Collective versus hub activation of epidemic phases on networks

We consider a general criterion to discern the nature of the threshold in epidemic models on scale-free (SF) networks. Comparing the epidemic lifespan of the nodes with largest degrees with the infection time between them, we propose a general dual scenario, in which the epidemic transition is either ruled by a hub activation process, leading to a null threshold in the thermodynamic limit, or given by a collective activation process, corresponding to a standard phase transition with a finite threshold. We validate the proposed criterion applying it to different epidemic models, with waning immunity or heterogeneous infection rates in both synthetic and real SF networks. In particular, a waning immunity, irrespective of its strength, leads to collective activation with finite threshold in scale-free networks with large degree exponent, at odds with canonical theoretical approaches.

Figure 1

Generic epidemic model in complex networks. (a) Nodes in the network can be either healthy or susceptible ($S$), infected ($I$), and immune or recovered ($R$). (b) Infected individuals recover spontaneously, $I\to R$, with rate $\beta$. Recovered individuals become again spontaneously susceptible, $R\to S$, with a rate $\alpha$. Heterogeneous transmission is implemented by infected nodes of degree $k$ transmitting the disease to adjacent susceptible nodes of degree $k^{\prime }$ with rate ${\lambda }_{k,k^{\prime }}$. (c) Schematic representation of a chain of $d$ vertices of arbitrary degrees $k_1,k_2,...,k_d$, connecting two hubs of degree $k$ and $k^{\prime }$. The infection starting in the center of the leftmost star (red) aims to reach the rightmost star (blue). Stubs represent edges that can transmit but cannot receive the infection.

Figure 2

(a) Lifespan for KJI model on star graphs with $\lambda \langle k^{1-\theta }\rangle /\langle k\rangle =0.2$ confirming the stretched exponential asymptotic behavior expected for $\theta <1/2$. (b) Epidemic activation thresholds for the KJI model with different values of $\theta$ and $\gamma =3.5$.

Figure 3

(a) Lifespan for SIRS or SIS dynamics on star graphs with $k$ leaves for $\lambda =0.05, \beta =1$, and different values of $\alpha$. Solid lines are power laws ${\tau }_k\sim k^{\alpha /\beta }$. (b) Infection times of vertices of degree $k^{\prime }$ in a network with $N={10}^5$ vertices where the epidemics starts in a vertex of degree $k=50$ which is never cured. The solid line represents the theoretical value of Eq. (15). (c) Epidemic thresholds against network size for different immunization times $1/\alpha$, using SF networks with degree exponents $\gamma =2.2$ (bottom curves) and $\gamma =2.7$ (top curves). Dashed lines correspond to ${\lambda }_c^{\mathrm{QMF}}=1/{\mathrm{\Lambda }}_m$, where ${\mathrm{\Lambda }}_m$ is the largest eigenvalue of the adjacency matrix.

Figure 4

Epidemic threshold for SIRS model against network size for UCM networks with distinct degree exponents (a) $\gamma =3.5$ and (b) $\gamma =4.0$ with minimal degree $k_0=3$. This SIS limit ($\alpha =\infty$) is also included for comparison.

Figure 5

Simulation of SIS and SIRS dynamics on real networks. Quasistationary density against infection rate in (a) Gowalla and (b) Amazon networks for different waning immunity rates. Arrows indicate the positions of the thresholds obtained via susceptibility method and QMF theory; see Appendix pp2. (c) Degree distribution for Gowalla ($N=196591, k_{\max }=14730, \gamma =2.4$) and Amazon networks ($N=334863, k_{\max }=549, \gamma =3.5$). Solid lines are power-law regressions.

Figure 6

Scatter plot of the largest eigenvalue ${\mathrm{\Lambda }}_m^D$ of the matrix $D_{ij}=A_{ij}/{\left(k_i{k}_j\right)}^{\theta }$ against the degree of the most connected vertex $k_{\max }$ for UCM networks with exponents (a) $\gamma =2.7$ and (b) $\gamma =3.5$ using minimal degree $k_0=3$ and structural cutoff $k_c\sim N^{1/2}$ [29]. The results were computed for five independent networks for $\gamma =3.5$ and 1 for $\gamma =2.7$ (largest degree fluctuates little). Sizes of $N={10}^3, 3\times {10}^3, {10}^4, 3\times {10}^4, {10}^5, 3\times {10}^5, {10}^6, 3\times {10}^6, {10}^7, 3\times {10}^7$, and ${10}^8$ were used. In the bottom panel, solid lines are power laws $k_{\max }^{1/2-\theta }$. In the top panel, solid lines are proportional to ${\mathrm{\Lambda }}_m/k_{\text{max}}^{\theta }$, with ${\mathrm{\Lambda }}_m$ being the numerically estimated largest eigenvalue of the adjacency matrix $A_{ij}$ for $\theta <1/2$.

Figure 7

Susceptibility against infection rate for SIRS dynamics with $\alpha =1.0$. We used UCM networks of different sizes (noted in the labels) with $k_0=3$ and $\gamma =2.7$.