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Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model

机译:全球Hopf分支和SEIRS流行病模型的持久性

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摘要

In this paper, an SEIRS system with two delays and the general nonlinear incidence rate is considered. The positivity and boundedness of solutions are investigated. The basic reproductive number, R_0, is derived. If R_0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable and the disease dies out. If R_0 > 1, then there exists a unique endemic equilibrium whose locally asymptotical stability and the existence of local Hopf bifurcations are established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived by using the center manifold and the normal form theory. Furthermore, there exists at least one positive periodic solution as the delay varies in some regions by using the global Hopf bifurcation result of Wu for functional differential equations. If R_0 > 1, then the sufficient conditions of the permanence of the system are obtained, i.e., the disease eventually persists in the population. Especially, the upper and lower boundaries that each population can coexist are given exactly. Some numerical simulations are performed to confirm the correctness of theoretical analyses.
机译:本文考虑具有两个时滞和一般非线性入射率的SEIRS系统。研究了溶液的正性和有界性。得出基本生殖数R_0。如果R_0≤1,则无病平衡在全局渐近稳定,并且疾病消失。如果R_0> 1,则存在一个独特的地方均衡,通过分析特征值的分布来建立其局部渐近稳定性和局部Hopf分叉的存在。利用中心流形和法则形式理论推导了确定Hopf分支分支方向和分支周期解的稳定性的显式算法。此外,通过将Wu的全局Hopf分叉结果用于泛函微分方程,随着某些区域的延迟变化,存在至少一个正周期解。如果R_0> 1,则获得系统持久性的充分条件,即该疾病最终在人群中持续存在。特别是,给出了每个人群可以共存的上限和下限。进行了一些数值模拟,以确认理论分析的正确性。

著录项

  • 来源
    《Mathematics and computers in simulation》 |2016年第4期|35-54|共20页
  • 作者单位

    Fundamental Science Department, North China Institute of Astronautic Engineering, Langfang, Hebei, 065000, PR China,Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, PR China;

    Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, PR China;

    Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, PR China;

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  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    SEIRS model; Stability; Global Hopf bifurcation; Permanence; Numerical simulations;

    机译:SEIRS模型;稳定性;全球Hopf分叉;永久性数值模拟;

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