Epidemic spreading on time-varying multiplex networks

Social interactions are stratified in multiple contexts and are subject to complex temporal dynamics. The systematic study of these two features of social systems has started only very recently, mainly thanks to the development of multiplex and time-varying networks. However, these two advancements have progressed almost in parallel with very little overlap. Thus, the interplay between multiplexity and the temporal nature of connectivity patterns is poorly understood. Here, we aim to tackle this limitation by introducing a time-varying model of multiplex networks. We are interested in characterizing how these two properties affect contagion processes. To this end, we study susceptible-infected-susceptible epidemic models unfolding at comparable timescale with respect to the evolution of the multiplex network. We study both analytically and numerically the epidemic threshold as a function of the multiplexity and the features of each layer. We found that higher values of multiplexity significantly reduce the epidemic threshold especially when the temporal activation patterns of nodes present on multiple layers are positively correlated. Furthermore, when the average connectivity across layers is very different, the contagion dynamics is driven by the features of the more densely connected layer. Here, the epidemic threshold is equivalent to that of a single layered graph and the impact of the disease, in the layer driving the contagion, is independent of the multiplexity. However, this is not the case in the other layers where the spreading dynamics is sharply influenced by it. The results presented provide another step towards the characterization of the properties of real networks and their effects on contagion phenomena.

Figure 1

(a) The average overlapping degree of the time-integrated multiplex networks as function of the fraction of the coupling nodes $p$. The red line is computed by Eq. (2). (b) The distribution of the integrated overlapping degree for different $p$. In both panels, the activities of the coupling nodes in the two layers are uncorrelated. The exponents for the distributions of activities are ${\gamma }_{\mathbf{x}}=2.1$. The simulations are performed on networks of ${10}^5$ nodes with $m_{\mathbf{x}}=1$, integrated over 100 time steps and averaged over 100 runs.

Figure 2

(a) The lifetime of SIS processes on temporal multiplex networks versus $\lambda$ for the uncorrelated, negative correlated, and positively correlated cases. Therein, ${\gamma }_{\mathbf{x}}=2.1$ and $m_{\mathbf{x}}=1$. The vertical dotted lines with the same colors to the simulation results are the corresponding analytical values obtained from Eq. (7). The simulation results are averaged by ${10}^2$ runs. (b) The analytical thresholds computed by Eq. (7) in the 2D plane (${\gamma }_A,{\gamma }_B$) when $m_{\mathbf{x}}=1$. (c) The analytical thresholds are calculated from Eq. (7) in the 2D plane ($m_A,m_B$) when ${\gamma }_{\mathbf{x}}=2.1$. In both bottom panels we considered uncorrelated activities.

Figure 3

(a) The analytical threshold computed from Eq. (8) versus the number of layers $M$ for the uncorrelated and the positively correlated cases. (b) The lifetime of SIS processes on temporal multiplex networks when $M=3$ for the uncorrelated and the positively correlated cases. The blue dotted line and the red dashed line are the corresponding analytical values. Other parameters are set as ${\gamma }_{\mathbf{x}}=2.1$, $p=1$, and $m_{\mathbf{x}}=1$.

Figure 4

(a) The lifetimes of the SIS processes on the temporal multiplex networks in which the activities of the coupling nodes in the two layers are uncorrelated for different fraction of coupling nodes. (b) The lifetimes of the SIS processes on the temporal multiplex networks in which the activities of the coupling nodes in the two layers are positively correlated for different fraction of coupling nodes. The vertical lines are the corresponding analytical values. Other parameters are set as ${\gamma }_{\mathbf{x}}=2.1$, $m_{\mathbf{x}}=1$.

Figure 5

The analytical threshold ${\lambda }_c$ [Eq. (A11)] is plotted as a function of $p$ for the uncorrelated, negatively correlated, and positively correlated cases. We set $M=2$, ${\gamma }_{\mathbf{x}}=2.1$, and $m_{\mathbf{x}}=1$.

Figure 6

The analytical threshold ${\lambda }_c$ is plotted as a function of $m_B$ when the activities of the coupling nodes in the two layers are uncorrelated and $m_A$ is fixed as 1. We set ${\gamma }_{\mathbf{x}}=2.1$.

Figure 7

(a) and (b), respectively present the asymptotic fraction of infected nodes in layers $A$ and $B$. The results are averaged by ${10}^2$ simulations. We set ${\gamma }_{\mathbf{x}}=2.1$, $m_A=1$, and $m_B=10$.