Contagion dynamics in time-varying metapopulation networks

The metapopulation framework is adopted in a wide array of disciplines to describe systems of well separated yet connected subpopulations. The subgroups or patches are often represented as nodes in a network whose links represent the migration routes among them. The connections have been so far mostly considered as static, but in general evolve in time. Here we address this case by investigating simple contagion processes on time-varying metapopulation networks. We focus on the SIR process and determine analytically the mobility threshold for the onset of an epidemic spreading in the framework of activity-driven network models. We find profound differences from the case of static networks. The threshold is entirely described by the dynamical parameters defining the average number of instantaneously migrating individuals and does not depend on the properties of the static network representation. Remarkably, the diffusion and contagion processes are slower in time-varying graphs than in their aggregated static counterparts, the mobility threshold being even two orders of magnitude larger in the first case. The presented results confirm the importance of considering the time-varying nature of complex networks.

Figure 1

Schematic representation of a metapopulation system and its global threshold. In panel (a), we show, on the left, the diffusion process on the macroscopic level. Red (gray) nodes represent subpopulations experiencing a local outbreak. White nodes represent subpopulation not yet reached by the disease. Nodes with red (gray) center and white outer layer instead represent subpopulations that have been seeded by diseased population through the thick links connecting them. On the right we show the microscopic diffusion process. Each subpopulation contains individuals divided into three compartments according to their disease status. White are susceptible (white), red (light gray) infectious, and dark gray recovered. The arrows represent the flows of migration coupling the internal dynamics. In panel (b), we show the critical value of the mobility rate $p^{*}$. Below this point, just an infinitesimal fraction of subpopulations is affected by the disease. For values equal or larger than the critical point, the system reaches a regime in which a finite fraction of subpopulations is affected by the spreading.

Figure 2

Schematic representation of the macroscopic dynamics in time-varying metapopulation networks. In the first time step, $T=1$, just the subpopulation $i$ is diseased (red/gray node). It is the source of the infection. By chance it becomes active and creates $m=1$ random links. Through this connection (thick line), some infected individuals move to the subpopulation $j$, which will start to experience the disease at the next time step. Next, for $T=2$, three other nodes are active. All the selected nodes are not diseased, so no infected individuals diffuse. At the time step, $T=3$, node $i$ is active, again infecting another node. The process is repeated for all the other time steps, driven by the activity of the each node.

Figure 3

In panel (a), we show the number of diseased nodes in a metapopulation network, $D_{\infty }$, as a function of mobility rate $p$. We consider $V={10}^5$ subpopulations, $\langle N\rangle ={10}^3$ individuals per subpopulation, $m=1$, $\epsilon ={10}^{-3}$, and $\eta =1$. We select a power-law distribution of activity potential $F\left(x\right)\sim x^{-\gamma }$ with $\gamma =2.2$. We then consider $\mu ={10}^{-2}$ and $R_0=1.1$. The triangle represents the analytical prediction of the invasion threshold, according to Eq. (32). In panel (b), we show the phase space of a SIR process. Considering $V={10}^4$ subpopulations, $\langle N\rangle ={10}^3$ individuals per subpopulation, $m=1$, $\epsilon ={10}^{-3}$, and $\eta =1$. We select a power-law distribution of activity potential $F\left(x\right)\sim x^{-\gamma }$ with $\gamma =2.2$. The 3D surface represents the value of the final number of diseased nodes in the metapopulation system as a function of the local threshold $R_0$ and of the diffusion rate $p$. The red (light gray) curve defines the global diffusion threshold $p_c$ for each $R_0$. Each plot is made averaging ${10}^2$ independent simulations. Each one of them started with 1$\%$ of random seeds.

Figure 4

In panel (a), we show the number of diseased nodes in a metapopulation network, $D_{\infty }$, as a function of mobility rate $p$. We consider $V={10}^5$ subpopulations, $\langle N\rangle ={10}^3$ individuals per subpopulation, $m=3$, $\epsilon ={10}^{-3}$, and $\eta =1$. We select a power-law distribution of activity potential $F\left(x\right)\sim x^{-\gamma }$ with $\gamma =2.2$. We then consider $\mu ={10}^{-2}$ and $R_0=1.1$. The triangle represents the analytical prediction of the invasion threshold according to Eq. (35). In panel (b), we show the phase space of a SIR process. Considering $V={10}^4$ subpopulations, $\langle N\rangle ={10}^3$ individuals per subpopulation, $m=3$, $\epsilon ={10}^{-3}$, and $\eta =1$. We select a power-law distribution of activity potential $F\left(x\right)\sim x^{-\gamma }$ with $\gamma =2.2$. The 3D surface represents the value of the final number of diseased nodes in the metapopulation system as a function of the local threshold $R_0$ and of the diffusion rate $p$. The red (light gray) curve defines the global diffusion threshold $p_c$ for each $R_0$. Each one of them started with 1$\%$ of random seeds.

Figure 5

The plot shows the number of diseased nodes in a metapopulation network, $D_{\infty }$, as a function of mobility rate $p$ on activity-driven networks (dots) and corresponding integrated, static ones (squares). The integration has been performed considering 50 time steps. We fix $V={10}^4$ subpopulations, $\langle N\rangle ={10}^3$ individuals per subpopulation, $m=3$, $\epsilon ={10}^{-3}$, and $\eta =1$. We select a power-law distribution of activity potential $F\left(x\right)\sim x^{-\gamma }$, with $\gamma =2.2$. We then consider $\mu ={10}^{-2}$ and $R_0=1.1$. Each plot is made averaging ${10}^2$ independent simulations. Each one of them started with 1$\%$ of random seeds.