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We analyze two alterations of the standard susceptible-infected-susceptible(SIS) dynamics that preserve the central properties of spontaneous healing andinfection capacity of a vertex increasing unlimitedly with its degree. Allmodels have the same epidemic thresholds in mean-field theories but dependingon the network properties, simulations yield a dual scenario, in which theepidemic thresholds of the modified SIS models can be either dramaticallyaltered or remain unchanged in comparison with the standard dynamics. Foruncorrelated synthetic networks having a power-law degree distribution withexponent $\gamma<5/2$, the SIS dynamics are robust exhibiting essentially thesame outcomes for all investigated models. A threshold in better agreement withthe heterogeneous rather than quenched mean-field theory is observed in themodified dynamics for exponent $\gamma>5/2$. Differences are more remarkablefor $\gamma>3$ where a finite threshold is found in the modified models incontrast with the vanishing threshold of the original one. This duality iselucidated in terms of epidemic lifespan on star graphs. We verify that theactivation of the modified SIS models is triggered in the innermost componentof the network given by a $k$-core decomposition for $\gamma<3$ while ithappens only for $\gamma<5/2$ in the standard model. For $\gamma>3$, theactivation in the modified dynamics is collective involving essentially thewhole network while it is triggered by hubs in the standard SIS. The dualityalso appears in the finite-size scaling of the critical quantities wheremean-field behaviors are observed for the modified, but not for the originaldynamics. Our results feed the discussions about the most proper conceptions ofepidemic models to describe real systems and the choices of the most suitabletheoretical approaches to deal with these models.

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