  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 $\alpha$ and the weight exponent $\beta$ into the
analytical SIR model, then examine the combination effects of $\alpha$ and
$\beta$ on the epidemic threshold and phase transition. We show that one can
adjust the values of $\alpha$ and $\beta$ 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 $\alpha$ is more sensitive than $\beta$ 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.
