含等级治疗率与不完全康复率的SIRS模型的分岔分析

doi:10.16088/j.issn.1001-6600.2020.06.009

广西师范大学学报（自然科学版） ›› 2020, Vol. 38 ›› Issue (6): 74-81.doi: 10.16088/j.issn.1001-6600.2020.06.009

含等级治疗率与不完全康复率的SIRS模型的分岔分析

黄春贤¹, 周效良^2*

1.闽南师范大学数学与统计学院, 福建漳州363000;
2.岭南师范学院数学与统计学院, 广东湛江524048

收稿日期:2019-09-05 发布日期:2020-11-30
通讯作者: 周效良(1965—), 男, 山西五台人, 岭南师范学院教授, 博士。 E-mail: Zhouxl@lingnan.edu.cn
基金资助:
国家自然科学基金(11561019);广东省创新强校重点项目(2019KZDXM032)

Bifurcation Analysis of an SIRS Epidemic Model with Graded Cure and Incomplete Recovery Rates

HUANG Chunxian¹, ZHOU Xiaoliang^2*

1. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou Fujian 363000, China;
2. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang Guangdong 524048, China

Received:2019-09-05 Published:2020-11-30

摘要/Abstract

摘要： 基于微分方程定性理论和分岔理论对一类带有等级治疗率与不完全康复率的SIRS海洛因模型的动力学性质进行探讨。先利用系统的极限集对系统进行降维;再应用特征根方法对系统的无病平衡点和地方病平衡点的拓扑类型进行详细分析;然后严格证明该系统随着分岔参数的变化在无病平衡点附近发生前向分岔;最后借助Matlab中ODE45软件包进行数值模拟,得到该系统在不同分岔参数条件下的相图以及状态变量曲线图,直观展示结论的正确性。

关键词: SIRS模型, 等级治疗率, 平衡点, 前向分岔, 数值模拟

Abstract: In this paper, an SIRS heroin model with graded cure and incomplete recovery rates is studied. Firstly, the limit set of the system is used to reduce a three-dimensional system to two-dimension system,and equilibrium topological types of the system are verified by analyzing the eigenvalues in detail. Then, it is shown that the system undergoes forward bifurcation with the change of bifurcation parameters. Finally, the phase diagram and state variable curve of the system under different bifurcation parameters are obtained through numerical simulations with the help of ODE45 software package in Matlab, which intuitively show the correctness of the conclusion.

Key words: SIRS model, graded cure, equilibrium, forward bifurcation, numerical simulation

中图分类号:

O175

黄春贤, 周效良. 含等级治疗率与不完全康复率的SIRS模型的分岔分析[J]. 广西师范大学学报（自然科学版）, 2020, 38(6): 74-81.

HUANG Chunxian, ZHOU Xiaoliang. Bifurcation Analysis of an SIRS Epidemic Model with Graded Cure and Incomplete Recovery Rates[J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(6): 74-81.

参考文献

[1] BRAUER F,CASTILLO-CHAVEZ C.Mathematical models in population biology and epidemiology[M]. New York:Springer,2012.
[2] 郝丽杰,蒋贵荣,鹿鹏.具垂直传染的SIRS传染病模型的脉冲控制和分岔分析[J].广西师范大学学报(自然科学版),2012,30(4):42-47.
[3] GOMES M G M,WHITE L J,MEDLEY G F.Infection,reinfection,and vaccination under suboptimal immune protection: epidemiological perspectives[J].Journal of Theoretical Biology,2004,228(4) :539-549.DOI:10.1016/j.jtbi.2004.02.015.
[4] 邢伟,高晋芳,颜七笙,等.具有非线性传染率及脉冲免疫接种的SIQR传染病模型[J].广西师范大学学报(自然科学版),2017,35(2):58-65. DOI:10.16088/j.issn.1001-6600.2017.02.009.
[5] KERMACK W O,MCKENDRICK A G.A contribution to the mathematical theory of epidemics[J].Proceedings of the Royal Society of London: Series A,1927,115:700-721.
[6] ZHOU J S.An SIS disease transmission model with recruitment-birth-death demographics[J].Mathematical &Computer Modelling,1995,21(11):1-11.DOI:10.1016/0895-7177(95)00074-C.
[7] ZHU C J. Critical result on the threshold of a stochastic SIS model with saturated incidence rate[J].Physica A:Statistical Mechanics and Its Applications,2019,523:426-437.DOI:10.1016/j.physa.2019.02.012.
[8] 杨鲲,林娇,蒋贵荣.具有脉冲生育的随机SIS传染病模型的动力学分析[J].广西师范大学学报(自然科学版),2015,33(4):81-86.DOI:10.16088/j.issn.1001-6600.2015.04.014.
[9] 冯金明,李遵先.一类具扩散的传染病模型的稳定性分析[J].广西师范大学学报(自然科学版),2018,36(2):63-68.DOI:10.16088/j.issn.1001-6600.2018.02.009.
[10] GOMES M G M,MARGHERI A,MEDLEY G F,et al.Dynamical behavior of epidemiological models with sub-optimal immunity and nonlinear incidence[J].Journal of Mathematical Biology,2005,51:414-430.
[11] 马知恩, 周义仓,王稳地,等.传染病动力学的数学建模与研究[M].北京:科学出版社,2004.
[12] LU M,HUANG J C,RUAN S G,et al.Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate[J].Journal of Differential Equations,2019,267(3):1859-1898.DOI:10.1016/j.jde.2019.03.005.
[13] WHITE E,COMISKEY C.Heroin epidemics,treatment and ODE modelling[J].Mathematical Biosciences,2007,208(1): 312-324.DOI:10.1016/j.mbs.2006.10.008.
[14] MULONE G,STRAUGHAN B.A note on heroin epidemics[J].Mathematical Biosciences,2009,218(2):138-141.DOI:10.1016/j.mbs.2009.01.006.
[15] LIU J L,ZHANG T L. Global behaviorof a heroin epidemic model with distributed delays[J].Applied Mathematics Letters,2011,24(10):1685-1692.DOI:10.1016/j.aml.2011.04.019.
[16] HUANG G,LIU A P. A note on global stability for a heroin epidemic model with distributed delay[J].Applied Mathematics Letters,2013,26(7):687-691.DOI:10.1016/j.aml.2013.01.010.
[17] MUROYA Y,LI H X,KUNIYA T.Complete global analysis of an SIRS epidemic model with graded cure and incomplete recovery rates[J].Journal of Mathematical Analysis and Applications,2014,410(2):719-732.DOI:10.1016/j.jmaa.2013.08.024.
[18] 史学伟,贾建文.一类具有信息变量和等级治愈率的SIR传染病模型的研究[J].山东大学学报(理学版),2016,51(3):51-59,69.DOI:10.6040/j.issn.1671-9352.0.2015.212.
[19] MA M J,LIU S Y,LI J. Bifurcation of a heroin model with nonlinear incidence rate[J].Nonlinear Dynamics,2017,88:555-565.DOI:10.1007/s11071-016-3260-9.
[20] LIU W M,LEVIN S A,IWASA Y.Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models[J].Journal of Mathematical Biology,1986,23:187-204.DOI:10.1007/BF00276956.
[21] LIZANA M,RIVERO J. Multiparametric bifurcations for a model in epidemiology[J].Journal of Mathematical Biology,1996,35:21-36.DOI:10.1007/s002850050040.
[22] CASTILLO-CHAVEZ C,SONG B J. Dynamical models of tuberculosis and their applications[J].Mathematical Biosciences and Engineering,2004,1(2):361-404.DOI:10.3934/mbe.2004.1.361.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed

含等级治疗率与不完全康复率的SIRS模型的分岔分析

Bifurcation Analysis of an SIRS Epidemic Model with Graded Cure and Incomplete Recovery Rates

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 5

Metrics

本文评价

推荐阅读 10

[1]	凌风如, 张超英, 陈燕雁, 覃章荣. LBM中基于半程反弹的统一边界条件研究[J]. 广西师范大学学报（自然科学版）, 2020, 38(1): 70-78.
[2]	冯金明,李遵先. 一类具扩散的传染病模型的稳定性分析[J]. 广西师范大学学报（自然科学版）, 2018, 36(2): 63-68.
[3]	陈春燕, 许志鹏, 邝华. 连续记忆效应的交通流跟驰建模与稳定性分析[J]. 广西师范大学学报（自然科学版）, 2017, 35(3): 14-21.
[4]	李谊纯, 董德信, 王一兵. 北仑河口及其邻近海域物质输运滞留时间研究[J]. 广西师范大学学报（自然科学版）, 2015, 33(2): 56-63.
[5]	冯春华. 一类含n个时滞神经网络模型解的周期振动性[J]. 广西师范大学学报（自然科学版）, 2012, 30(3): 48-53.