一类具有接种和潜伏期的传染病模型及动力学分析

数学物理学报 ›› 2019, Vol. 39 ›› Issue (5): 1247-1259.

一类具有接种和潜伏期的传染病模型及动力学分析

张鑫喆,贺国峰,黄刚*()

中国地质大学数学与物理学院武汉 430074

收稿日期:2018-11-28 出版日期:2019-10-26 发布日期:2019-11-08
通讯作者: 黄刚 E-mail:huanggang@cug.edu.cn
基金资助:
国家自然科学基金(11571326)

Dynamical Properties of a Delayed Epidemic Model with Vaccination and Saturation Incidence

Xinzhe Zhang,Guofeng He,Gang Huang*()

School of Mathematics and Physics, China University of Geosciences, Wuhan 430074

Received:2018-11-28 Online:2019-10-26 Published:2019-11-08
Contact: Gang Huang E-mail:huanggang@cug.edu.cn
Supported by:
the NSFC(11571326)

摘要/Abstract

摘要：

该文提出并研究了一类具有接种和非线性感染率的SVEIR传染病模型，其中时滞用来刻画疾病的潜伏期.讨论了模型平衡点的存在性和局部稳定性以及系统的一致持续性.进一步，通过构造合适的Lyapunov泛函得到平衡点的全局渐近稳定性：当基本再生数小于或等于1，无病平衡点全局渐近稳定，此时疾病将会消除；当基本再生数大于1时，地方病平衡点全局渐近稳定，此时疾病将会持续流行.最后，通过数值模拟验证了前面的理论分析结果，并对疾病的传播和控制给出了合理建议.

关键词: 疫苗接种, 潜伏期, Lyapunov泛函, 全局渐近稳定

Abstract:

In this paper, we propose and study a delayed SVEIR epidemic model with vaccination and saturation incidence. The existence and local stability of equilibria are addressed. By using Lyapunov functionals and Lyapunov-LaSalle invariance principle, it shows that if the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable and the disease will disappear; and if the basic reproduction number is greater than one, the endemic equilibrium is globally asymptotically stable and the disease will persist. Some numerical simulations are performed to illustrate our analytic results.

Key words: Vaccination, Latent period, Lyapunov functional, Globally asymptotically stable

中图分类号:

O175.1

张鑫喆,贺国峰,黄刚. 一类具有接种和潜伏期的传染病模型及动力学分析[J]. 数学物理学报, 2019, 39(5): 1247-1259.

Xinzhe Zhang,Guofeng He,Gang Huang. Dynamical Properties of a Delayed Epidemic Model with Vaccination and Saturation Incidence[J]. Acta mathematica scientia,Series A, 2019, 39(5): 1247-1259.

图/表 2

参考文献 18

1	Buonomo B , Lacitignola D , Vargas-De-León C . Qualitative analysis and optimal control of an epidemic model with vaccination and treatment. Mathematics and Computers in Simulation, 2014, 100: 88- 102 doi: 10.1016/j.matcom.2013.11.005
2	Diekmann O , Heesterbeek J A , Metz J A . On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 1990, 28: 365- 382
3	Gumel A B , Moghadas S M . A qualitative study of a vaccination model with nonlinear incidence. Applied Mathematics and Computation, 2003, 143: 409- 419 doi: 10.1016/S0096-3003(02)00372-7
4	Gao D P , Huang N J , Kang S M , Zhang C . Global stability analysis of an SVEIR epidemic model with general incidence rate. Boundary Value Problems, 2008, 2008: 42
5	Huang G , Takeuchi Y , Ma W B , Wei D . Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. Bulletin of Mathematical Biology, 2010, 72: 1192- 1207 doi: 10.1007/s11538-009-9487-6
6	Huang G , Ma W B , Takeuchi Y . Global properties for virus dynamics model with Beddington-DeAngelis functional response. Applied Mathematics Letters, 2006, 22: 1690- 1693
7	Hale J K , Waltman P . Persistence in infinite-dimensional systems. Society for Industrial and Applied Mathematics, 1989, 20: 388- 395
8	Kuniya T . Global stability of a multi-group SVIR epidemic model. Nonlinear Analysis:Real World Applications, 2013, 14 (2): 1135- 1143 doi: 10.1016/j.nonrwa.2012.09.004
9	Liu S Q , Wang S K , Wang L . Global dynamics of delay epidemic models with nonlinear incidence rate and relapse. Nonlinear Analysis:Real World Applications, 2011, 12: 119- 127 doi: 10.1016/j.nonrwa.2010.06.001
10	Liu W M , Hethcote H W , Levin S A . Dynamical behavior of epidemiological models with nonlinear incidence rates. Journal of Mathematical Biology, 1987, 25: 359- 380 doi: 10.1007/BF00277162
11	Liu X , Takeuchi Y , Iwami S . SVIR epidemic models with vaccination strategies. Journal of Theoretical Biology, 2008, 253 (1): 1- 11
12	Rost G , Wu J . Seir epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences and Engineering, 2017, 5: 389- 402
13	Wang J L , Zhang R , Kuniya T . The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes. Journal of Biological Dynamics, 2015, 9: 73- 101 doi: 10.1080/17513758.2015.1006696
14	Wang J L . Sveir epidemiological model with varying infectivity and distributed delays. Mathematical Biosciences and Engineering, 2012, 8: 875- 888
15	Wang J L , Dong X , Sun H Q . Analysis of an SVEIR model with age-dependence vaccination, latency and relapse. Journal of Nonlinear Sciences and Application, 2017, 10 (7): 3755- 3776 doi: 10.22436/jnsa.010.07.31
16	Wang L W , Liu Z J , Zhang X A . Global dynamics for an age-structured epidemic model with media impact and incomplete vaccination. Nonlinear Analysis:Real World Applications, 2016, 32: 136- 158 doi: 10.1016/j.nonrwa.2016.04.009
17	Yamada K , Yoshimura M . A possible association of Y chromosome heterochromatin with stature. Human Genetics, 1981, 58 (3): 268- 270 doi: 10.1007/BF00294920
18	Zhang T L , Teng Z D . Global behavior and permanence of SIRS epidemic model with time delay. Nonlinear Analysis:Real World Applications, 2008, 9: 1409- 1424 doi: 10.1016/j.nonrwa.2007.03.010

[1]	罗日才, 许弘雷, 王五生. 一类新的变时滞中立型神经网络的全局渐近稳定性条件[J]. 数学物理学报, 2015, 35(3): 634-640.
[2]	吴事良, 刘三阳. 反应扩散系统双稳波前解的全局渐近稳定性[J]. 数学物理学报, 2010, 30(2): 440-448.
[3]	赵丽琴. 一类微分方程零解全局弱吸引和全局吸引到充要条件[J]. 数学物理学报, 2009, 29(3): 529-537.
[4]	徐为坚;. 具有种群Logistic增长及饱和传染率的SIS模型的稳定性和Hopf分支[J]. 数学物理学报, 2008, 28(3): 578-584.
[5]	高淑京; 陈兰荪. 具有三个成长阶段的单种群时滞模型的永久持续生存和全局稳定性[J]. 数学物理学报, 2006, 26(4): 527-533.
[6]	陈福来;文贤章. 脉冲泛函微分方程的渐近性态[J]. 数学物理学报, 2006, 26(2): 287-296.
[7]	张强, 马润年. 时滞细胞神经网络的时滞相关指数稳定性[J]. 数学物理学报, 2004, 4(6): 694-698.
[8]	滕志东. 一类无穷时滞周期Lotka-Volterra型系统的正周期解[J]. 数学物理学报, 2001, 21(1): 94-101.
[9]	滕志东陈兰荪. 具有时滞的周期Lotka-Volterra型系统的全局渐近稳定性[J]. 数学物理学报, 2000, 20(3): 296-303.