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CN 34-1054/N

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Dynamic analysis of an SIQR model with saturation contact rate and hybrid strategies

  • Department of Common Course, Anhui Xinhua University, Hefei 230088, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.05.016
  • Received Date: 26 June 2019
  • Accepted Date: 02 August 2019
  • Rev Recd Date: 02 August 2019
  • Publish Date: 31 May 2020
    Abstract

  • Abstract

    Considering vaccination, quarantine and elimination hybrid strategies, an SIQR epidemic model with saturation contact rate was established. And the global stability of the model was studied by means of both theoretical and numerical ways. Firstly, the threshold-basic reproductive number R0 which determines whether the disease is extinct or not and the conditions for the existence of equilibriums were obtained by the calculation. Secondly, by Liapunov function, it was proved that the disease-free equilibrium P0 is globally asymptotically stable when R0<1. Thirdly, by constructing a suitable Dulac function, it was obtained that the unique endemic equilibrium P* is globally asymptotically stable when R0>1. Finally, some numerical simulations were presented to illustrate the analysis results.

    Abstract

    Considering vaccination, quarantine and elimination hybrid strategies, an SIQR epidemic model with saturation contact rate was established. And the global stability of the model was studied by means of both theoretical and numerical ways. Firstly, the threshold-basic reproductive number R0 which determines whether the disease is extinct or not and the conditions for the existence of equilibriums were obtained by the calculation. Secondly, by Liapunov function, it was proved that the disease-free equilibrium P0 is globally asymptotically stable when R0<1. Thirdly, by constructing a suitable Dulac function, it was obtained that the unique endemic equilibrium P* is globally asymptotically stable when R0>1. Finally, some numerical simulations were presented to illustrate the analysis results.
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    • References(18)
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    叶志勇HYPERLINK"http://www.cnki.com.cn/Article/%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20http:/yuanjian.cnki.com.cn/Search/Result?author=%E5%8F%B6%E5%BF%97%E5%8B%87"\t"_blank"，刘原HYPERLINK"http://www.cnki.com.cn/Article/%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20http:/yuanjian.cnki.com.cn/Search/Result?author=%E5%88%98%E5%8E%9F"\t"_blank"，吴用HYPERLINK"http://www.cnki.com.cn/Article/%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20http:/yuanjian.cnki.com.cn/Search/Result?author=%E5%90%B4%E7%94%A8"\t"_blank"．具有非单调传染率的SIQR传染病模型的稳定性分析[J]．生物数学学报，2014，29(1)：105-112．
    [15]
    张珍，靳祯．一类带脉冲接种和脉冲剔除的SIR传染病模型的稳定性态[J]．太原师范学院学报(自然科学版)，2006，5(4)：8-10．
    [16]
    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，2004:3-8．
    [17]
    马知恩，周义仓．常微分方程定性与稳定性方法[M] ．北京：科学出版社，2001: 41-50．
    [18]
    徐文雄，张仲华，成芳．一类SIS流行病传播数学模型全局渐近稳定性[J]．四川师范大学学报(自然科学版)，2004，27(6)：585-588．)
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    [1]
    XU R, MA Z E.Global stability of a delayed SEIRS epidemic model with saturation incidence rate[J]. Nonlinear Dynamics, 2010, 61: 229-239.
    [2]
    MA Yanli, LIU Jiabao, LI Haixia . Global dynamics of an SIQR model with vaccination and elimination hybrid strategies[J]. Mathematics, 2018, 6(12): 328-339.
    [3]
    马艳丽，张仲华，刘家保，等．一类具有脉冲接种与脉冲剔除的SIQR模型[J]．中国科学技术大学学报，2018，48(2)：111-117．
    [4]
    LI Guihua, WANG Wendi, JIN Zhen. Global stability of an SEIR model with constant immigration[J]. Chaos, Solitons and Fractals, 2006, 30(4): 1012-1019.
    [5]
    ECKALBAR J C, ECKALBAR W L. Dynamics of an SIR model with vaccination dependent on past prevalence with high-order distributed delay[J]. Biosystems, 2015, 129(1): 50-65.
    [6]
    徐金瑞，王美娟，张拥军． 一类具有标准发生率的SIS型传染病模型的全局稳定性[J]．生物数学学报，2010，25(2)：249-256．
    [7]
    马艳丽，张仲华．潜伏类和移出类具有传染性的SEIR模型的渐近分析[J]．中国科学技术大学学报，2016，46(2)：95-103．
    [8]
    LI Jianquan, ZHANG Juan, MA Zhi’en. Global analysis of some epidemic models with general contact rate and constant immigration [J]. Applied Mathematics and MechanicsHYPERLINK"https://link.springer.com/journal/10483"\o"AppliedMathematicsandMechanics", 2004, 25(4): 396-404.
    [9]
    LI GuihuaHYPERLINK"https://www.sciencedirect.com/science/article/pii/S0960077904003558"\l"!", JIN ZhenHYPERLINK"https://www.sciencedirect.com/science/article/pii/S0960077904003558"\l"!". Global stability of an SEI epidemic model with general contact rate [J]. Chaos, Solitons and Fractals, 2005, 23(3): 997-1004.
    [10]
    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(9)：737-744．
    [11]
    TAN X X, LI S J, DAI Q W, et al. An epidemic model with isolated intervention based on cellular automata [J]. Advanced Materials Research, 2014, 926(1): 1065-1068.
    [12]
    ECKALBAR J C, ECKALBAR W L. Dynamics of an SIR model with vaccination dependent on past prevalence with high-order distributed delay[J]. Biosystems, 2015, 129(1): 50-65.
    [13]
    SHI PEILIN, DONG LINGZHEN. Dynamical models for infectious diseases with varying population size and vaccination[J]. Journal of Applied Mathematics, 2012, 12(1), 253-273.
    [14]
    叶志勇HYPERLINK"http://www.cnki.com.cn/Article/%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20http:/yuanjian.cnki.com.cn/Search/Result?author=%E5%8F%B6%E5%BF%97%E5%8B%87"\t"_blank"，刘原HYPERLINK"http://www.cnki.com.cn/Article/%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20http:/yuanjian.cnki.com.cn/Search/Result?author=%E5%88%98%E5%8E%9F"\t"_blank"，吴用HYPERLINK"http://www.cnki.com.cn/Article/%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20http:/yuanjian.cnki.com.cn/Search/Result?author=%E5%90%B4%E7%94%A8"\t"_blank"．具有非单调传染率的SIQR传染病模型的稳定性分析[J]．生物数学学报，2014，29(1)：105-112．
    [15]
    张珍，靳祯．一类带脉冲接种和脉冲剔除的SIR传染病模型的稳定性态[J]．太原师范学院学报(自然科学版)，2006，5(4)：8-10．
    [16]
    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，2004:3-8．
    [17]
    马知恩，周义仓．常微分方程定性与稳定性方法[M] ．北京：科学出版社，2001: 41-50．
    [18]
    徐文雄，张仲华，成芳．一类SIS流行病传播数学模型全局渐近稳定性[J]．四川师范大学学报(自然科学版)，2004，27(6)：585-588．)
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