Epidemic threshold and topological structure of susceptible-infectious-susceptible epidemics in adaptive networks

The interplay between disease dynamics on a network and the dynamics of the structure of that network characterizes many real-world systems of contacts. A continuous-time adaptive susceptible-infectious-susceptible (ASIS) model is introduced in order to investigate this interaction, where a susceptible node avoids infections by breaking its links to its infected neighbors while it enhances the connections with other susceptible nodes by creating links to them. When the initial topology of the network is a complete graph, an exact solution to the average metastable-state fraction of infected nodes is derived without resorting to any mean-field approximation. A linear scaling law of the epidemic threshold ${\ensuremath{\tau}}_{c}$ as a function of the effective link-breaking rate $\ensuremath{\omega}$ is found. Furthermore, the bifurcation nature of the metastable fraction of infected nodes of the ASIS model is explained. The metastable-state topology shows high connectivity and low modularity in two regions of the $\ensuremath{\tau},\ensuremath{\omega}$ plane for any effective infection rate $\ensuremath{\tau}>{\ensuremath{\tau}}_{c}$: (i) a ``strongly adaptive'' region with very high $\ensuremath{\omega}$ and (ii) a ``weakly adaptive'' region with very low $\ensuremath{\omega}$. These two regions are separated from the other half-open elliptical-like regions of low connectivity and high modularity in a contour-line-like way. Our results indicate that the adaptation of the topology in response to disease dynamics suppresses the infection, while it promotes the network evolution towards a topology that exhibits assortative mixing, modularity, and a binomial-like degree distribution.