Epidemic spread on interconnected metapopulation networks

Numerous real-world networks have been observed to interact with each other, resulting in interconnected networks that exhibit diverse, nontrivial behavior with dynamical processes. Here we investigate epidemic spreading on interconnected networks at the level of metapopulation. Through a mean-field approximation for a metapopulation model, we find that both the interaction network topology and the mobility probabilities between subnetworks jointly influence the epidemic spread. Depending on the interaction between subnetworks, proper controls of mobility can efficiently mitigate epidemics, whereas an extremely biased mobility to one subnetwork will typically cause a severe outbreak and promote the epidemic spreading. Our analysis provides a basic framework for better understanding of epidemic behavior in related transportation systems as well as for better control of epidemics by guiding human mobility patterns.

Figure 1

Schematic illustration of an interconnected network composed of two subnetworks which are connected by interconnected links (black dashed). Nodes in each subnetwork follow a specific intradegree distribution and an interdegree distribution. Inside each node, individuals mix homogeneously. Each individual takes one of two states, $S$ or $I$.

Figure 2

Time courses of prevalences in subnetwork $M_1$ (lower lines) and in the entire system (upper lines) with different mobility probabilities ${\tau }^{M_1}$ and ${\tau }^{M_2}$. The prevalence in subnetwork $M_2$ is equivalent to that in subnetwork $M_1$. The population density is fixed at ${\rho }_0=3$. The basic epidemiological parameters in the two subnetworks are the same and set at $R_0^{M_1}=R_0^{M_2}(=\frac{{\beta }^{M_2}}{{\mu }^{M_2}})=1$. The two isolated subnetworks follow the same degree distribution $P\left(k\right)\sim k^{-2.1}$ with $k_{\min }=3$. Interconnected links are added at random with the average connectivity $\langle k_{M_1{M}_2}\rangle =5$.

Figure 3

Final prevalence in an interconnected network composed of two identical subnetworks for different combinations of mobility probabilities ${\tau }^{M_1}$ and ${\tau }^{M_2}$. The solutions of mean-field Eq. (4) (solid) are in good agreement with those of the agent-based model. The insets show the enlargement scale around the threshold. (a) $R_0^{M_1}(=\frac{{\beta }^{M_1}}{{\mu }^{M_1}})=R_0^{M_2}(=\frac{{\beta }^{M_2}}{{\mu }^{M_2}})=1$; (b) $R_0^{M_1}=2R_0^{M_2}=1$. The other parameters are the same as in Fig. 2.

Figure 4

Final prevalence vs. ${\tau }^{M_1}$ and ${\tau }^{M_2}$ in an interconnected network composed of two identical subnetworks. (a) $R_0^{M_1}(=\frac{{\beta }^{M_1}}{{\mu }^{M_1}})=R_0^{M_2}(=\frac{{\beta }^{M_2}}{{\mu }^{M_2}})=1$; (b) $R_0^{M_1}=2R_0^{M_2}=1$. The other parameters are the same as in Fig. 2.

Figure 5

Distribution of infectious individuals at patches with degree vector $(k_{\mathrm{IN}},k_{\mathrm{BW}})$ on two interconnected networks. The two subnetworks are generated with the configuration network model by following the degree distribution $P\left(k\right)\sim k^{-2.1}$ with $k_{\min }=3$, while the interconnected links are added at random with the average interconnectivity $\langle k_{M_1{M}_2}\rangle =15$. The population density is set at ${\rho }_0=2$. The other parameters are the same as in Fig. 2.

Figure 6

Distribution of infectious individuals at patches with degree vector $(k_{\mathrm{IN}},k_{\mathrm{BW}})$ on two interconnected networks. Both the two subnetworks and the interconnected links are generated with the configuration network model following the degree distribution $P\left(k\right)\sim k^{-2.1}$ with $k_{\min }=3$. The population density is set at ${\rho }_0=2$. The other parameters are the same as in Fig. 2.

Figure 7

The invasion thresholds vs ${\tau }^{M_2}$ and ${\tau }^{M_1}$ in an interconnected network composed of two identical subnetworks for different $Q_{12}=\frac{q_{12}}{q_{11}}$ with $q_{11}=\frac{k_{\max }^{M_1{M}_1}}{\langle k_{M_1{M}_1}\rangle }$ and $q_{12}=\frac{k_{\max }^{M_1{M}_2}}{\langle k_{M_1{M}_2}\rangle }$. (a) $Q_{12}=0.5$; (b) $Q_{12}=1$; (c) $Q_{12}=1.5$; (d) $Q_{12}=2$; (e) $Q_{12}=2.5$; (f) $Q_{12}=3$. In the two isolated networks, the maximum connectivity is $k_{\max }=44$, the average connectivity is $\langle k\rangle =6$, and ${\rho }_0^{M_1}={\rho }_0^{M_2}$. The basic reproductive number for the epidemiological process in each subnetwork is set to be the same, i.e., $R_0^{M_1}=R_0^{M_2}=1$. The other parameters are the same as in Fig. 2.

Figure 8

Final prevalence versus ${\tau }^{M_1}$ and ${\tau }^{M_2}$ on two interconnected networks with $Q_{12}=0.8$. The color represents the final prevalence and the cross points are the theoretical values obtained from Eq. (13). The population densities are set at ${\rho }_0^{M_1}={\rho }_0^{M_2}={\rho }_0$ with ${\rho }_0=1.2$; the population densities are less than the invasion thresholds in isolated networks, i.e., ${\rho }_0^{M_1}<{\widehat{\rho }}_{0c}^{M_1}$ and ${\rho }_0^{M_2}<{\widehat{\rho }}_{0c}^{M_2}$. The cross points separate regions of the disease-free state, the endemic state only in subnetwork $M_1$, and the endemic state only in subnetwork $M_2$, respectively, due to the mobility from the neighboring subnetwork. The two same subnetworks are generated with the configuration network model following the degree distribution $P\left(k\right)\sim k^{-2.1}$ with $k_{\min }=3$. The interconnected links are added at random with the average interconnectivity $\langle k_{M_1{M}_2}\rangle =6.6$. Each color value and point correspond to the average results over 50 simulations based on five pairs of subnetworks with 10 times generation of randomly interconnected links. The epidemic parameters are set at $\beta =0.2$ and $\mu =1$.

Figure 9

Final prevalence versus ${\tau }^{M_1}$ and ${\tau }^{M_2}$ on two interconnected networks with $Q_{12}=4.2$. The color represents the final prevalence and the cross points are the theoretical values obtained from Eq. (13). The population densities are set at ${\rho }_0^{M_1}={\rho }_0^{M_2}={\rho }_0$ with ${\rho }_0=2.2$; these densities are less than the invasion thresholds in isolated networks, i.e., ${\rho }_0^{M_1}<{\widehat{\rho }}_{0c}^{M_1}$ and ${\rho }_0^{M_2}<{\widehat{\rho }}_{0c}^{M_2}$. The cross points separate regions of the disease-free state, the endemic states in subnetworks $M_1$ and $M_2$, or in both due to mobility from the neighboring subnetwork. Two same subnetworks are generated with the Erdős-Rényi (ER) network model with $\langle k\rangle =6.6$. The interconnected links are generated with the configuration network model following the degree distribution $P\left(k\right)\sim k^{-2.1}$ with $k_{\min }=3$. Each color value and point correspond to the average results over 50 simulations based on five pairs of subnetworks with 10 times generation of randomly interconnected links. The epidemic parameters are set at $\beta =0.2$ and $\mu =1$.

Figure 10

Schematic illustration of an interconnected network composed of a regular two-dimensional lattice with length $L$ and a subnetwork with long-range links. These long-range links are part of the network generated with a geographically embedded scale-free network. (a) Embedded scale-free networks on the lattice following the degree distribution $P\left(k\right)\sim k^{-\gamma }$. (b) Subnetwork $M_2$ which is the lattice. (c) Subnetwork $M_1$ where the nodes that are connected with long-range links. (d) The cumulative degree distribution of subnetwork $M_1$ with different exponents $\gamma =2.5$ and $\gamma =3.5$ in log-log scale with $A=10$. Individuals within each patch may move to the neighboring patches located on the lattice (subnetwork $M_2$) or to those connected by long-range links (subnetwork $M_1$).

Figure 11

Final prevalence in an interconnected network composed of a regular lattice (subnetwork $M_2$) and a subnetwork $M_1$ with long-range links. These long-range links are part of the network generated from a geographically embedded scale-free network (subnetwork $M_1$ with $A=10$, $P\left(k\right)\sim k^{-\gamma }$, $k_{\min }=6$, and $\gamma =2.5$). The epidemiological parameters are set at $R_0^{M_1}=R_0^{M_2}=1$. Initially, $10\%$ of the individuals in each network get infected at random. The average connectivities in subnetworks $M_1$ and $M_2$ are $\langle k_{M_1}\rangle =9.5$ and $\langle k_{M_2}\rangle =4$, respectively.

Figure 12

Final prevalence vs ${\tau }^{M_1}$ and ${\tau }^{M_2}$ in an interconnected network composed of a regular lattice (subnetwork $M_2$) and a subnetwork $M_1$ with long-range links. The other parameters are the same as in Fig. 11.