We analyze the contact process on random graphs generated according to the
preferential attachment scheme as a model for the spread of viruses in the Internet.
We show that any virus with a positive rate of spread from a node to its neighbors has
a non-vanishing chance of becoming epidemic. Quantitatively, we discover an interesting dichotomy:
for a virus with effective spread rate $\lambda$, if the infection starts at a typical vertex
then it develops into an epidemic with probability
$\lambda^{\Theta(\log(1/\lambda)/\log\log(1/\lambda))}$, but on average the epidemic probability is
$\lambda^{\Theta(1)}$.