strength , whose behavior instead depends strongly on the type of weight considered. Broader tails are observed when considering the total number of animals displaced out of a given holding. The same representation of Figure 1 is adopted, with symbols representing the result of a particular snapshot, and grey lines the results obtained for a subset of the other snapshots.

Figure 3. Relation between the number of bovine traffic movements of a holding and its number of connections for different values of .

Panels A to D report the average in-strength of nodes with a given value of in-degree, whereas panels E to H present the average out-strength of nodes with given out-degree. The same representation of Figure 1 is adopted, with symbols representing the result of a particular snapshot, and grey lines the results obtained for a subset of the other snapshots.

Figure 4. Probability distributions of the time interval between two consecutive displacements of a bovine.

corresponds to the time during which the bovine stays at given premises. The seasonality behavior of breeding is clearly shown by the peaks at 3 and 6 months, while at shorter times the distribution behaves as . The global distribution is a convolution of the time distributions obtained for different farm types, shown in panels B to H.

Figure 5. Neighborhoods of a selected node in three consecutive monthly networks.

The subgraphs are obtained by showing all nodes within distance 3 from a selected node (in red in the figure), for consecutive monthly snapshots. The visualization highlights how the neighborhood of a given node may strongly change its structure in time. It is important to note that nodes that disappear from the plots may still be present in the network, but are not shown as they may be at distance larger than 3 from the seed, thus not belonging to its neighborhood.

Figure 6. Probability distributions of the duration of activity and of the duration of inactivity of nodes and links.

Results are reported for daily (panels A to D) and weekly (panels E to H) networks. In the daily case, weekend breaks are neglected as they are characterized by a much lower activity and clear weekly patterns (see Figure S3). The observed peaks in of the daily networks correspond to inactivity periods of multiples of a week.

Figure 7. Fraction of appearing/disappearing links as a function of the weight associated to the link.

The weight considered here counts the number of animals, . Results for daily, weekly, and monthly networks are shown (panels A, B, C, respectively). As a reference, the weight distribution is also shown with a grey histogram.

Figure 8. Distributions of the growth rates of the number of bovines displaced along a connection.

The solid line represents the distribution of the growth rates considering all networks of a given aggregating time window . Symbols corresponds a selection of snapshots.

Figure 9. Fluctuations of the total outgoing traffic of bovines of a given holding for various aggregating time windows.

The plot shows, for each holding of the system, the fluctuations of the values of assumed by each node during all snapshots of the under study. The median (black dots) and the 95% confidence interval (brown shaded area) of outgoing traffic are shown.

Figure 10. Evolution of monthly network backbones.

Top: Distributions of the growth rates of the weights of the backbone links, where the network backbone is obtained under different filtering procedures. In each case, growth rates r are measured only for links that are present in two successive backbones. Center and Bottom: Overlap between the backbones of monthly networks. The overlap measures the number of links common to the pair of networks under consideration, normalized by their total number of links. Backbones are obtained either with a global threshold filter (center row) or using a disparity filter (bottom row). Three values of the significance parameter are considered.

Figure 11. Percolation analysis on consecutive monthly networks.

Two consecutive monthly snapshots ( and ) have been considered. A list of nodes with decreasing degree is calculated on the snapshot , and is applied as a removal strategy for both networks. The same procedure has been performed on the corresponding network backbones obtained for two values of the significance parameter .

Figure 12. Motifs: schematic representation and their occurrence.

A schematic example of the dynamics of a subset of the mobility networks is shown in panel A through three successive snapshots. The connections are color-coded according to the time at which they are active. A temporal motif is a temporal sequence of links such that the destination node of a link at time is the origin of another link at time . Two examples of motifs, of respective lengths 2 and 3, are shown below. We restrict the present study to the case of day. Panel B shows the results on the presence of motifs, analyzed by counting the number of occurrences during the timeframe under study. The longer the motifs, the smaller the number of times they appear. By focusing only on the set of motifs that occur at least twice, panel C compares the size of this set (expressed as a fraction of the total) obtained from the empirical dataset with the sizes obtained through various randomization procedures (see main text). The results are shown as functions of the motifs length. In panel D the median and confidence intervals of the number of motifs passing through a farm depending on the farms type are shown, together with the same computation for a null model in which the farm types are reshuffled at random.