Epidemic spreading and risk perception in multiplex networks: A self-organized percolation method

In this paper we study the interplay between epidemic spreading and risk perception on multiplex networks. The basic idea is that the effective infection probability is affected by the perception of the risk of being infected, which we assume to be related to the fraction of infected neighbors, as introduced by Bagnoli et al. [Phys. Rev. E 76, 061904 (2007)]. We rederive previous results using a self-organized method that automatically gives the percolation threshold in just one simulation. We then extend the model to multiplex networks considering that people get infected by physical contacts in real life but often gather information from an information network, which may be quite different from the physical ones. The similarity between the physical and the information networks determines the possibility of stopping the infection for a sufficiently high precaution level: if the networks are too different, there is no means of avoiding the epidemics.

Figure 1

Example of a multiplex generated with our method. We start with the physical and the virtual networks, both symmetric and with the same average connectivity, and then we build the information network by choosing, for each node, outgoing links with probability $q$ from the virtual network and with probability $1-q$ from the real network.

Figure 2

Comparison between mean-field approximation and simulations for random networks with different values of $\langle k\rangle$.

Figure 3

Comparison between results of the mean-field approximation (curves) and results of simulations (symbols) for three instances of scale-free networks with $\gamma =2.4,N=10000,m=2$, and $K=300$. Although the simulations give similar results, the mean-field computations (that coincide with the simulation only for $J_c=0$) are very dependent on the details of the network.

Figure 4

Evolution of the local minimum value of the percolation parameter $p_i$ for a 1D regular network with $k=2$.

Figure 5

Asymptotic number of infected individuals $c$ versus the bare infection probability $\tau$ for the SIS dynamics for different networks. From left to right, for $c=0$: scale-free (SF), random (Poisson), and regular. Here $N=10000$.

Figure 6

Critical precaution threshold $J_c$ (gray intensity or color code) as a function of the bare infection $\tau$ and of the difference $q$ between the physical and the information network. Here the physical and virtual networks are Poissonian (random), with $\langle k\rangle =6$ and $N=1000$. In the darker region there is always a value of $J_c$ able to stop the epidemic, while in the white region the epidemic cannot be stopped.

Figure 7

Critical precaution threshold $J_c$ versus the difference between the physical and the information network $q$ for some values of the bare infection $\tau$ (from right to left $\tau =0.1$, 0.2, 0.3). Random physical network and scale-free virtual network, both with $\langle k\rangle =6,N=10000$.

Figure 8

Critical precaution threshold $J_c$ versus the difference between the physical and the information network $q$ for some values of the bare infection $\tau$ (from right to left: $\tau =0.2,0.3,\cdots ,0.6$). Random physical and virtual networks, both with $\langle k\rangle =6$ and $N=10000$.