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An SIQR mode with impulsive vaccination and impulsive elimination

  • 1.

    General Education Department, Anhui Xinhua University, Hefei 230088, China

  • 2.

    School of Sciences, Xi’an University of Science and Technology, Xi’an 710049, China

  • 3.

    School of Sciences, Anhui Jianzhu University, Hefei 230601, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2018.02.004
  • Received Date: 16 March 2016
  • Rev Recd Date: 25 April 2017
  • Publish Date: 28 February 2018
    Abstract

  • Abstract

    Impulsive vaccination, impulsive elimination and quarantine strategies were considered in an SIQR epidemic model. The dynamical behavior of an SIQR epidemic model was discussed both theoretically and numerically. Firstly, the disease-free T periodic solution and the basic reproductive number R0 were obtained. Secondly, the local asymptotic stability of the disease-free T periodic solution with Floquet theorem was proved and the global asymptotic stability of the disease-free T periodic solution was also proved by impulsive differential equation. Thirdly, numerical simulation was conducted to illustrate the

    Abstract

    Impulsive vaccination, impulsive elimination and quarantine strategies were considered in an SIQR epidemic model. The dynamical behavior of an SIQR epidemic model was discussed both theoretically and numerically. Firstly, the disease-free T periodic solution and the basic reproductive number R0 were obtained. Secondly, the local asymptotic stability of the disease-free T periodic solution with Floquet theorem was proved and the global asymptotic stability of the disease-free T periodic solution was also proved by impulsive differential equation. Thirdly, numerical simulation was conducted to illustrate the
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    • References(12)
  • [1]
    张珍，靳祯．一类带有脉冲接种和脉冲剔除的SIR传染病模型的稳定形态[J]．太原师范学院学报(自然科学版)，2006，5(4)：8-10．
    ZHANG Zhen, JIN Zhen. Stability properties in an SIR epidemic model with pulse vaccination and pulse elimination[J]. Journal of Taiyuan Normal University (Natural Science Edition), 2006，5(4)：8-10．
    [2]
    马艳丽，徐文雄，张仲华．具有一般形式接触率的SEIR模型的稳定性分析[J]．中国科学技术大学学报，2015，45(1)：737-744.
    MA Yanli, XU Wenxiong , ZHANG Zhonghua. Stability analysis of SEIR model with general contact rate[J]. Journal of University of Science and Technology of China, 2015, 45(1): 737-744.
    [3]
    STONE L, SHULGIN B, AGUR Z. Theoretical examination of the pulse vaccination policy in the SIR epidemic model[J]. Mathematical and Computer Modelling, 2000, 31(4-5): 207-215.
    [4]
    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[J]．兰州大学学报(自然科学版)，2011，47(1)：82-86．
    ZHANG Peijun, LI Weide, ZHU Lingfeng. Comparison between the continuous and pulse vaccinations about SIRS infectious disease model[J]. Journal of Lanzhou University（Natural Sciences）, 2011，47(1)：82-86．
    [5]
    朱玑，李维德，朱凌峰．基于SIR传染病模型的不同控制策略比较[J]．北华大学学报(自然科学版)，2011，3(1)：15-21．
    ZHU Ji, LI Weide, ZHU Lingfeng. Comparison among different control strategies on SIR epidemic model[J]．Journal of Beihua University (Natural Science), 2011，3(1)：15-21．
    [6]
    SAMANTA G P. Analysis of a delayed epidemic model with pulse vaccination[J]. Chaos, Solitions and Fractals, 2014, 66(1): 74-85.
    [7]
    QIN W, TANG S, ROBERT A. Nonlinear pulse vaccination in an SIR epidemic model with resource limitation[J]. Abstract and Applied Analysis, 2013, 2013(1): 24-37.
    [8]
    PEI Y, LI S, LI C, et al. The effect of constant and pulse vaccination on an SIR epidemic model with infectious period[J]. Applied Mathematical Modeling, 2011, 35(8): 3866-3878.
    [9]
    LI J, YANG Y. SIV-SVS epidemic models with continuous and impulsive vaccination strategies[J]. Journal of Theoretical Biology, 2011, 280(1):108-116.
    [10]
    周艳丽，王贺桥，王美娟，等．具有脉冲预防接种的SIQR 流行病数学模型[J]．上海理工大学学报，2007，29(1)：11-16．
    ZHOU Yanli, WANG Heqiao, WANG Meijuan, et al. SIQR epidemical model with impulsive vaccination[J]. Journal of University of Shanghai for Science and Technology, 2007，29(1)：11-16．
    [11]
    宋燕，刘薇，张宇．具有垂直传染及脉冲免疫接种的SIQR传染病模型[J]．兰州大学学报(自然科学版)，2014，50(2)：455-459．
    SONG Yan, LIU Wei, ZHANG Yu. An SIQR epidemic model with vertical transmission and impulsive vaccination[J]．Journal of Lanzhou University (Natural Sciences), 2014，50(2)：455-459．
    [12]
    张珍．不同步进行脉冲接种和脉冲剔除的SIR模型的动力学性态研究[J]．山西师范大学学报(自然科学版)，2012，26(2)：8-11．
    ZHANG Zhen.Stability properties in an SIR model with asynchronous pulse vaccination and pulse elimination[J]. Journal of Shanxi Normal University (Natural Science Edition), 2012, 26(2)：8-11.
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    [1]
    张珍，靳祯．一类带有脉冲接种和脉冲剔除的SIR传染病模型的稳定形态[J]．太原师范学院学报(自然科学版)，2006，5(4)：8-10．
    ZHANG Zhen, JIN Zhen. Stability properties in an SIR epidemic model with pulse vaccination and pulse elimination[J]. Journal of Taiyuan Normal University (Natural Science Edition), 2006，5(4)：8-10．
    [2]
    马艳丽，徐文雄，张仲华．具有一般形式接触率的SEIR模型的稳定性分析[J]．中国科学技术大学学报，2015，45(1)：737-744.
    MA Yanli, XU Wenxiong , ZHANG Zhonghua. Stability analysis of SEIR model with general contact rate[J]. Journal of University of Science and Technology of China, 2015, 45(1): 737-744.
    [3]
    STONE L, SHULGIN B, AGUR Z. Theoretical examination of the pulse vaccination policy in the SIR epidemic model[J]. Mathematical and Computer Modelling, 2000, 31(4-5): 207-215.
    [4]
    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[J]．兰州大学学报(自然科学版)，2011，47(1)：82-86．
    ZHANG Peijun, LI Weide, ZHU Lingfeng. Comparison between the continuous and pulse vaccinations about SIRS infectious disease model[J]. Journal of Lanzhou University（Natural Sciences）, 2011，47(1)：82-86．
    [5]
    朱玑，李维德，朱凌峰．基于SIR传染病模型的不同控制策略比较[J]．北华大学学报(自然科学版)，2011，3(1)：15-21．
    ZHU Ji, LI Weide, ZHU Lingfeng. Comparison among different control strategies on SIR epidemic model[J]．Journal of Beihua University (Natural Science), 2011，3(1)：15-21．
    [6]
    SAMANTA G P. Analysis of a delayed epidemic model with pulse vaccination[J]. Chaos, Solitions and Fractals, 2014, 66(1): 74-85.
    [7]
    QIN W, TANG S, ROBERT A. Nonlinear pulse vaccination in an SIR epidemic model with resource limitation[J]. Abstract and Applied Analysis, 2013, 2013(1): 24-37.
    [8]
    PEI Y, LI S, LI C, et al. The effect of constant and pulse vaccination on an SIR epidemic model with infectious period[J]. Applied Mathematical Modeling, 2011, 35(8): 3866-3878.
    [9]
    LI J, YANG Y. SIV-SVS epidemic models with continuous and impulsive vaccination strategies[J]. Journal of Theoretical Biology, 2011, 280(1):108-116.
    [10]
    周艳丽，王贺桥，王美娟，等．具有脉冲预防接种的SIQR 流行病数学模型[J]．上海理工大学学报，2007，29(1)：11-16．
    ZHOU Yanli, WANG Heqiao, WANG Meijuan, et al. SIQR epidemical model with impulsive vaccination[J]. Journal of University of Shanghai for Science and Technology, 2007，29(1)：11-16．
    [11]
    宋燕，刘薇，张宇．具有垂直传染及脉冲免疫接种的SIQR传染病模型[J]．兰州大学学报(自然科学版)，2014，50(2)：455-459．
    SONG Yan, LIU Wei, ZHANG Yu. An SIQR epidemic model with vertical transmission and impulsive vaccination[J]．Journal of Lanzhou University (Natural Sciences), 2014，50(2)：455-459．
    [12]
    张珍．不同步进行脉冲接种和脉冲剔除的SIR模型的动力学性态研究[J]．山西师范大学学报(自然科学版)，2012，26(2)：8-11．
    ZHANG Zhen.Stability properties in an SIR model with asynchronous pulse vaccination and pulse elimination[J]. Journal of Shanxi Normal University (Natural Science Edition), 2012, 26(2)：8-11.
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