Abstract When the role of network topology is taken into consideration, one of the objectives is to understand the possible implications of topological structure on epidemic models. As most real networks can be viewed as complex networks, we propose a new delayed SE^{τ}IR^{ω}S epidemic disease model with vertical transmission in complex networks. By using a delayed ODE system, in a small-world (SW) network we prove that, under the condition R_{0} ≤ 1, the disease-free equilibrium (DFE) is globally stable. When R_{0} > 1, the endemic equilibrium is unique and the disease is uniformly persistent. We further obtain the condition of local stability of endemic equilibrium for R_{0} > 1. In a scale-free (SF) network we obtain the condition R_{1} > 1 under which the system will be of non-zero stationary prevalence.

Abstract：
When the role of network topology is taken into consideration, one of the objectives is to understand the possible implications of topological structure on epidemic models. As most real networks can be viewed as complex networks, we propose a new delayed SE^{τ}IR^{ω}S epidemic disease model with vertical transmission in complex networks. By using a delayed ODE system, in a small-world (SW) network we prove that, under the condition R_{0} ≤ 1, the disease-free equilibrium (DFE) is globally stable. When R_{0} > 1, the endemic equilibrium is unique and the disease is uniformly persistent. We further obtain the condition of local stability of endemic equilibrium for R_{0} > 1. In a scale-free (SF) network we obtain the condition R_{1} > 1 under which the system will be of non-zero stationary prevalence.

.Analysis of SE^{τ}IR^{ω}S Epidemic Disease Models with Vertical Transmission in Complex Networks[J] Acta Mathematicae Applicatae Sinica, English Serie, 2012,V28(1): 63-74

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