Traffic-driven epidemic spreading in multiplex networks

Recent progress on multiplex networks has provided a powerful way to abstract the diverse interaction of a network system with multiple layers. In this paper, we show that a multiplex structure can greatly affect the spread of an epidemic driven by traffic dynamics. One of the interesting findings is that the multiplex structure could suppress the outbreak of an epidemic, which is different from the typical finding of spread dynamics in multiplex networks. In particular, one layer with dense connections can attract more traffic flow and eventually suppress the epidemic outbreak in other layers. Therefore, the epidemic threshold will be larger than the minimal threshold of the layers. With a mean-field approximation, we provide explicit expressions for the epidemic threshold and for the onset of suppressing epidemic spreading in multiplex networks. We also provide the probability of obtaining a multiplex configuration that suppresses the epidemic spreading when the multiplex is composed of: (i) two Erdős-Rényi layers and (ii) two scale-free layers. Therefore, compared to the situation of an isolated network in which a disease may be able to propagate, a larger epidemic threshold can be found in multiplex structures.

Figure 1

Schematic of the multiplex networks. Without loss of generality, here the multiplex network is assumed to be double layered, labeled $A$ and $B$, respectively. For traffic dynamics, elements travel from the origins to the destinations following the shortest paths. For example, if an element travels from $i$ to $j$, the shortest path is $i^A\to j^A$ with length 1. Since only layer $A$ is used, this is an intralayer path. Another example, the path length is 3 from $j$ to $k$, i.e., $j^A\to i^A\to i^B\to k^B$. Obviously, this is an interlayer path.

Figure 2

Comparison of the threshold ${\beta }_c$ obtained by theoretical analysis and by experimental simulation. The multiplex networks used in the simulation are formed by two Erdős-Rényi networks ($L=2$) with $N=500$. For the traffic dynamics, the element generation rate is set as $r=1$. The results are obtained by Eq. (8) (main figure), Eq. (6) (lower right), and Eq. (8) with $\lambda \approx 1$ (upper left), respectively. $R^2$ is the coefficient of determination for the linear fitting. Equation (8) shows the best prediction with $R^2=0.973$.

Figure 3

Fraction of infected nodes $\rho$ as a function of infected probability $\beta$ for two typical Erdős-Rényi multiplex networks formed with two layers of different average degrees: (a) $\langle k^A\rangle =10$ and $\langle k^B\rangle =20$; (b) $\langle k^A\rangle =15$ and $\langle k^B\rangle =30$. Other parameters are $N=500$ and $r=1$.

Figure 4

Probability of obtaining a multiplex configuration that suppresses epidemic spreading. The number of points is ${70}^2$, and for each point, 100 configurations are generated with fixed $\langle k^A\rangle$ and $\langle k^B\rangle$. The colors indicate the probability of ${\beta }_c>min({\beta }_c^{\left(A\right)},{\beta }_c^{\left(B\right)})$. The lines show the result of Eq. (13) to detect the boundaries [the solid lines are obtained by Eq. (13) with the real value of $\lambda$, and the dashed lines are obtained with $\lambda =1]$. Other parameters are $L=2,N=500$, and $r=1$.

Figure 5

Probability of obtaining a scale-free multiplex configuration that suppresses epidemic spreading. The number of simulation points is ${30}^2$, and, for each point, we generate 100 configurations fixing $k_{min}^A$ and $k_{min}^B$. The colors indicate the probability of observing that the epidemic threshold of the multiplex satisfies ${\beta }_c>min({\beta }_c^{\left(A\right)},{\beta }_c^{\left(B\right)})$. The lines show the results of Eq. (13) in detecting the region where the multiplex structure induces an epidemic.