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Phys Rev Lett. 2010 Nov 19;105(21):218701. Epub 2010 Nov 17.

Thresholds for epidemic spreading in networks.

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Istituto dei Sistemi Complessi (CNR-ISC), UOS Sapienza and Dip. di Fisica, Sapienza Università di Roma, P.le A. Moro 2, I-00185 Roma, Italy.


We study the threshold of epidemic models in quenched networks with degree distribution given by a power-law. For the susceptible-infected-susceptible model the activity threshold λ(c) vanishes in the large size limit on any network whose maximum degree k(max) diverges with the system size, at odds with heterogeneous mean-field (HMF) theory. The vanishing of the threshold has nothing to do with the scale-free nature of the network but stems instead from the largest hub in the system being active for any spreading rate λ>1/√k(max) and playing the role of a self-sustained source that spreads the infection to the rest of the system. The susceptible-infected-removed model displays instead agreement with HMF theory and a finite threshold for scale-rich networks. We conjecture that on quenched scale-rich networks the threshold of generic epidemic models is vanishing or finite depending on the presence or absence of a steady state.

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