An SEIR epidemic model with an arbitrarily distributed exposed stage is revisited to study the impact of heterogeneity on the spread of infectious diseases. The heterogeneity may come from age or behavior and disease stages, resulting in multi-group and multi-stage models, respectively. For each model, Lyapunov functionals are used to show that the basic reproduction number $\mathcal{R}_0$ gives a sharp threshold. If $\mathcal{R}_0\leq 1$, then the disease-free equilibrium is globally asymptotically stable and the disease dies out from all groups or stages. If $\mathcal{R}_0>1$, then the disease persists in all groups or stages, and the endemic equilibrium is globally asymptotically stable.
Citation: Zhisheng Shuai, P. van den Driessche. Impact of heterogeneity on the dynamics of an SEIR epidemic model[J]. Mathematical Biosciences and Engineering, 2012, 9(2): 393-411. doi: 10.3934/mbe.2012.9.393
Abstract
An SEIR epidemic model with an arbitrarily distributed exposed stage is revisited to study the impact of heterogeneity on the spread of infectious diseases. The heterogeneity may come from age or behavior and disease stages, resulting in multi-group and multi-stage models, respectively. For each model, Lyapunov functionals are used to show that the basic reproduction number $\mathcal{R}_0$ gives a sharp threshold. If $\mathcal{R}_0\leq 1$, then the disease-free equilibrium is globally asymptotically stable and the disease dies out from all groups or stages. If $\mathcal{R}_0>1$, then the disease persists in all groups or stages, and the endemic equilibrium is globally asymptotically stable.
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