Ranking of influential spreaders at early times from the geographic spreading centrality (GSC).

<p>The GSC metric predictions are in quantitative agreement with the results from the Monte Carlo study on the empirical model.</p

Late-time spreading ability of different airports, measured by the global attack of an SIR epidemic that originates at each airport.

<p>(<i>a</i>) Global attack as a function of reproductive number, for five different airports (see inset). We initialize the disease by infecting 10 randomly chosen individuals inside the subpopulation of consideration. We use days. Each point is the result of a Monte Carlo study averaging over 200 reaction and 20 mobility realizations and using individuals. (<i>b</i>) Ranking of the 40 major airports in US in terms of their spreading ability measured by the normalized global attack. We compare the normalized global-attack ranking curve (black diamonds) to the ones that result from considering the airport’s normalized degree (magenta squares) and the airport’s normalized traffic (brown triangles). Also shown is the ranking of the airports shown in (<i>a</i>). Both degree and traffic provide effective rankings of influential late-time spreaders, which in this case can be understood from the good cross-correlation between the two (inset).</p

Monte Carlo study of the global attack of an epidemic as a function of the reproductive number , for the different models explained in the text.

<p>We used a value of the recovery rate days. We initialized the epidemic with 10 infected individuals chosen randomly across the network. We used a population of individuals, and average our results over 200 realizations. (Inset) The global attack for larger values of exhibits smaller differences among models, except for those between annealed and quenched transition rates at the nodes, as evidenced by the simulation results of Model 1 vs. the other models.</p

Ranking of influential early-time spreaders by existing metrics.

<p>Shown are the results from the model simulations (black triangles), and comparison with the ranking provided by existing metrics of centrality and late-time influential spreading. (<i>a</i>) Normalized degree. (<i>b</i>) Normalized traffic. (<i>c</i>) Normalized betweenness centrality. (<i>d</i>) Normalized -shell centrality.</p

Ranking of influential spreaders by the normalized early-time mean square displacement of infectious individuals.

<p>We initialize the disease by infecting 10 individuals from each specific airport (see inset), and use days. Each point is the result of a Monte Carlo study averaging over 100 reaction and 20 mobility realizations and using individuals. (Inset) Graphical representation of the mean position of infected individuals, 10 days after the outbreak from three different locations. The circle radius denotes the geographic extension of the infectious cloud (as measured by the square root of the Mean Square Displacement <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040961#pone.0040961-Nicolaides1" target="_blank">[12]</a> of infected individuals) while their color represents the number of infected at the same time (dark colors denote large number of infected).</p

Ranking of influential spreaders by the normalized early-time Total Square Displacement.

<p>(<i>a</i>) for different reproductive numbers, 10 days after the disease is initiated. (<i>b</i>) at different times after the initiation of the disease. We use and days. Each point in the above plots is the result of a Monte Carlo study averaging over 100 reaction and 20 mobility realizations and using individuals.</p

Role of spatial organization, traffic quenched disorder, and mobility patterns, on early-time spreading.

<p>(<i>a</i>) Shown is the TSD-ranking for individual realizations of two null networks testing the influence of (1) geographic locations of the nodes, and (2) heterogeneity in the traffic of the links. The dissimilarity between those rankings and that from the original network model strongly suggests that any effective measure of influential early-time spreaders must incorporate geography and traffic quenched disorder. (<i>b</i>) TSD-ranking for a simplified model of human mobility. Removing the detailed patterns of mobility affects the evolution of the predicted TSD (see inset for HNL airport) but does not affect the early-time spreading ranking significantly.</p