Many real-world networks are characterized by adaptive changes in their topology depending on the state of their nodes. Here we study epidemic dynamics on an adaptive network, where the susceptibles are able to avoid contact with the infected by rewiring their network connections. This gives rise to assortative degree correlation, oscillations, hysteresis, and first order transitions. We propose a low-dimensional model to describe the system and present a full local bifurcation analysis. Our results indicate that the interplay between dynamics and topology can have important consequences for the spreading of infectious diseases and related applications. DOI: 10.1103/PhysRevLett.96.208701 PACS numbers: 89.75.Hc, 87.19.Xx, 89.75.Fb In the physical literature the dynamics of complex networks has recently received much attention, with many applications in social, biological, and technical systems [1,2]. In particular, most research has been directed in two distinct directions. On the one hand, attention has been paid to the structure of the networks, revealing that simple dynamical rules, such as preferential attachment or selective rewiring, can be used to generate complex topologies [3][4][5][6]. Many of these rules are not only a useful tool for the generation of model graphs, but are also believed to shape real-world networks like the internet or the network of social contacts. On the other hand, research has focused on large ensembles of dynamical systems, where the interaction between individual units is described by a complex graph [7][8][9][10][11][12][13][14][15]. These studies have shown that the network topology can have a strong impact on the dynamics of the nodes, e.g., the absence of epidemic thresholds on scale free networks [7,8] or the detrimental effect of assortative degree correlations on targeted vaccination [12]. In the past the cross fertilization between these two lines of thought has led to considerable advances. However, the dynamics of networks and the dynamics on networks are still generally studied separately. In doing so, a characteristic features of many real-world networks is not taken into account, namely, the ability to adapt the network topology dynamically in response to the dynamic state of nodes [16 -19].Consider, for example, the spreading of an infectious disease on a social network. Humans tend to respond to the emergence of an epidemic by avoiding contacts with infected individuals. Such rewiring of the local connections can have a strong effect on the dynamics of the disease, which in turn influences the rewiring process. Thus, a complicated mutual interaction between a time varying network topology and the dynamics of the nodes emerges.In this Letter we study a susceptible-infectedsusceptible (SIS) model on an adaptive network. We demonstrate that a simple intuitive rewiring rule for the network connections has a profound impact on the emerging network, and is able to generate specific network properties such as a wide degree distribution, assortative degree correlatio...
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