Fig. 5.
Evolving flow patterns in epidemic spread. We used the susceptible-infected-recovered (SIR) model to track the spread of disease in a weighted scale-free network and measured the flow through all nodes. The system exhibits evolving flow patterns: a vs. S i at t = 0 exhibits a positive scaling, representing degree-driven flow (red). b At later times the role of the hubs gradually diminishes and begins to decay in the limit of large S i, a lack of scaling resembling homogeneous flow (green). c As the system approaches the pandemic state (large t) begins to sharply decrease with S i, entering a strongly degree-averting flow regime (blue). d Susceptibility vs. t of a hub node (black) and a low degree node (gray). The hubs become infected (non susceptible) at earlier times, and hence cease to contribute to the spread—leading to the transition from degree-driven (red) to degree-averting (blue) flow patterns. e The flow through the empirical weighted international air-traffic network (nodes—international airports; edges—volume of human travel on route) under SIR, as represented by node size at t = 0, namely at the start of the outbreak. f vs. S i for the aviation network at t = 0. The positive scaling confirms the degree driven flow. g, h At a later time we find, on the same network, a different flow pattern, in which the flow through the hubs begins to decline. i, j Finally, for large t the flow enters the degree averting regime, as strongly avoids the hubs. In j we show also the flow curve obtained at t = 3 (green watermark) for comparison. Indeed for t = 10 (blue) we observe a much stronger decline in hub-flow than that observed at t = 3, demonstrating the gradual evolution towards degree-averting flow. These evolving flow patterns illustrate the non-trivial mapping of the static topology to the observed dynamic behavior. See detailed description in Supplementary Note 5. Error bars represent 95% confidence intervals (Supplementary Note 3)