Many real networks exhibit a layered structure in which links in each layer reflect the function of nodes on different environments. These multiple types of links are usually represented by a multiplex network in which each layer has a different topology. In real-world networks, however, not all nodes are present on every layer. To generate a more realistic scenario, we use a generalized multiplex network and assume that only a fraction of the nodes are shared by the layers. We develop a theoretical framework for a branching process to describe the spread of an epidemic on these partially overlapped multiplex networks. This allows us to obtain the fraction of infected individuals as a function of the effective probability that the disease will be transmitted . We also theoretically determine the dependence of the epidemic threshold on the fraction of shared nodes in a system composed of two layers. We find that in the limit of the threshold is dominated by the layer with the smaller isolated threshold. Although a system of two completely isolated networks is nearly indistinguishable from a system of two networks that share just a few nodes, we find that the presence of these few shared nodes causes the epidemic threshold of the isolated network with the lower propagating capacity to change discontinuously and to acquire the threshold of the other network.

-In many real-world complex systems, individuals have many kind of interactions among them, suggesting that it is necessary to consider a layered structure framework to model systems such as social interactions. This structure can be captured by multilayer networks and can have major effects on the spreading of process that occurs over them, such as epidemics. In this Letter we study a targeted immunization strategy for epidemic spreading over a multilayer network. We apply the strategy in one of the layers and study its effect in all layers of the network disregarding degree-degree correlation among layers. We found that the targeted strategy is not as efficient as in isolated networks, due to the fact that in order to stop the spreading of the disease it is necessary to immunize more than the 80% of the individuals. However, the size of the epidemic is drastically reduced in the layer where the immunization strategy is applied compared to the case with no mitigation strategy. Thus, the immunization strategy has a major effect on the layer were it is applied, but does not efficiently protect the individuals of other layers.Introduction. -The new insights in the complex networks analysis, is no further considering networks as isolated entities, but characterizing how networks interact with other networks and how this interaction affects processes that occurs on top of them. A system composed by many networks is called Network of Networks (NoN), a terminology introduced a few years ago [1][2][3][4]. In NoN, there are connectivity links within each individual network, and external links that connect each network to other networks in the system. A particular class of Network of Networks in which the nodes have multiple types of links across different layers [5][6][7][8][9][10][11], are called Multiplex or Multilayer Networks [12]. The multiplex network approach has proven to be a successful tool in modeling a number of very wide real-world systems, such as the Indian air and train transportation networks [13] and the International Trade Network [14,15].In the last couple of years, the study of the effect of multiplexity of networks in propagation processes such as epidemics has been the focus of many recent researches [12,[16][17][18][19][20]. In Ref [21] the research concentrated in the propagation of a disease in partially overlapped multilayer networks, because the fact that individuals are not necessarily present in all the layers of a society impacts the propagation of the epidemic. For the epidemic model they used the susceptible-infected-recovered (SIR) model [22][23][24] that describes the propagation of non recurrent diseases for which ill individuals either die or, after recovery, become im-

We explore how heterogeneity in the intensity of interactions between people affects epidemic spreading. For that, we study the susceptible-infected-susceptible model on a complex network, where a link connecting individuals i and j is endowed with an infection rate β ij = λw ij proportional to the intensity of their contact w ij , with a distribution P (w ij ) taken from face-to-face experiments analyzed in Cattuto et al. (PLoS ONE 5, e11596, 2010). We find an extremely slow decay of the fraction of infected individuals, for a wide range of the control parameter λ.Using a distribution of width a we identify two large regions in the a − λ space with anomalous behaviors, which are reminiscent of rare region effects (Griffiths phases) found in models with quenched disorder. We show that the slow approach to extinction is caused by isolated small groups of highly interacting individuals, which keep epidemic alive for very long times.A mean-field approximation and a percolation approach capture with very good accuracy the absorbing-active transition line for weak (small a) and strong (large a) disorder, respectively.