School of Mathematics and Physics, China University of Geosciences, Wuhan, China.
Math Biosci Eng. 2012 Apr;9(2):297-312. doi: 10.3934/mbe.2012.9.297.
In recent years many delay epidemiological models have been proposed to study at which stage of the epidemics the delays can destabilize the disease free equilibrium, or the endemic equilibrium, giving rise to stability switches. One of these models is the SEIR model with constant latency time and infectious periods [2], for which the authors have proved that the two delays are harmless in inducing stability switches. However, it is left open the problem of the global asymptotic stability of the endemic equilibrium whenever it exists. Even the Lyapunov functions approach, recently proposed by Huang and Takeuchi to study many delay epidemiological models, fails to work on this model. In this paper, an age-infection model is presented for the delay SEIR epidemic model, such that the properties of global asymptotic stability of the equilibria of the age-infection model imply the same properties for the original delay-differential epidemic model. By introducing suitable Lyapunov functions to study the global stability of the disease free equilibrium (when R0 ≤ 1) and of the endemic equilibria (whenever R0 > 1) of the age-infection model, we can infer the corresponding global properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in [2] is globally asymptotically stable whenever it exists.
近年来，已经提出了许多延迟流行病学模型来研究在疾病爆发的哪个阶段，延迟可以使无病平衡点或地方病平衡点失稳，从而引发稳定性转换。其中一个模型是具有恒定潜伏期和传染性期的 SEIR 模型[2]，作者已经证明这两个延迟在诱导稳定性转换方面是无害的。然而，对于存在的地方病平衡点的全局渐近稳定性问题仍然没有解决。即使是 Huang 和 Takeuchi 最近提出的用于研究许多延迟流行病学模型的 Lyapunov 函数方法，也不适用于这个模型。在本文中，为延迟 SEIR 传染病模型提出了一个年龄感染模型，使得年龄感染模型平衡点的全局渐近稳定性的性质暗示了原始延迟微分传染病模型的相同性质。通过引入合适的 Lyapunov 函数来研究无病平衡点（当 R0 ≤ 1 时）和年龄感染模型的地方病平衡点（当 R0 > 1 时）的全局稳定性，我们可以推断出[2]中延迟 SEIR 模型平衡点的相应全局性质，从而证明[2]中地方病平衡点在存在时是全局渐近稳定的。
