Epidemic outbreaks in two-scale community networks

We consider a model for the diffusion of epidemics in a population that is partitioned into local communities. In particular, assuming a mean-field approximation, we analyze a continuous-time susceptible-infected-susceptible (SIS) model that has appeared recently in the literature. The probability by which an individual infects individuals in its own community is different from the probability of infecting individuals in other communities. The aim of the model, compared to the standard, nonclustered one, is to provide a compact description for the presence of communities of local infection where the epidemic process is faster compared to the rate at which it spreads across communities. Ultimately, it provides a tool to express the probability of epidemic outbreaks in the form of a metastable infection probability. In the proposed model, the spatial structure of the network is encoded by the adjacency matrix of clusters, i.e., the connections between local communities, and by the vector of the sizes of local communities. Thus, the existence of a nontrivial metastable occupancy probability is determined by an epidemic threshold which depends on the clusters' size and on the intercommunity network structure.

Figure 1

The two-scale community model: the contagion spreads within each cluster node with intracluster infection rate ${\beta }_L$ and among clusters with intercluster infection rate ${\beta }_G$. A link between two clusters means that each node in one cluster is linked with all nodes in the other cluster.

Figure 2

(a) Fraction of infected nodes for different values of $k$ and ${\tau }_G={\beta }_G/\delta$, with fixed ratio ${\beta }_L/{\beta }_G=2$ and the value $\delta =1$, for a network of regular degree $d=10$ and $N=500$. Both the NIMFA and the exact $\varepsilon ={10}^{-3}$ SIS model are shown. The inset plot represents the root mean square error between the simulated and the approximated fraction of infected nodes. (b) The corresponding value of the epidemic threshold for the NIMFA and the exact $\varepsilon$-SIS model.

Figure 3

Difference between the exact $\varepsilon$-SIS model and the NIMFA fractions of infected nodes as a function of ${\tau }_G$ and for different values of $k$, for a network of regular degree $d=10$ and $N=500$.

Figure 4

(a) Fraction of infected nodes for different values of $k$ and ${\tau }_G$, with fixed ratio ${\beta }_L/{\beta }_G=2$, order $c=10$, and the value $\delta =1$. (b) The corresponding value of the epidemic threshold. All graphs have been obtained averaging over 300 instances of Erdős-Rényi random graphs for $p=0.3$; the level of confidence is set to $98\%$.

Figure 5

(a) Fraction of infected nodes as a function of ${\tau }_G$ and ${\tau }_L$ for $k=5$ and $c=10$. (b) Detail of the epidemic threshold.

Figure 6

(a) Fraction of infected nodes for different values of $k$ and ${\tau }_G={\beta }_G/\delta$, with fixed ratio ${\beta }_L/{\beta }_G=2$ and the value $\delta =1$, for an Erdős-Rényi graph of order $c=20$ and $p=0.3$. Both the NIMFA and the exact $\varepsilon ={10}^{-3}$ SIS model are shown. The inset plot represents the root mean square error between the simulated and the approximated fraction of infected nodes. (b) The corresponding value of the epidemic threshold for the NIMFA and the exact $\varepsilon$-SIS model.

Figure 7

Difference between the exact $\varepsilon$-SIS model and the NIMFA fractions of infected nodes as a function of ${\tau }_G$ and for different values of $k$, for an Erdős-Rényi graph of order $c=20$, $p=0.3$.

Figure 8

Difference ${\Delta }_{{\tau }_G}$ between the epidemic threshold in the cases of homogeneous and inhomogeneous cloud distributions for different values of $k$ (5,10,15), with fixed ratio ${\beta }_L/{\beta }_G=8$.

Figure 9

The epidemic threshold in the cases of homogeneous and inhomogeneous cloud distributions for different values of $k$, for a spanning tree of an Erdős-Rényi graph of order $c=10$ and $p=0.3$. Both the NIMFA and the $\varepsilon$-SIS thresholds are shown.

Figure 10

Ring clusters vs fully connected clusters: the difference ${\Delta }_{{\tau }_G}$ between the epidemic threshold in the case of ring clusters and fully connected clusters for different values of $k$.

Figure 11

Epidemic threshold in the case of ring and clique clusters for different values of $k$, for an instance of Erdős-Rényi random graph of order $c=10$ and $p=0.3$. Both the NIMFA and the exact $\varepsilon$-SIS model thresholds are shown.