  We investigate the effects of risk perception in a simple model of epidemic
spreading. We assume that the perception of the risk of being infected depends
on the fraction of neighbors that are ill. The effect of this factor is to
decrease the infectivity, that therefore becomes a dynamical component of the
model. We study the problem in the mean-field approximation and by numerical
simulations for regular, random and scale-free networks.
  We show that for homogeneous and random networks, there is always a value of
perception that stops the epidemics. In the ``worst-case'' scenario of a
scale-free network with diverging input connectivity, a linear perception
cannot stop the epidemics; however we show that a non-linear increase of the
perception risk may lead to the extinction of the disease. This transition is
discontinuous, and is not predicted by the mean-field analysis.
