Abstract. In this paper, we investigate the epidemic spreading for SIR model in weighted scale-free networks with nonlinear infectivity, where the transmission rate in our analytical model is weighted. Concretely, we introduce the infectivity exponent α and the weight exponent β into the analytical SIR model, then examine the combination effects of α and β on the epidemic threshold and phase transition. We show that one can adjust the values of α and β to rebuild the epidemic threshold to a finite value, and it is observed that the steady epidemic prevalence R grows in an exponential form in the early stage, then follows hierarchical dynamics. Furthermore, we find α is more sensitive than β in the transformation of the epidemic threshold and epidemic prevalence, which might deliver some useful information or new insights in the epidemic spreading and the correlative immunization schemes.

Many empirical studies reveal that the weights and community structure are ubiquitous in various natural and artificial networks. In this paper, based on the SI disease model, we investigate the epidemic spreading in weighted scale-free networks with community structure. Two exponents, α and β, are introduced to weight the internal edges and external edges, respectively; and a tunable probability parameter q is also introduced to adjust the strength of community structure. We find the external weighting exponent β plays a much more important role in slackening the epidemic spreading and reducing the danger brought by the epidemic than the internal weighting exponent α. Moreover, a novel result we find is that the strong community structure is no longer helpful for slackening the danger brought by the epidemic in the weighted cases. In addition, we show the hierarchical dynamics of the epidemic spreading in the weighted scale-free networks with communities which is also displayed in the famous BA scale-free networks.

Recently, the study of dynamical behaviors of the susceptible-infected (SI) disease model in complex networks, especially in Barabási-Albert (BA) scalefree networks, has attracted much attention. Although some interesting phenomena have been observed, the formative reasons for those particular dynamical behaviors are still not well understood, despite the speculation that topological properties (for example the degree distribution) have a strong impact on epidemic spreading. In this paper, we study the evolution behaviors of epidemic spreading on a class of scale-free networks sharing identical degree sequence, and observe significantly different evolution behaviors in the whole family of networks. We show that the power-law degree distribution does not suffice to characterize the dynamical behaviors of disease diffusion on scale-free networks.