Epidemic phase transition of the SIS type in networks

By making only one approximation of a mean-“eld type, we determine the nature of the SIS type of epidemic phase transition in any network: the steady-state fraction of infected nodes y∞ is linear in (� Š1 c Š� Š1 ) for effective infection rates �↓�c, the derivative of y∞ at the epidemic thresholdc= 1 �1 is exactly computed and depends on the largest eigenvalue �1 of the adjacency matrix and on the “rst- and third-order moments of the corresponding eigenvector. Since coupled oscillators in a network synchonize at a coupling strength proportional to 1 �1 ,a similar characterization of the phase transition is suggested. The behavior of y∞ aroundc was the missing part in the general steady-state theory of a SIS-type epidemic ona network. Copyright c �EPLA, 2012