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Dynamical systems running on the top of complex networks has been extensivelyinvestigated for decades. But this topic still remains among the most relevantissues in complex network theory due to its range of applicability. The contactprocess (CP) and the susceptible-infected-susceptible (SIS) model are usedquite often to describe epidemic dynamics. Despite their simplicity, thesemodels are robust to predict the kernel of real situations. In this work, wereview concisely both processes that are well-known and very applied examplesof models that exhibit absorbing-state phase transitions. In the epidemicscenario, individuals can be infected or susceptible. A phase transitionbetween a disease-free (absorbing) state and an active stationary phase (wherea fraction of the population is infected) are separated by an epidemicthreshold. For the SIS model, the central issue is to determine this epidemicthreshold on heterogeneous networks. For the CP model, the main interest is torelate critical exponents with statistical properties of the network.

An overview of epidemic models with phase transitions to absorbing states running on top of complex networks