The structure of coevolving infection networks

Disease awareness in infection dynamics can be modeled with adaptive contact networks whose rewiring rules reflect the attempt by susceptibles to avoid infectious contacts. Simulations of this type of models show an active phase with constant infected node density in which the interplay of disease dynamics and link rewiring prompts the convergence towards a well defined degree distribution, irrespective of the initial network topology. We develop a method to study this dynamic equilibrium and give an analytic description of the structure of the characteristic degree distributions and other network measures. The method applies to a broad class of systems and can be used to determine the steady-state topology of many other adaptive networks. Introduction. – Research on the influence of network topology on the global dynamics of systems composed of many interacting units has been very active for the last ten years. More recently, adaptive and coevolving networks, that is, networks whose topology changes along with the dynamics, have begun to be considered on two standings. The first one is to explore the consequences of this interplay from the viewpoint of global dynamics [1]. The second is to understand what sort of interactions may give rise to certain network architectures [2]. Understanding the origin of degree distributions and correlations of real networks was first addressed by considering networks that evolve independently of the dynamics [3], and it is natural to extend this idea to networks whose evolution is determined by the dynamics they support. Infection dynamics is a natural setting to investigate the coupled evolution of the interacting elements and the network of interactions, because disease awareness translates into susceptibles that try to evade infection by changing their contact patterns according to the disease status of their neighbors. The SIS (susceptible, infected, susceptible) and the SIR (susceptible, infected, recovered) models are the simplest models for the spread of epidemic infections [4]. They are based on the idea that an infected individual infects susceptibles at a certain rate for a certain period, and then recovers and becomes susceptible again (if the disease does not confer immunity), or recovers (a)E-mail: [email protected] and becomes immune. Stochastic versions of these models on networks can be mapped onto bond percolation [5]. Simple epidemic models like these exhibiting trivial global dynamics on static networks have been shown to display interesting behavior on adaptive networks, in which links between nodes may be created or removed according to rewiring rules dictated by disease awareness. In the network scheme of SIS dynamics, susceptible nodes (S-nodes, a fraction [S] of the total number of nodes) are infected with rate p along links connecting them to infected nodes (I-nodes, the remaining fraction [I] = 1 − [S]), whereas the latter recover with rate r. Gross et al. [6] incorporated disease awareness into the conventional SIS model by introducing a topology-changing, yet link-number preserving, rewiring mechanism: S-nodes try to evade infection by retracting links from infected neighbors with rate w, and rewiring them to randomly selected S-nodes. The description of the model in the framework of the standard pair approximation yields a complex phase diagram. In addition to a frozen phase (where the disease dies out) and to a stationary active phase (where [I] 6= 0 is constant and [AB], the number per node of links connecting nodes of type A and B, A,B ∈ {S, I}, is also constant), other active phases emerge that are absent in the SIS model without rewiring, namely an oscillatory phase and a bistable phase [7]. Simulations in the stationary active phase show the states of the nodes and the links coevolving to produce and maintain a dynamic network topology characterized by well defined degree distributions not