  We present a detailed analytical and numerical study for the spreading of
infections in complex population networks with acquired immunity. We show that
the large connectivity fluctuations usually found in these networks strengthen
considerably the incidence of epidemic outbreaks. Scale-free networks, which
are characterized by diverging connectivity fluctuations, exhibit the lack of
an epidemic threshold and always show a finite fraction of infected
individuals. This particular weakness, observed also in models without
immunity, defines a new epidemiological framework characterized by a highly
heterogeneous response of the system to the introduction of infected
individuals with different connectivity. The understanding of epidemics in
complex networks might deliver new insights in the spread of information and
diseases in biological and technological networks that often appear to be
characterized by complex heterogeneous architectures.
