Caveats on epidemic processes model on complex networks

Theoretical and computational approaches to epidemic-like models have met with outstanding success on complex networks. However, some questions remains opened for the dynamics of the Susceptible-Infected-Susceptible (SIS) model on networks with a heavy-tailed degree distribution. This archetypal disease spreading model undergoes an absorbing phase transition revealing its epidemic threshold at some value for the effective spreading rate. We numerically test the limits and validity of two competing theoretical approaches proposed to describe the SIS epidemic threshold on uncorrelated networks with degree distribution $P(k)\sim k^{-\gamma}$. On one hand, we have the Heterogeneous Mean-Field (HMF) approach that assumes the statistical equivalence for those nodes of the same degree class (same number of contacts). On the other hand, the Quenched Mean-Field (QMF) considers the entire connectivity pattern of the network but treating the state (infected or not) of neighboring nodes as being statistically independent. We investigate network properties in which analytical results derived under the theoretical assumptions of HMF and QMF are or not verified. We show that the presence of outliers and formation of densely connected cores on these networks drastically change the outcomes predicted by these theoretical approaches. Moreover, resorting to extensive simulations for a large ensemble of network realizations, we show that network samples (for the regime of $\gamma>3$) with the presence of outliers can produce a diverging average lifetime for the model, showing that the epidemic
lifetime averaged over the ensemble is infinite even for finite sizes independently of the infection rate.

The authors would like to thank FAPEMIG.