Robustness and fragility of the susceptible-infected-susceptible epidemic models on complex networks

Abstract

We analyze two alterations of the standard susceptible-infected-susceptible (SIS) dynamics that preserve the central properties of spontaneous healing and infection capacity of a vertex increasing unlimitedly with its degree. All models have the same epidemic thresholds in mean-field theories but depending on the network properties, simulations yield a dual scenario, in which the epidemic thresholds of the modified SIS models can be either dramatically altered or remain unchanged in comparison with the standard dynamics. For uncorrelated synthetic networks having a power-law degree distribution with exponent γ<5/2, the SIS dynamics are robust exhibiting essentially the same outcomes for all investigated models. A threshold in better agreement with the heterogeneous rather than quenched mean-field theory is observed in the modified dynamics for exponent γ>5/2. Differences are more remarkable for γ>3, where a finite threshold is found in the modified models in contrast with the vanishing threshold of the original one. This duality is elucidated in terms of epidemic lifespan on star graphs. We verify that the activation of the modified SIS models is triggered in the innermost component of the network given by a k-core decomposition for γ<3 while it happens only for γ<5/2 in the standard model. For γ>3, the activation in the modified dynamics is collective involving essentially the whole network while it is triggered by hubs in the standard SIS. The duality also appears in the finite-size scaling of the critical quantities where mean-field behaviors are observed for the modified but not for the original dynamics. Our results feed the discussions about the most proper conceptions of epidemic models to describe real systems and the choices of the most suitable theoretical approaches to deal with these models.

Onset of synchronization of Kuramoto oscillators in scale-free networks

“…Nevertheless, in this regime of γ, the latter approximation estimates correctly secondary peaks in susceptibility curves associated to localization effects due to the epidemic activation of the largest hub in the network -a phenomenon for which we have not observed a counterpart in the synchronization dynamics of large SF networks. Synchronization thresholds, on the other hand, present the same behavior as observed in most dynamical processes that exhibit a phase transition from active to inactive states, such as contact process and SIRS model [28,29]. In fact, the phase transition observed in the Kuramoto model is a standard phase transition associated to a collective phenomenon (i.e., the activation of the entire network), whereas the phase transition in the SIS model is related to a mutual reinfection of hubs.…”

Complex Networks: a Mini-review

“…Due to its interdisciplinarity, we are capable of investigating many dynamical processes that really affect our daily lives, from biological until technological contexts. From this knowledge, we can make useful predictions involving, for example, epidemic dynamics [ 6 , 79 , 80 , 92 – 95 ], vector-borne or livestock diseases [ 40 , 41 , 82 , 84 , 85 , 96 ], spreading rumors [ 97 – 101 ], and synchronization [ 29 , 30 , 70 , 102 ].…”

Nonmassive immunization to contain spreading on complex networks

“…The activation mechanisms of epidemic process and, in particular, of SIS can be quite tricky to analyze [35][36][37][38]. We can classify the activation into motifdriven and collective processes [37,39]. In the former, a subextensive fraction is responsible for the triggering the epidemics and spreading it out to the rest of the network infecting an extensive fraction the population and the epidemic threshold vanishes as N → ∞.…”

“…This is the case of the SIS model on power-law networks for which activation can be triggered by either hubs or a densely connected subgraph given by the maximal index of a k-core decomposition, depending on the degree exponent [38]. In the case of collective activation, an extensive part of the network is directly involved [37,39]. This happens, for example, in the Harris contact process for any value of the degree exponent [39] and in the susceptible-infectedrecovered-susceptible (SIRS) model for γ > 3 [37].…”