  We present a finite-size scaling theory of a contact process with permanent
immunity on uncorrelated scale-free networks. We model an epidemic outbreak by
an analog of the susceptible-infected-removed model where an infected
individual attacks only one susceptible in a time unit in a way we can expect a
non-vanishing critical threshold at scale-free networks. As we already know,
the susceptible-infected-removed model can be mapped in a bond percolation
process, allowing us to compare the critical behavior of site and bond
universality classes on networks. We used the external field finite-scale
theory, where the dependence on the finite size enters the external field
defined as the initial number of infected individuals. We can impose the scale
of the external field as $N^{-1}$. The system presents an epidemic-endemic
phase transition where the critical behavior obeys the mean-field universality
class, as we show theoretically and by simulations.
