Bursting endemic bubbles in an adaptive network

The spread of an infectious disease is known to change people's behavior, which in turn affects the spread of disease. Adaptive network models that account for both epidemic and behavioral change have found oscillations, but in an extremely narrow region of the parameter space, which contrasts with intuition and available data. In this paper we propose a simple susceptible-infected-susceptible epidemic model on an adaptive network with time-delayed rewiring, and show that oscillatory solutions are now present in a wide region of the parameter space. Altering the transmission or rewiring rates reveals the presence of an endemic bubble—an enclosed region of the parameter space where oscillations are observed.

Figure 1

Comparison between the solution of (4) and numerical simulation. Three sets of results are shown, $\omega =0$ (top), $\omega =1$ (middle), and $\omega =1.4$ (bottom). Other parameters are $\beta =0.6$, $\gamma =1$, $\tau =6$, and $\langle k\rangle =10$. Simulation results are averaged across 100 iterations on random networks of 1000 nodes. All simulations begin by randomly selecting a node to infect at time $t=0$. Simulation runs which die out are discarded and performed again.

Figure 2

(a) A two-parameter bifurcation diagram indicating different dynamical regimes in the behavior of model (4) with $\beta =0.6$. (b) The endemic bubble for model (4) for $\tau =6$. The endemic equilibrium is stable on the gray surface and unstable on the green surface. The green surface is constructed using the minima and maxima of oscillations and shows the shape of the endemic bubble. (c) The value of the endemic equilibrium plotted against the rewiring delay $\tau$ for $\beta =0.6$ and $\omega =2$. Increasing $\tau$ decreases the expected number of infected individuals at endemic equilibrium until the Hopf bifurcation point, beyond which the amplitude of oscillations grows. (d) The average behavior from 100 numerical simulations on random networks of 1000 nodes compared to the mean-field model (4). The solid black line (circles) denotes the prevalence of the disease in the mean-field model (simulations), and the red dashed line (diamonds) denotes the normalized mean degree calculated from (6). Parameter values are $\beta =0.55$, $\omega =1.5$, $\tau =5.5$, $\gamma =1$, $\langle k\rangle =10$. Simulations in which epidemic outbreaks died out were discarded and performed again.

Figure 3

Real part of the maximum characteristic eigenvalue of the endemic equilibrium of (4) for $\beta =0.2$ (a), 0.4 (b), and 0.6 (c). Other parameters are the same as in Fig. 2. The endemic equilibrium is unstable in the red/yellow region, stable in the green/blue region, and biologically infeasible in the white region.

Figure 4

Snapshots of the degree distribution for networks of ${10}^4$ nodes. In each plot the black (solid) line is the initial degree distribution, the blue (dashed) line is for the early growth phase, red (dotted) shows an early peak, and green (dash-dotted) and magenta (squares) show later snapshots. Disease parameters are $\beta =0.6$, $\gamma =1$, $\omega =1.4$, $\tau =6$. (a) Erdős-Rényi network with $\langle k\rangle =10$. (b) Homogeneous network with $k=10$ for all nodes. (c), (d) Truncated scale-free networks with the scaling exponents $\alpha =2$ and 3, respectively.