Probing empirical contact networks by simulation of spreading dynamics

Disease, opinions, ideas, gossip, etc. all spread on social networks. How these networks are connected (the network structure) influences the dynamics of the spreading processes. By investigating these relationships one gains understanding both of the spreading itself and the structure and function of the contact network. In this chapter, we will summarize the recent literature using simulation of spreading processes on top of empirical contact data. We will mostly focus on disease simulations on temporal proximity networks -- networks recording who is close to whom, at what time -- but also cover other types of networks and spreading processes. We analyze 29 empirical networks to illustrate the methods

Study of the Anatomical Association between Morton’s Neuroma and the Space Inferior to the Deep Transverse Metatarsal Ligament Using Ultrasound

Morton’s neuroma (MN) is a common condition in clinical practice. The compressive etiology is the most accepted, in which compression occurs in the tunnel formed by the adjacent metatarsals, the deep transverse metatarsal ligament (DTML) and the plantar skin. Ultrasound (US) is a reliable method of study. The presence of insufficient space under the DTML may be related to the appearance of MN. Objectives: To verify the relationship between MN and the space under the DTML between the metatarsal heads of the third (M3) and the fourth (M4) metatarsals using US. Methods: This is a cross-sectional epidemiological study. The research study using the ultrasound (US) technique was carried out on 200 feet belonging to 100 patients aged 18 to 65 of both sexes, with a control group formed by 62 patients and a study group formed by 38 patients diagnosed with MN. Results: The presence of MN and the factors associated with it were studied in 100 patients using ultrasound (US). The assessment and comparison with US of the space inferior to the DTML between M3 and M4 in control groups and patients with MN show that patients with MN have a smaller size in the variable “h” (height or distance DTML-plantar skin), in the variable “b” (base or intermetatarsal distance M3 and M4) and in the variable “s” (surface of the parallelogram “h” × “b”). The predictors of MN are a decrease in dimension “b” and an increase in weight. Sitting in an office chair and the use of a bicycle, due to equinus, have an influence on the space below the DTML, reducing it and promoting the appearance of MN. Conclusions: The two US measurements (“h” and “b”) in the space below the DTML are smaller in patients with MN than in the asymptomatic group. A shorter distance between M3 and M4, and an increase in BMI are predictors of MN

Comparison between the dynamic out-component and static measures.

<p>The relative size of the intersection of the top nodes is based on their value of the dynamic out-component and on static measures of centrality. In the upper panel the comparison for a fixed infectious period of days is shown and in the lower one for a fixed infectious period of days.</p

On the Robustness of In- and Out-Components in a Temporal Network

<div><h3>Background</h3><p>Many networks exhibit time-dependent topologies, where an edge only exists during a certain period of time. The first measurements of such networks are very recent so that a profound theoretical understanding is still lacking. In this work, we focus on the propagation properties of infectious diseases in time-dependent networks. In particular, we analyze a dataset containing livestock trade movements. The corresponding networks are known to be a major route for the spread of animal diseases. In this context chronology is crucial. A disease can only spread if the temporal sequence of trade contacts forms a chain of causality. Therefore, the identification of relevant nodes under time-varying network topologies is of great interest for the implementation of counteractions.</p> <h3>Methodology/Findings</h3><p>We find that a time-aggregated approach might fail to identify epidemiologically relevant nodes. Hence, we explore the adaptability of the concept of centrality of nodes to temporal networks using a data-driven approach on the example of animal trade. We utilize the size of the in- and out-component of nodes as centrality measures. Both measures are refined to gain full awareness of the time-dependent topology and finite infectious periods. We show that the size of the components exhibit strong temporal heterogeneities. In particular, we find that the size of the components is overestimated in time-aggregated networks. For disease control, however, a risk assessment independent of time and specific disease properties is usually favored. We therefore explore the disease parameter range, in which a time-independent identification of central nodes remains possible.</p> <h3>Conclusions</h3><p>We find a ranking of nodes according to their component sizes reasonably stable for a wide range of infectious periods. Samples based on this ranking are robust enough against varying disease parameters and hence are promising tools for disease control.</p> </div

Distribution of for an exemplary infectious period of days.

<p>Panel A shows the size of the out-component for an exemplary node as a function of . For many times , the size of the out-component has similar values close to , but for some we also find to vanish. Panel B shows the distribution for all nodes, i.e. the top view of panel A for all nodes of the network. Each horizontal line represents one node, the example node chosen for panel A is indicated by the dotted orange line. For the sake of clarity, only every 100th node is plotted. Nodes are arranged according to their averaged value of over all from top to bottom, i.e. the node with the largest averaged out-component is displayed as the top line of the panel.</p

Robustness of samples based on the out-component size of nodes.

<p>Shown are the mean intersections and for three different sample sizes (red), (blue), and (green) of the network representing approximately , , or nodes, respectively. The sampling is calculated based on the mean largest out-component over all and (see text for details). is based on averaging over all pairs with or respectively. Confidence intervals are given by the shaded areas.</p

Outbreak probabilities (A) and out-component sizes (B) for different infectious periods .

<p>Panel A: Outbreak probability as given by the fraction of primary infections causing at least one secondary infection. The dashed line shows the outbreak probability of the time-aggregated network, i.e. the fraction of nodes with non-vanishing out-degree. Panel B: Average out-components of primary infections, i.e. the <i>number</i> of follow-up infections. The 50% confidence interval is indicated by the shaded area. Only for a significant fraction of the network can be infected. For increasing , both values approach a saturation. For days, approximately every second primary infection will cause follow-up infections which will reach on average of the network. Both numbers are significantly lower than their counterparts in the static network, as indicated by the dashed line. Here approximately of all primary infections cause follow-up infections with a mean size of epidemic of almost of the network.</p

Ranking of nodes according to their mean out-component size .

<p>Each curve corresponds to one node. The top hundred nodes with the largest out-components are shown. Curves representing nodes with higher ranking are darker than those with lower rankings. For illustration purposes an arbitrarily chosen node is displayed in red.</p