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CN 34-1054/N

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Dynamics analysis of an impulsive stochastic SIS epidemic model with standard incidence

  • 1.

    School of Mathematics and Computing Science,Guilin University of Electronic Technology,Guilin 541004,China

  • 2.

    College of Science,Guilin University of Aerospace Technology,Guilin 541004,China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2018.08.005
  • Received Date: 17 July 2017
  • Accepted Date: 08 January 2018
  • Rev Recd Date: 08 January 2018
  • Publish Date: 31 August 2018
    Abstract

  • Abstract

    Random disturbance, birth pulse and pulse treatments were simultaneously considered to build a class of SIS epidemic model with standard incidence．By applying the theory of stochastic differential equation， the sufficient conditions for the stochastic stability of the trivial solution were obtained.The sufficient conditions for the existence of the disease free solution were obtained by using a discrete map. The stochastic extinction of the disease was investigated by using the It formula.Moreover， numerical simulation was conducted to verify the theoretical analysis.

    Abstract

    Random disturbance, birth pulse and pulse treatments were simultaneously considered to build a class of SIS epidemic model with standard incidence．By applying the theory of stochastic differential equation， the sufficient conditions for the stochastic stability of the trivial solution were obtained.The sufficient conditions for the existence of the disease free solution were obtained by using a discrete map. The stochastic extinction of the disease was investigated by using the It formula.Moreover， numerical simulation was conducted to verify the theoretical analysis.
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    • References(13)
  • [1]
    GRAY A, GREENHALGH D, MAO X, et al. The SIS epidemic model with Markovian switching[J]. Journal of Mathematical Analysis and Applications, 2012, 394(2): 496-516.
    [2]
    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,2004:3-24.
    [3]
    马艳丽, 张仲华. 潜伏类和移出类具有传染性的SEIR模型的渐近性分析[J]中国科学技术大学学报, 2016, 46(2): 95-103.
    MA Yanli, ZHANG Zhonghua. Asymptotical analysis of SEIR model with infectious force in latent and immune periods[J]. Journal of University of Science and Technology of China, 2016, 46(2): 95-103.
    [4]
    蒋贵荣, 刘期怀, 龙腾飞,等. 脉冲动力系统的分岔混沌理论及其应用[M]. 北京: 科学出版社, 2015: 167-169.
    [5]
    刘开源, 陈兰荪. 一类具有垂直传染与脉冲免疫的SEIR传染病模型的全局分析[J]. 系统科学与数学, 2010, 30(3): 323-333.
    LIU Kaiyuan, CHEN Lansun. Global analysis of an SEIR epidemic disease mode with vertical transmission and pulse vaccination[J]. Journal of Systems Science and Mathematical Sciences, 2010, 30(3): 323-333.
    [6]
    JIANG Guirong, YANG Qigui. Periodic solutions and bifurcation in an SIS epidemic model with birth pulse[J]. Mathematical and Computer Modeling, 2009, 50: 498-508.
    [7]
    HU Zhixing, LIU Sheng, WANG Hui. Backward bifurcation of an epidemic model with standard incidence rate and treatment rate[J]. Nonlinear Analysis: Real World Applications, 2008, 9(5): 2302-2312.
    [8]
    马淑芳, 仇晓芬, 钟秋慧. 具有脉冲出生的SIS传染病模型的生存性[J]. 黑龙江大学自然科学学报, 2010, 27(1): 34-37.
    MA Shufang, QIU Xiaofen, ZHONG Qiuhui. Persistence of SIS epidemic model with birth pulse[J]. Journal of Natural Science of Heilongjiang University, 2010, 27(1): 34-37.
    [9]
    王克.随机生物数学模型[M].北京:科学出版社,2010.
    [10]
    LIN Yuguo, JIANG Daqing, WANG Shuai. Stationary distribution of a stochastic SIS epidemic model with vaccination [J]. Physica A: Statistical Mechanics and Its Applications, 2014, 394: 187-197.
    [11]
    周艳丽, 张卫国．非线性传染率的随机SIS传染病模型的持久性和灭绝性[J].山东大学学报(理学版), 2013, 48(10): 68-77.
    ZHOU Yanli, ZHANG Weiguo. Persistence and extinction in stochastic SIS epidemic model with nonlinear incidence rate[J]. Journal of Shandong University (Natural Science), 2013, 48(10): 68-77.
    [12]
    XU C. Global threshold dynamics of a stochastic differential equation SIS model[J]. Journal of Mathematical Analysis and Applications, 2017, 447(2): 736-757.
    [13]
    ANDERSON R, MAY R. Infectious Diseases of Humans: Dynamics and Control[M]. Oxford: Oxford University Press, 1991.)
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    [1]
    GRAY A, GREENHALGH D, MAO X, et al. The SIS epidemic model with Markovian switching[J]. Journal of Mathematical Analysis and Applications, 2012, 394(2): 496-516.
    [2]
    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,2004:3-24.
    [3]
    马艳丽, 张仲华. 潜伏类和移出类具有传染性的SEIR模型的渐近性分析[J]中国科学技术大学学报, 2016, 46(2): 95-103.
    MA Yanli, ZHANG Zhonghua. Asymptotical analysis of SEIR model with infectious force in latent and immune periods[J]. Journal of University of Science and Technology of China, 2016, 46(2): 95-103.
    [4]
    蒋贵荣, 刘期怀, 龙腾飞,等. 脉冲动力系统的分岔混沌理论及其应用[M]. 北京: 科学出版社, 2015: 167-169.
    [5]
    刘开源, 陈兰荪. 一类具有垂直传染与脉冲免疫的SEIR传染病模型的全局分析[J]. 系统科学与数学, 2010, 30(3): 323-333.
    LIU Kaiyuan, CHEN Lansun. Global analysis of an SEIR epidemic disease mode with vertical transmission and pulse vaccination[J]. Journal of Systems Science and Mathematical Sciences, 2010, 30(3): 323-333.
    [6]
    JIANG Guirong, YANG Qigui. Periodic solutions and bifurcation in an SIS epidemic model with birth pulse[J]. Mathematical and Computer Modeling, 2009, 50: 498-508.
    [7]
    HU Zhixing, LIU Sheng, WANG Hui. Backward bifurcation of an epidemic model with standard incidence rate and treatment rate[J]. Nonlinear Analysis: Real World Applications, 2008, 9(5): 2302-2312.
    [8]
    马淑芳, 仇晓芬, 钟秋慧. 具有脉冲出生的SIS传染病模型的生存性[J]. 黑龙江大学自然科学学报, 2010, 27(1): 34-37.
    MA Shufang, QIU Xiaofen, ZHONG Qiuhui. Persistence of SIS epidemic model with birth pulse[J]. Journal of Natural Science of Heilongjiang University, 2010, 27(1): 34-37.
    [9]
    王克.随机生物数学模型[M].北京:科学出版社,2010.
    [10]
    LIN Yuguo, JIANG Daqing, WANG Shuai. Stationary distribution of a stochastic SIS epidemic model with vaccination [J]. Physica A: Statistical Mechanics and Its Applications, 2014, 394: 187-197.
    [11]
    周艳丽, 张卫国．非线性传染率的随机SIS传染病模型的持久性和灭绝性[J].山东大学学报(理学版), 2013, 48(10): 68-77.
    ZHOU Yanli, ZHANG Weiguo. Persistence and extinction in stochastic SIS epidemic model with nonlinear incidence rate[J]. Journal of Shandong University (Natural Science), 2013, 48(10): 68-77.
    [12]
    XU C. Global threshold dynamics of a stochastic differential equation SIS model[J]. Journal of Mathematical Analysis and Applications, 2017, 447(2): 736-757.
    [13]
    ANDERSON R, MAY R. Infectious Diseases of Humans: Dynamics and Control[M]. Oxford: Oxford University Press, 1991.)
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