具有饱和接触率和垂直传染的SIRS传染病模型分岔分析

数学物理学报 ›› 2014, Vol. 34 ›› Issue (6): 1415-1425.

具有饱和接触率和垂直传染的SIRS传染病模型分岔分析

凌琳|刘苏雨|蒋贵荣

桂林电子科技大学数学与计算科学学院桂林 541004

收稿日期:2013-10-13 修回日期:2014-11-27 出版日期:2014-12-25 发布日期:2014-12-25
基金资助:
国家自然科学基金(11162004)、广西自然科学基金(2012GXNSFAA053006)和广西研究生教育创新计划项目(YCSZ2012072)资助.

Bifurcation Analysis of a SIRS Epidemic Model with Saturating Contact Rate and Vertical Transmission

LING Lin, LIU Su-Yu, JIANG Gui-Rong

School of Mathematics and Computational Science, Guilin University of Electronic Techology, Guilin 541004

Received:2013-10-13 Revised:2014-11-27 Online:2014-12-25 Published:2014-12-25
Supported by:
国家自然科学基金(11162004)、广西自然科学基金(2012GXNSFAA053006)和广西研究生教育创新计划项目(YCSZ2012072)资助.

摘要/Abstract

摘要：

研究了一类具有饱和发生率、脉冲生育、脉冲接种和垂直传染的SIRS传染病模型的复杂动力学行为, 首先构造了一个庞卡莱映射, 然后利用映射的不动点及其特征值, 得到了系统无病周期解的存在和稳定的条件, 接着详细讨论了系统的跨临界分岔、超临界分岔和倍周期分岔现象, 最后给出了能很好验证理论分析的数值结果.

关键词: SIRS模型, 饱和发生率, 超临界分岔, 跨临界分岔, 倍周期分岔

Abstract:

The complex dynamics of a SIRS epidemic model with saturate incidence rate, birth pulse, pulse vaccination and vertical transmission was studied. First, a Poincar\'e map was formulated, the existence and stability of the infection-free periodic
solution were obtained with the help of the fixed point of the map and its eigenvalues. Then transcritical bifurcation,
supercritical bifurcation and flip bifurcation were discussed in detail. Finally, numerical results, which are in good agreement with the theoretical analysis, were presented.

Key words: SIRS epidemic model, Saturating contact rate, Transcritical bifurcation, Supercritical bifurcation, Flip bifurcation

中图分类号:

34A37

凌琳, 刘苏雨, 蒋贵荣. 具有饱和接触率和垂直传染的SIRS传染病模型分岔分析[J]. 数学物理学报, 2014, 34(6): 1415-1425.

LING Lin, LIU Su-Yu, JIANG Gui-Rong. Bifurcation Analysis of a SIRS Epidemic Model with Saturating Contact Rate and Vertical Transmission[J]. Acta mathematica scientia,Series A, 2014, 34(6): 1415-1425.

参考文献

[1] Yang Y, Xiao Y. Threshold dynamics for compartmental epidemic models with impulses. Nonlinear Analysis: Real World Applications, 2012, 13: 224--34
[2] 周艳丽, 王贺桥, 王美娟, 徐长永. 具有脉冲预防接种的SIQR流行病数学模型. 上海理工大学学报, 2007, 29(1): 11--16
[3] Hu Z X, Liu S, Wang H. Backward bifurcation of an epidemic model with standard incidence rate and treatment rate.
Nonlinear Analysis: Real World Applications, 2008, 9: 2302--2312
[4] Zhang J, Ma Z E. Global dynamics of an SEIR epidemic model with saturating contact rate. Mathematical Biosciences,
2003, 185: 15--32
[5] 章培军, 李维德, 朱凌峰. SIRS传染病模型的连续接种和脉冲接种的比较. 兰州大学学报(自然科学版),2011, 47(1): 82--86
[6] 庞国萍, 陈兰荪. 具有饱和传染率的脉冲免疫接种SIRS模型. 系统科学与数学, 2007, 27(4): 563--572
[7] 蒋钰,梅立泉, 宋新宇, 田卫军, 丁雪梅. 具有时滞和脉冲接种的非线性发生率的流行病模型分析. 数学物理学报, 2012, 32A(4): 670--684
[8] Lakmeche A, Arino O. Bifurcation of nontrivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment.Dynamics of Continuous, Discrete and Impulsive System, 2000, 7: 265--287
[9] Tang S Y, Chen L S. Density-dependent birth rate, birth pulses and their population dynamics consequences. Journal of Mathematical Biology, 2002, 190: 185--199
[10] Gakkhar S, Negi K. Pulse vaccination in SIRS epidemic model with non monotonic incidence rate, Chaos, Solitons and Fractals, 2008, 35: 626--638
[11] 钱临宁, 陆启韶. 一类自治脉冲微分方程的动力学研究. 动力学与控制学报, 2008, 6(2): 97--101
[12] Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1983