  We investigate mechanisms of the typically observed recoverable prevalence in
epidemic spreading. Assuming the time-independent connectivity correlations, we
analyze the dynamics of spreading on linearly growing scale-free (SF) networks,
and derive the extinction condition related to the rates of network growth,
infection, and immunization of viruses. The behavior is consistent with the
previous results for SF networks by a mean-field approximation without
connectivity correlations. In particular, it is suggested that the growing must
be stopped to prevent the spreading of infection. This insight helps to
understand the spreading phenomena on communication or social networks.
