Epidemic spreading in evolving networks

A model for epidemic spreading on rewiring networks is introduced and analyzed for the case of scale free steady state networks. It is found that contrary to what one would have naively expected, the rewiring process typically tends to suppress epidemic spreading. In particular it is found, that as in static networks under a mean-field approximation, rewiring networks with degree distribution exponent $\gamma >3$ exhibit a threshold in the infection rate below which epidemics die out in the steady state. However the threshold is higher in the rewiring case. For $2<\gamma \le 3$ no such threshold exists, but for small infection rate the steady state density of infected nodes (prevalence) is smaller for rewiring networks.

Figure 1

Rewiring of a link from a node with degree $k$ to a node with degree $k^{\prime }$ with a rate $u\left(k\right)v\left(k^{\prime }\right)$.

Figure 2

(Color online) Top: $r_k$ is plotted against $k$ for various rewiring rates. The curves represent increasing rewiring rates from left to right. The set of Eqs. (26) is solved numerically with $b=3.5$ and $k_0=15$ for the critical case, which corresponds to $⟨u⟩=1$, $⟨k⟩=4.917454$, and $⟨k^2⟩=99.39410$. Bottom: The threshold ${\lambda }_c$ as a function of $\nu$. It was calculated numerically using Eq. (27) with the same parameters. The analytically derived $\nu ⪢1$ limiting formula (33) is also plotted (full line) with $C=50$, which gives a good fit.

Figure 3

(Color online) $F\left(x\right)=\left(r_k-1\right){\nu }^{-1/2}$ is plotted against $k{\nu }^{-1/2}$ using the data of Fig. 2. The data collapse confirms the scaling form. The analytically obtained limiting forms [Eq. (31)] are plotted with dashed lines.

Figure 4

The prevalence $\rho$ (a) and the derivative of the prevalence with respect to the infection rate $\text{d}\rho /\text{d}\lambda$ (b) are plotted as a function of the infection rate for a network of size $N=200$, $⟨k⟩=2$ and rewiring dynamics [Eq. (11)] with $b=2.5$ and $k_0=1$ for which $⟨k^2⟩\approx 58$. Curves for rewiring rates $\nu =0,10,{10}^2,{10}^3,{10}^4,{10}^5$ are represented by full lines from left to right respectively. For clarity, the derivative was calculated using smoothed interpolated data of the prevalence.

Figure 5

The prevalence $\rho$ (a) and the derivative of the prevalence with respect to the infection rate $\text{d}\rho /\text{d}\lambda$ (b) are plotted as a function of the infection rate for a network of size $N=200$, $⟨k⟩=4.9$ and rewiring dynamics [Eq. (11)] with $b=3.5$ and $k_0=15$ for which $⟨k^2⟩\approx 73$. Curves for rewiring rates $\nu =0,10,{10}^2,{10}^3,{10}^4$ are represented by full lines from left to right, respectively. For clarity, the derivative was calculated using smoothed interpolated data of the prevalence.

Figure 6

The prevalence as a function of the scaled infection rate ${\lambda }_c⟨k^2⟩$ for a network of different sizes $N$, $⟨k⟩=2$ and rewiring dynamics [Eq. (11)] with $b=2.5$ and $k_0=1$ plotted for different rewiring rates where for each network of size $N$ the rewiring rate is $\nu =⟨k^2⟩$. The behavior near the threshold is replotted on a finer scale and is given in the inset.

Figure 7

The prevalence as a function of the infection rate for a network of size $N=200$, $⟨k⟩=2$ and rewiring dynamics [Eq. (11)] with $b=2.5$ and $k_0=1$. The simulation results for $\nu =0$ and $\nu ={10}^4$ are compared with the numerical solution of Eq. (14).

Figure 8

The prevalence as a function of the infection rate for a network of size $N=200$, $⟨k⟩\approx 4.917$ and rewiring dynamics [Eq. (11)] with $b=3.5$ and $k_0=15$. The simulation results for $\nu =0$ and $\nu ={10}^4$ are compared with the numerical solution of Eq. (14).