Epidemic Dynamics on an Adaptive Network

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.

Figure 1

Structure of adaptive networks. Plotted is the mean nearest-neighbor degree $⟨k_{nn}⟩$ (top) and the degree distribution ${\rho }_k$ for susceptibles (bottom, circles) and infected (bottom, dots) depending on the degree $k$. (Left) Indiscriminate rewiring: the network is a random graph with Poissonian degree distributions and vanishing degree correlation. (Center) No local dynamics ($p=r=0$): the infected and the susceptibles separate into two unconnected random subgraphs. (Right) Adaptive network with rewiring and local dynamics ($w=0.3$, $r=0.002$, $p=0.008$): the degree distributions are broadened considerably and a strong assortative degree correlation appears. The plots correspond to $N={10}^5$, $K={10}^6$.

Figure 2

Degree correlation index $r_{\mathrm{corr}}$ [22] as a function of the rewiring rate (top). Furthermore, the mean $\mu$ (center) and the variance ${\sigma }^2$ (bottom) of the degree distributions for the susceptibles (circles) and the infected (dots) are shown. Both quantities have been normalized with respect to their values in a random graph without rewiring, $\overline{\mu }$ and ${\overline{\sigma }}^2$, respectively. The plots correspond to $N={10}^5$, $K={10}^6$, $r=0.002$, $p=0.008$.

Figure 3

Bifurcation diagram of the density of the infected $i^{*}$ as a function of the infection probability $p$ for different values of the rewiring rate $w$. In each diagram $i^{*}$ has been computed analytically from Eqs. (2, 3, 4) (thin lines). Along the stable branches these results have been confirmed by the explicit simulation of the full network (circles). Without rewiring only a single continuous transition occurs at $p^{*}=0.0001$ (a). By contrast, with rewiring a number of discontinuous transitions, bistability, and hysteresis loops (indicated by arrows) are observed (b), (c), (d). Fast rewiring (c), (d) leads to the emergence of limit cycles (thick lines indicate the lower turning point of the cycles), which have been computed numerically with the bifurcation software AUTO [24]. Parameter values $N={10}^5$, $K={10}^6$, and $r=0.002$.

Figure 4

Two parameter bifurcation diagram showing the dependence on the rewiring rate $w$ and the infection probability $p$. In the white and light gray regions there is only a single attractor, which is a healthy state in the white region and an endemic state in the light gray region. In the medium gray region both of these states are stable. Another smaller region of bistability is shown in dark gray. Here, a stable healthy state coexists with a stable epidemic cycle. The transition lines between these regions correspond to transcritical (dash-dotted), saddle-node (dashed), Hopf (continuous), and cycle fold (dotted) bifurcations. The transcritical bifurcation line agrees very well with Eq. (1). Note that the saddle-node and transcritical bifurcation lines emerge from a cusp bifurcation at $p=0.0001$, $w=0$. Parameters are as in Fig. 3.