Infection dynamics on scale-free networks.

We discuss properties of infection processes on scale-free networks, relating them to the node-connectivity distribution that characterizes the network. Considering the epidemiologically important case of a disease that confers permanent immunity upon recovery, we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual's degree of connectivity within the population. As in an earlier study [R. Pastor-Satorras and A. Vesipignani, Phys. Rev. Lett. 86, 3200 (2001); Phys. Rev. E. 63, 006117 (2001)] for an infection that did not confer immunity upon recovery, the epidemic process--in contrast with many traditional epidemiological models--does not exhibit threshold behavior, and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale-free network. Finally, we discuss effects that arise from finite population sizes, showing that networks of finite size do exhibit threshold effects: infections cannot spread for arbitrarily low transmission probabilities.