Multiple transitions of the susceptible-infected-susceptible epidemic model on complex networks

Phase transitions involving equilibrium and nonequilibrium processes on complex networks have begun drawing an increasing interest soon after the boom of network science in the late 1990s. Percolation, epidemic spreading, and spin systems are only a few examples of breakthrough in
the investigation of critical phenomena in complex networks. Absorbing state phase transitions have become a paradigmatic issue in the interplay between nonequilibrium systems and complex networks with epidemic spreading being a prominent example where high complexity emerges from very simple dynamical rules on heterogeneous substrates. The existence or absence of finite epidemic thresholds involving an endemic phase of the susceptible-infected-susceptible (SIS) model on scale-free networks with a degree distribution $P (k) \sim k^{-\gamma}$, where $\gamma$ is the degree exponent, has been target of a intense investigation. Distinct theoretical approaches for the SIS model were devised to determine an epidemic threshold $\lambda_c$ separating an absorbing, disease-free state from an active phase. The quenched mean-field (QMF) theory explicitly includes the entire structure of the network through its adjacency matrix while the heterogeneous mean-field (HMF) theory performs a coarse-graining of the network grouping vertices accordingly their degrees. The HMF theory predicts a vanishing threshold for the SIS model for the range $2 < \gamma < 3$, while a finite threshold is expected for $\gamma > 3$. Conversely, the QMF theory states a threshold inversely proportional to the largest eigenvalue of the adjacency matrix, implying that the threshold vanishes for any value of $\gamma$. For $\gamma < 3$, there exists a consensus for SIS thresholds. However, for $\gamma > 3$ the different mean-field approaches predict different outcomes. Therefore, in this work, we performed extensive simulations in the quasistationary state of the SIS dynamics on random networks having a power law degree distribution with $\gamma > 3$, for a comparison with these mean-field theories. We observed concomitant multiple transitions in finite networks presenting large gaps in the degree distribution and the obtained multiple epidemic thresholds are well described by different mean-field theories. We observed that the transitions involving thresholds which vanish at the thermodynamic limit involve localized states, in which a vanishing fraction of the network effectively contributes to epidemic activity, whereas an endemic state, with a finite density of infected vertices, can occur at a finite threshold.

Acknowledgements and Financial Support: FAPEMIG, CAPES and CNPq.