Collective versus hub activation of endemic states on networks

Heterogeneous topological patterns of contacts, such as those exhibited by many
natural and artificial complex networks, can utterly alter the behavior of
dynamical processes mediated by them, as compared with the classical
expectations in regular lattices. One of most remarkable results is that the
epidemic threshold of the susceptible-infected-susceptible (SIS) model on
scale-free (SF) networks with a degree distribution $P(k)\sim k^{-\gamma}$,
where $\gamma$ is the degree exponent, is zero in the infinite size limit.
Theoretical approaches showing that hubs in a network (identified as star
graphs) can have an exceedingly large lifetime of activity in SIS model and that
stars can reinfect each other through connected paths imply in an active endemic
state for any infection rate $\lambda>0$. These results highlight the role the
the large survival infection time of hubs in the SIS model have on the
sustaining of the endemic active state. Here we will discuss the extension of
these concepts to more generic epidemic processes. Particularly, we considered
temporary immunization conferred after the infective period in simple
susceptible-infected-removed-susceptible (SIRS) epidemic model, which is
equivalent to SIS in the basic theoretical approaches for epidemic
processes. However, our theoretical analysis predicts that the interplay between
lifetime of hubs and th reinfection mechanism in SIRS model leads to different
outcomes depending on the degree exponent $\gamma$. For $\gamma<3$, a SIS-like
behavior having a null threshold in the infinite size limit is predicted. For
$\gamma>3$ a finite threshold, where endemic state is possible only through a
collective activation of the whole network, is obtained for any finite
immunization period, drastically contrasting with the SIS model. Extensive numerical
simulations support our predictions. \\
Financial support: FAPEMIG, CAPES and CNPq