Global Stability of Virus Spreading in Complex Heterogeneous Networks

Various networks are possessed of an obvious heterogeneity in the connectivity properties, and it is of practical significance to study epidemic spreading in networks of this kind. Pastor-Satorras and Vespignani established the dynamical mean-field reaction rate equations for the spreading of infections in complex heterogeneous networks based on the well-known SIS model, and figured out an epidemic threshold λc such that if λ (effective spreading rate) is above λc, the infection spreads and becomes endemic. The significance of this result is far-reaching; however, the authors have not found a strict mathematical proof of their conclusion in the literature. In this paper, we approach this problem by proving that if λ is above λc, the infection spreads and approaches the unique positive stationary point of the reaction rate equations as long as there exist infected nodes in the network initially; i.e., the virus infection process is globally stable.