Epidemic spread in directed interconnected networks. (English) Zbl 1515.92074
The authors are concerned with the spread of epidemic in directed interconnected networks. The research is motivated by the fact that many networks interact, including those describing local and global spread of various infections. Furthermore, the spread of some infections like brucellosis is directional because no cases of transmission of the infection between humans were recorded and they are very unlikely to infect animals, but humans can be infected by exposure to infected livestock and livestock products. Therefore, it makes sense to develop models describing the dynamics of epidemic on directed networks in addition to existing models suggested for undirected networks.
In this paper, an SIS model is developed using the mean-field approach. It includes undirected networks, single-layer networks, and bipartite networks as particular cases. The network under study consists of two interconnected directed subnetworks A (for humans) and B (for animals). The nodes in A are of the same type but distinct from the type in B. Connections in intra-subnetworks and inter-subnetworks are realized as directed edges. For any node in the network, the attached directed edges fall into one of the following four types: (1) outgoing edges pointing at the nodes in the subnetwork A; (2) incoming edges directed from the nodes in the subnetwork A; (3) outgoing edges pointing at the nodes in the subnetwork B; and (4) incoming edges directed from the nodes in the subnetwork B. The resulting directed interconnected network is described by the system of four differential equations.
The analysis starts with the calculation of the basic reproduction number \(\mathcal{R}_{0}\) based on the use of a \(4\times4\) next generation matrix \(\Gamma.\) Two particular cases are considered, (i) an undirected network and (ii) the network where the joint degree distribution is independent. In both cases \(\Gamma\) is an irreducible non-negative matrix which, in virtue of Perron-Frobenius theorem, should have a positive real eigenvalue equal to the spectral radius \(\rho\left( \Gamma\right) .\) Theorem 1 describes the dynamics of the system in terms of \(\mathcal{R}_{0}\) – when \(\mathcal{R} _{0}\leq1,\) the disease free equilibrium is globally asymptotically stable and for \(\mathcal{R}_{0}>1\) the unique endemic equilibrium is globally asymptotically stable. Theorem 2 deals with the case when the disease is prevalent in one of the subnetworks. Numerical simulations on different networks are performed to illustrate theoretical results and demonstrate the impact of the network’s topological structure on the spread of disease.
In this paper, an SIS model is developed using the mean-field approach. It includes undirected networks, single-layer networks, and bipartite networks as particular cases. The network under study consists of two interconnected directed subnetworks A (for humans) and B (for animals). The nodes in A are of the same type but distinct from the type in B. Connections in intra-subnetworks and inter-subnetworks are realized as directed edges. For any node in the network, the attached directed edges fall into one of the following four types: (1) outgoing edges pointing at the nodes in the subnetwork A; (2) incoming edges directed from the nodes in the subnetwork A; (3) outgoing edges pointing at the nodes in the subnetwork B; and (4) incoming edges directed from the nodes in the subnetwork B. The resulting directed interconnected network is described by the system of four differential equations.
The analysis starts with the calculation of the basic reproduction number \(\mathcal{R}_{0}\) based on the use of a \(4\times4\) next generation matrix \(\Gamma.\) Two particular cases are considered, (i) an undirected network and (ii) the network where the joint degree distribution is independent. In both cases \(\Gamma\) is an irreducible non-negative matrix which, in virtue of Perron-Frobenius theorem, should have a positive real eigenvalue equal to the spectral radius \(\rho\left( \Gamma\right) .\) Theorem 1 describes the dynamics of the system in terms of \(\mathcal{R}_{0}\) – when \(\mathcal{R} _{0}\leq1,\) the disease free equilibrium is globally asymptotically stable and for \(\mathcal{R}_{0}>1\) the unique endemic equilibrium is globally asymptotically stable. Theorem 2 deals with the case when the disease is prevalent in one of the subnetworks. Numerical simulations on different networks are performed to illustrate theoretical results and demonstrate the impact of the network’s topological structure on the spread of disease.
Reviewer: Yuriy V. Rogovchenko (Kristiansand)
MSC:
| 92D30 | Epidemiology |
| 34B45 | Boundary value problems on graphs and networks for ordinary differential equations |
| 92C42 | Systems biology, networks |
Keywords:
directed interconnected network; epidemic; basic reproduction number; equilibria; global asymptotic stabilityReferences:
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