Enhancement of large fluctuations to extinction in adaptive networks

During an epidemic, individual nodes in a network may adapt their connections to reduce the chance of infection. A common form of adaption is avoidance rewiring, where a noninfected node breaks a connection to an infected neighbor and forms a new connection to another noninfected node. Here we explore the effects of such adaptivity on stochastic fluctuations in the susceptible-infected-susceptible model, focusing on the largest fluctuations that result in extinction of infection. Using techniques from large-deviation theory, combined with a measurement of heterogeneity in the susceptible degree distribution at the endemic state, we are able to predict and analyze large fluctuations and extinction in adaptive networks. We find that in the limit of small rewiring there is a sharp exponential reduction in mean extinction times compared to the case of zero adaption. Furthermore, we find an exponential enhancement in the probability of large fluctuations with increased rewiring rate, even when holding the average number of infected nodes constant.

Figure 1

Large fluctuations in adaptive networks. (a) Fraction of infected nodes, $x_i$, at time $t$ in a single stochastic trajectory. (b) Histogram of the trajectory, $\rho$, as a function of $x_i$, shown in blue. (c) Histogram of per-capita number of susceptible-infected edges, $x_{si}$, shown in blue. Parameters for (a)–(c) are $N=1000$, $\alpha =1$, $\beta =0.03$, $w=0.03$, and $〈k〉=40$, for which the endemic state is the only deterministically stable state. Predictions from Eqs. (5)–(9) are shown in red for (b) and (c). (d) Histograms of $x_i$ for several $w$, and $x_i^{*}$ held constant. Thirty stochastic trajectories are averaged with the following parameters: ($w=0.10,\beta =0.03113$, blue $\circ$), ($w=0.25,\beta =0.033033$, red $\square$), ($w=0.40,\beta =0.035051$, green $▵$), and ($w=0.55,\beta =0.03711$, magenta $♦$). Predictions are shown with solid lines; $N=1000$, $\alpha =1$, and $〈k〉=40$.

Figure 2

Two-dimensional projections of the density function of extinction paths computed from stochastic simulations, shown on log scale. (a) Number of infected nodes and the number of susceptible-infected edges are specified on the vertical and horizontal axes, respectively. Parameters are $N=1000$, $\alpha =1$, $\beta =0.03$, $w=0.03$, and $〈k〉=40$. (b) Number of infected nodes and the number of infected-infected edges are specified on the vertical and horizontal axes, respectively. Parameters are $N=1000$, $\alpha =1$, $\beta =0.103$, $w=5.10$, and $〈k〉=40$. Predictions from Eq. (8) are shown in blue with endemic states (red $*$) and a saddle (blue $\circ$).

Figure 3

Extinction schematics showing the fraction of infected nodes at the endemic state, $x_i^{*}$, versus the infection rate. (a) E-I: Large fluctuation (dashed arrow) leads from a deterministically stable (red) endemic state, $x_i^{*}$, to a deterministically unstable (blue) state $x_i=0$. Red $\square$'s are $〈x_i〉$ measured from simulations. Parameters are $N=1000$, $\alpha =1$, $w=0.10$, and $〈k〉=40$. (b) E-II: Large fluctuation (dashed arrow) leads from a deterministically stable endemic state to a saddle, $x_i^{\mathrm{sadd}}$ (blue dashed line), after which the network follows the deterministic dynamics from $x_i^{\mathrm{sadd}}$ to $x_i=0$ (solid arrow). Parameters are $N=1000$, $\alpha =1$, $w=5.0$, and $〈k〉=40$.

Figure 4

Log of the mean extinction time vs. rewiring rate. Mean times were taken from 400 stochastic realizations of the dynamics. A single Erdős-Rényi network was used as an initial condition for each series with a uniformly random distribution of infection such that $x_i=x_i^{*}$. $N=1000$ and $\alpha =1$ for all networks, with average degrees: $〈k〉=40(\circ )$ and $〈k〉=20(\square )$. (a) E-I: $\beta =0.03(\circ )$ and $\beta =0.06(\square )$. (b) E-II: $\beta =0.103(\circ )$ and $\beta =0.206(\square )$. Predictions from Eqs. (9) and (10) are shown in red (dashed). The constant in Eq. (10) was fit to match simulations.

Figure 5

Average susceptible degree distribution for several rewiring rates in E-I. The distribution is an average of 200 stochastic realization at $t\sim 〈T〉$, with initial conditions chosen as in Fig. 4. (a) Fraction of susceptible nodes with degree $k$, $g_k^s$, vs. $k$. (b) $〈z〉$ vs. $w$. Parameters are $N=1000$, $\alpha =1$, $〈k〉=20$, and $\beta =0.06$.

Figure 6

Histogram of $z-〈z〉$ for slow (red) and fast (blue) rewiring, Eq. (3). The parameters are $N=1000$, $〈k〉=40$, and $\alpha =1$. Slow rewiring: $\beta =0.03$ and $w=0.06$. Fast rewiring: $\beta =0.103$ and $w=5.0$.