Disease spreading with social distancing: A prevention strategy in disordered multiplex networks

The frequent emergence of diseases with the potential to become threats at local and global scales, such as influenza A(H1N1), SARS, MERS, and recently COVID-19 disease, makes it crucial to keep designing models of disease propagation and strategies to prevent or mitigate their effects in populations. Since isolated systems are exceptionally rare to find in any context, especially in human contact networks, here we examine the susceptible-infected-recovered model of disease spreading in a multiplex network formed by two distinct networks or layers, interconnected through a fraction $q$ of shared individuals (overlap). We model the interactions through weighted networks, because person-to-person interactions are diverse (or disordered); weights represent the contact times of the interactions. Using branching theory supported by simulations, we analyze a social distancing strategy that reduces the average contact time in both layers, where the intensity of the distancing is related to the topology of the layers. We find that the critical values of the distancing intensities, above which an epidemic can be prevented, increase with the overlap $q$. Also we study the effect of the social distancing on the mutual giant component of susceptible individuals, which is crucial to keep the functionality of the system. In addition, we find that for relatively small values of the overlap $q$, social distancing policies might not be needed at all to maintain the functionality of the system.

Figure 1

Scheme of a disordered multiplex network formed by two overlapped layers, $A$ and $B$. The size of the layers is $N_A=N_B=15$, and the fraction of nodes present in both layers is $q=3/15=0.2$ (vertical lines are used as a guide to show the shared nodes, which are represented by boxes). The thickness of the segments represents the diversity of the normalized contact times ${\omega }_{ij}$ between individuals. (a) Initially, all the individuals are in the susceptible (S) state, except for an infected (I) node, which kick-starts the propagation of the disease. (b) At the final stage, the recovered (R) individuals are connected by the branches of infection, which originate from the link denoted by a red arrow. One of the branches, denoted by dotted lines, corresponds to the spread of the disease only through layer $A$ and is represented by the first term in Eq. (1). The other branch, denoted by dash-dotted lines, is a branch of infection that spreads through both layers and is represented by the second term in Eq. (1).

Figure 2

Phase diagrams for $R$ on the $(q,a_A)$ and $(q,a_B)$ planes. (a) Diagrams for an ER layer with different values of $\langle k_A\rangle$ and a SF layer with ${\lambda }_B=2.5$ and cutoff $k_c^B=20$. The solid and black curves represent the critical disorder intensity $a_{Ac}$ (left-hand) and $a_{Bc}$ (right-hand) for $\langle k_A\rangle =3,4,5,6,7$, from bottom to top. (b) Diagrams for an ER layer wih $\langle k_A\rangle =5$ and a SF layer with cutoff $k_c^B=20$ and ${\lambda }_B=2.1,2.3,2.5,2.7,2.9$, from top to bottom. Colored regions below the critical curves indicate epidemic phases ($R>0$), which expand as the overlap $q$ increases. Note that in panel (a) the increase of $\langle k_A\rangle$ causes a major increment on the critical intensities of layer $A$, while in $B$ the increment is smaller. A similar effect occurs in panel (b), when decreasing ${\lambda }_B$. The remaining parameters of the layers are $k_{\min }^A=0$, $k_{\max }^A=20$, $k_{\min }^B=2$, and $k_{\max }^B=250$.

Figure 3

Phase diagrams for GCS on the $(q,a_A)$ and $(q,a_B)$ planes. (a) Diagrams for an ER layer with different values of $\langle k_A\rangle$ and a SF layer with ${\lambda }_B=2.5$ and cutoff $k_c^B=20$. The solid and black curves represent the critical disorder intensity $a_A^{*}$ (left-hand) and $a_B^{*}$ (right-hand) for $\langle k_A\rangle =3,4,5,6,7$, from bottom to top. (b) Diagrams for an ER layer with $\langle k_A\rangle =5$ and a SF layer with cutoff $k_c^B=20$ and ${\lambda }_B=2.1,2.3,2.5,2.7,2.9$, from top to bottom. Colored regions below the critical curves indicate nonfunctional phases ($\text{GCS}=0$), and functional phases ($\text{GCS}>0$) above the curves. Unlike the results shown in the phase diagrams for $R$, in this case the critical intensities may vanish, i.e., $a_A^{*}=a_B^{*}=0$, if the overlap $q$ is relatively small. This means that the functionality of the GCS is ensured, independently of the intensity of the implemented distancing strategy.

Figure 4

Typical outcomes of the social distancing strategy applied to a multiplex network. Solid and dashed lines represent the critical curves for $R$ and GCS, respectively. In region I of the diagram the social distancing policies are rather weak, leading not only to an epidemic, but also to the crumbling of the GCS, which disrupts the functionality of the system (E-NF phase). As the policies are intensified, in region II, the GCS is prevented from falling apart, while the emergence of an epidemic becomes less likely (E-F phase). In region III, a strict enough social distancing policy (i.e., high disorder intensities in both layers) ensures the system lies on a nonepidemic and functional phase (NE-F phase), which is the best possible scenario. For the presented results, we use an ER layer with $\langle k_A\rangle =5$ and a SF layer with ${\lambda }_B=2.5$ and cutoff $k_c^B=20$. Also, we consider a disease with $\beta =0.1$ and $t_r=5$.

Figure 5

Agreement between theoretical and simulated results. Solid and black lines correspond to theoretical results, while simulations are shown in symbols. (a) Fraction $R$ of recovered individuals (red circles) and the size GCS of the mutual giant component of susceptible individuals (green squares) as functions of the disorder intensity $a_A$ in layer $A$, for $q=0.5$. The critical disorder intensities are $a_{Ac}\approx 2.5$, $a_{Bc}\approx 4.5$ for $R$ and $a_A^{*}\approx 0.26$, $a_B^{*}\approx 0.46$ for GCS. (b) $R$ and GCS as functions of the overlap $q$ between the layers of the multiplex network, for $a_A=0.5$. We observe that for $q\approx 0.81$ the GCS collapses, which accounts for the overlap effect over the spread of the disease. For the simulations, we used two layers of size $N={10}^5$. $P_A\left(k\right)$ is a Poisson distribution with $\langle k_A\rangle =4$, $k_{\min }^A=0$ and $k_{\max }^A=40$, while $P_B\left(k\right)$ is a power-law distribution with cutoff $k_C=20$, $\lambda =2$, $k_{\min }^B=2$ and $k_{\max }^B=250$.