Many real networks are not isolated from each other but form networks of networks, often interrelated in non trivial ways. Here, we analyze an epidemic spreading process taking place on top of two interconnected complex networks. We develop a heterogeneous mean field approach that allows us to calculate the conditions for the emergence of an endemic state. Interestingly, a global endemic state may arise in the coupled system even though the epidemics is not able to propagate on each network separately, and even when the number of coupling connections is small. Our analytic results are successfully confronted against large-scale numerical simulations. PACS numbers: 89.75.Fb, 05.45.Df, 64.60.al Epidemic spreading is one of the most successful application areas of the new science of networks [1, 2] . Indeed, the general acceptance within the scientific community that many diseases, like sexually transmitted diseases or the H1N1 virus, spread over networked systems represents a major step toward their understanding and control [3] [4] [5] . From a physics perspective, epidemic processes have been widely studied as a paradigm of nonequilibrium phase transitions with absorbing states [6] . When applied to complex networks, these processes have become a source of new and striking phenomena that do not have a counterpart in regular lattices. Germane examples are the absence of epidemic and percolation thresholds in scale-free networks with a power law degree distribution P (k) ∼ k −γ with γ ∈ (2, 3], and an anomalous critical behavior when γ ∈ (3, 4) [7] [8] [9] . We currently have a solid understanding of epidemic processes when they take place on single isolated networks. In contrast, our comprehension is very limited when epidemics happen on coupled interconnected networks. For example, sexually transmitted diseases can propagate both in heterosexual and homosexual networks of sexual contacts [3] . These two networks are not completely isolated due to the existence of bisexual individuals, which act as an effective coupling between the two networks and potentially affect their epidemic properties [10] . To the best of our knowledge, a theory describing these type of systems has not yet been fully developed. In this paper, we fill this gap and present a rigorous heterogeneous mean field study of the susceptibleinfected-susceptible (SIS) model taking place on two interconnected complex networks. Our analysis reveals a highly non-trivial behavior of the epidemic process depending on the strength and nature of the coupling between the networks. We calculate the global epidemic threshold of the process, which turns out to be smaller than the epidemic thresholds of the two networks separately under certain conditions. This implies that an endemic state may arise even if the epidemics is not endemic in any of the two networks separately, as we prove analytically and with large-scale computer simulations. To begin our analysis, we have to specify the topological properties of our networks. Let A and B be two interconnected random networks with given local properties and given two-points correlations. Each node in network A (respectively B) is characterized by a vector degree k a ≡ (k aa , k ab ) representing the number of its internal connections k aa to other nodes in A, and the number of external connections k ab with nodes of network B. The analogous to the degree distribution in single networks is then the probability that a randomly chosen node of network A has a vector degree k a , P A (k a ) = P A (k aa , k ab ), where, in general, k aa and k ab may be correlated. A similar definition applies to P B (k b ) for network B. These two distributions are arbitrary except for the consistency condition N A /N B = k ba / k ab , where N A /N B is the ratio between the sizes of the two networks and the right hand side is the ratio between their external average degrees. This condition states that the total number of edges leaving network A towards B must be the same going from network B to network A. Two-points correlations, on the other hand, are encoded by the transition probabilities P AA (k a |k a ), P AB (k b |k a ), P BA (k a |k b ),and P BB (k b |k b ). For instance, P AB (k b |k a ) is the probability that, being in a node of network A with vector degree k a a randomly chosen neighbor in network B has vector degree k b . Similar definitions applies to the rest of the transition probabilities. As the epidemic spreading model, we consider the SIS model that, together with the susceptible-infectedrecovered model (SIR), is one of the best studied models in epidemiology [11] . The model has a non-equilibrium phase transition between an endemic state with sustained epidemic activity and a healthy phase where the epidemics dies out. Individuals can be in two different states, either susceptible (S) or infected (I). Infected individuals decay spontaneously to the susceptible state at rate δ (that, without loss of generality, we set to δ = 1) whereas susceptible ones get infected at a rate proporarXiv:1202.4087v1 [cond-mat.dis-nn]
Recently, REF studied the susceptible-infectious-susceptible (SIS) epidemic model in an interdependent network.
Epidemic spreading on interconnected networks
Physical review. E, Statistical, nonlinear, and soft matter physics
Physical review. E, Statistical, nonlinear, and soft matter physics
Physics Computer Science Medicine