We study the persistent epidemic prevalence with oscillatory behavior and the extinction of computer viruses via e-mails on a contact relational network growing with new users, for which scale-free structure is estimated from real data. Typical oscillatory phenomenon is simulated in a stochastic model for the execution and detection of viruses. The conditions of extinction by random and targeted immunizations for hubs are derived through bifurcation analysis for simpler deterministic models by using a mean-field approximation without the connectivity correlations. We can qualitatively understand the mechanisms of the spread in linearly growing scale-free networks.