Multistage onset of epidemics in heterogeneous networks

We develop a theory for the susceptible-infected-susceptible (SIS) epidemic model on networks that incorporate both network structure and dynamic correlations. This theory can account for the multistage onset of the epidemic phase in scale-free networks. This phenomenon is characterized by multiple peaks in the susceptibility as a function of the infection rate. It can be explained by that, even under the global epidemic threshold, a hub can sustain the epidemics for an extended period. Moreover, our approach improves theoretical calculations of prevalence close to the threshold in heterogeneous networks and also can predict the average risk of infection for neighbors of nodes with different degree and state on uncorrelated static networks.

Figure 1

Schematic illustration of one state cycle of the center node on a star subnetwork with a nonzero fraction of permanently infected leaf nodes.

Figure 2

$p_k$ as a function of degree $k$ on a scale-free network. These panels show simulations on one realization of uncorrelated configuration model with minimum degree 3, $\gamma =4.5,$ and $N={10}^7$. Parameters: $g=1$, (a) $r=0.22$; (b) $r=0.25$; (c) $r=0.28$; (d) $r=0.3$; (e) $r=0.8$, the relaxation time $t_r={10}^6$ in panels (a–d) and $t_r=5\times {10}^3$ in (e). The shaded areas in panels (a–d) indicate the position where Eq. (11) is valid.

Figure 3

(a–d) The prevalence $\rho$ and (e) the ratio $\rho /x$ as a function of infection rate $r$ on uncorrelated static scale-free networks. Parameters are $g=1$, minimum degree 3, $N={10}^7$; $\gamma =4.5$ (a) and (c); $\gamma =3.5$ (b) and (d). Panels (c) and (d) are blow-ups of the boxed regions of panels (a) and (b), respectively. Simulation results are averaged over 20 independent runs. The values of panel (e) are from our theoretical results in panels (a) and (b).

Figure 4

Panel (a) illustrates a network consisting of two star subnetworks a distance $l$ apart. Dashed lines show the paths between hubs. Panels (b) and (c) show the susceptibility $\chi$ and prevalence $\rho$ as a function of the effective infection rate $\lambda$ for the network in panel (a). The parameters are $k_1=1000$ and $k_2=300$. Simulations on one realization with the relaxation time $t_r={10}^6$.

Figure 5

The average lifetime $T$ as a function of the $\lambda$ on a $k$-star network. The degree of the center node is $k=300$. Simulation results are averaged over $5\times {10}^4$ independent runs.

Figure 6

The susceptibility $\chi$ as a function of the effective infection rate $\lambda$ on single realizations of a scale-free network. Parameters: minimum degree 3; (a) $\gamma =4.5$, $N={10}^7$, $k_{max}=314$, and $\langle l\rangle =6.841$; (b) $\gamma =3.5$, $N=3\times {10}^7$, $k_{max}=3611$, and $\langle l\rangle =2.789$. The network of (a) is the same as that in Fig. 2. To get smooth simulation curves, the relaxation time $t_r$ are varied from ${10}^5$ to ${10}^9$ depending on $N$ and $\lambda$. We show several analytical estimates of the transition point.

Figure 7

The epidemic prevalence $\rho$ in the SIS process is plotted as a function of the effective infection rate on annealed and static scale-free networks. Parameters: $N={10}^5$, minimum degree 3 and $\gamma =4.5$. Lines are from Eq. (A1), while the points are simulation results.

Figure 8

Epidemic spreading on a star network with one permanently infected leaf node. The probabilities $P\left(m\right)$ and $P\left(n\right)$ is plotted as a function of the number of infected leaf nodes when the center node is cured and is infected. Parameters: $r=0.2$, $g=1$, network size $N=1000$.