Effects of community structure on epidemic spread in an adaptive network

When an epidemic spreads in a population, individuals may adaptively change the structure of their social contact network to reduce risk of infection. Here we study the spread of an epidemic on an adaptive network with community structure. We model two communities with different average degrees. The disease model is susceptible-infected-susceptible (SIS), and adaptation is rewiring of links between susceptibles and infectives. Locations of rewired links are selected so that the community structure will be preserved if susceptible-infective links are homogeneously distributed. The bifurcation structure is obtained, and a mean field model is developed that accurately predicts the steady-state behavior of the system. In a static network, weakly connected heterogeneous communities can have significantly different infection levels. In contrast, adaptation promotes similar infection levels and alters the network structure so that communities have more similar average degrees. We estimate the time for network restructuring to allow infection incursion from one community to another and show that it is inversely proportional to the number of cross-links between communities. In extremely heterogeneous systems, periodic oscillations in infection level can occur due to repeated infection incursions.

Figure 1

Bifurcation diagram for infected fraction in a static network ($w=0$) as a function of infection rate $p$ for different fractions of cross-links $f$. Black: community A; gray: community B. Solid curves: mean field solutions (stable branches); dashed curves: mean field solutions (unstable branches); circles: Monte Carlo simulations. Average degrees $\langle k_a\rangle \approx 28$ and $\langle k_b\rangle \approx 12$. The plots correspond to $d=0.7$. In (d), in order to obtain better statistics, we averaged over 100 network realizations, each of which is similar to (b).

Figure 2

Effective threshold infection rate from mean field vs fraction of cross-links $f$ for $d=0.7$. Black curve: community A; gray curve: community B. The infection rate $p$ at which the fraction of infecteds is less than ${10}^{-3}$ is considered the effective threshold rate.

Figure 3

Bifurcation diagram for infected fraction in an adaptive network as a function of infection rate $p$ for different fractions of cross-links $f$. Black: community A; gray: community B. Solid curves: mean field solutions (stable branches); dashed curves: mean field solutions (unstable branches); circles: Monte Carlo simulations. Average degrees $\langle k_a\rangle \approx 28$ and $\langle k_b\rangle \approx 12$. The plots correspond to $d=0.7$ and $w=0.04$. The simulations are done similarly as in Fig. 1, but here we run the system for $5\times {10}^6$ time units in order to approach the endemic branch.

Figure 4

Time series from Monte Carlo simulation of full system. (a) Infected node fractions. Black curve: fraction of infecteds in community A; gray curve: fraction of infecteds in community B. (b) Fraction of links. Black curve: fraction $f_{aa}$ of AA type links; gray curve: fraction $f_{bb}$ of BB type links. Initial condition is as described in the text. $f={10}^{-4}$, $d=0.7$, $w=0.04$, and $p=0.0014$.

Figure 5

The critical time for infection incursion in community B vs fraction of cross-links $f$. The critical time values are averaged over 5 runs except for $f={10}^{-6}$. In each run, the time when the fraction of infecteds in community B exceeded $0.1\%$ is considered the critical time. Initial condition for each run is a two-community network randomly generated as in Fig. 4. The symbols show the simulation results; the curve is the best fit line with slope $-1$.

Figure 6

Fraction of links vs infection rate $p$. Curves: mean field steady states; points: Monte Carlo steady state averages. Black: fraction $f_{aa}$ of AA type links; gray: fraction $f_{bb}$ of BB type links. (The fraction $f_{ab}$ of AB links is less than $1\%$.) $f={10}^{-4}$, $d=0.7$, and $w=0.04$. The data are from the simulations done in Fig. 3.

Figure 7

Time series from Monte Carlo simulation of full system. (a) Infected node fractions. Black curve: fraction of infecteds in community A; gray curve: fraction of infecteds in community B. (b) Fraction of links. Black curve: fraction $f_{aa}$ of AA type links; gray curve: fraction $f_{bb}$ of BB type links. Initial condition is as described in the text. $d=0.9$, $w=0.04$, and $p=0.0014$.