College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, PR China.
J Biol Dyn. 2008 Jan;2(1):64-84. doi: 10.1080/17513750801894845.
A delayed SEIRS epidemic model with pulse vaccination and saturation incidence rate is investigated. Using Krasnoselskii's fixed-point theorem, we obtain the existence of infection-free periodic solution of the impulsive delayed epidemic system. We define some new threshold values R(1), R(2) and R(3). Further, using the comparison theorem, we obtain the explicit formulae of R(1) and R(2). Under the condition R(1) < 1, the infection-free periodic solution is globally attractive, and that R(2) > 1 implies that the disease is permanent. Theoretical results show that the disease will be extinct if the vaccination rate is larger than θ* and the disease is uniformly persistent if the vaccination rate is less than θ(*). Our results indicate that a long latent period of the disease or a large pulse vaccination rate will lead to eradication of the disease. Moreover, we prove that the disease will be permanent as R(3) > 1.
我们研究了具有脉冲接种和饱和发生率的时滞 SEIRS 传染病模型。利用 Krasnoselskii 的不动点定理，我们得到了脉冲时滞传染病系统无感染周期解的存在性。我们定义了一些新的阈值 R(1)、R(2) 和 R(3)。进一步，利用比较定理，我们得到了 R(1) 和 R(2) 的显式公式。在 R(1) < 1 的条件下，无感染周期解是全局吸引的，而 R(2) > 1 意味着疾病是持久性的。理论结果表明，如果接种率大于 θ*，疾病将被消灭，如果接种率小于 θ(*),疾病将均匀持续存在。我们的结果表明，疾病的长潜伏期或大脉冲接种率将导致疾病的消灭。此外，我们证明了当 R(3) > 1 时，疾病将是永久性的。
