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Asymptotic analysis of an SIQS epidemic model with varying total population size and quarantine measures

  • Minnan Science and Technology University, Quanzhou 362332, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.03.004
  • Received Date: 17 January 2019
  • Accepted Date: 22 May 2020
  • Rev Recd Date: 22 May 2020
  • Publish Date: 31 March 2020
    Abstract

  • Abstract

    A class of SIQS epidemic model with vertical infection and varying total population size and quarantine measures was established, by employing the epidemic dynamic theory, and considering both the input and output of the population. The threshold conditions which guarantee the global asymptotic stable disease-free equilibrium and endemic equilibrium of the SIQS epidemic model are obtained using methods including Routh-Hurwitz bounded, Lyapunov function and generalized Bendixson-Dulac function. The results show that the spread and prevalence of a disease can be controlled within a certain range, and that quarantine measures can accelerate the extinction of the disease.

    Abstract

    A class of SIQS epidemic model with vertical infection and varying total population size and quarantine measures was established, by employing the epidemic dynamic theory, and considering both the input and output of the population. The threshold conditions which guarantee the global asymptotic stable disease-free equilibrium and endemic equilibrium of the SIQS epidemic model are obtained using methods including Routh-Hurwitz bounded, Lyapunov function and generalized Bendixson-Dulac function. The results show that the spread and prevalence of a disease can be controlled within a certain range, and that quarantine measures can accelerate the extinction of the disease.
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    • References(13)
  • [1]
    TENG Z, WANG L. Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate[J]. Physica A, 2016，451: 507-518.
    [2]
    YOSHIDA N, HARA T. Global stability of a delayed SIR epidemic model with density dependent birth and death rates[J]. Computational and Applied Mathematics, 2007, 201(2): 339-347.
    [3]
    ZHAO Y., JIANG, D., MAO X, et al. The threshold of a stochastic SIRS epidemic model in a population with varying size[J]. Discrete Contin. Dyn. Syst., 2015, 20(4): 1277-1295.
    [4]
    ZHANG X, SHI Q, MA S. Dynamic behavior of a stochastic SIQS epidemic model with Lévy jumps[J]. Nonlinear Dynamics, 2018, 93(3): 1481-1493.
    [5]
    庞国萍，陈兰荪. 具饱和传染率的脉冲免疫接种SIRS模型[J]. 系统科学与数学, 2007, 27(4): 563-572.
    PANG G P, CHEN L S. The SIRS epidemic model with saturated contact rate and pulse vaccination [J]. J. Sys. Sci. & Math. Scis., 2007, 27(4): 563-572.
    [6]
    LIU Q, CHEN Q. Dynamics of a stochastic SIR epidemic model with saturated incidence[J]. Appl. Math. Comput. 2016，282: 155-166.
    [7]
    徐为坚. 具常数输入率及饱和发生率的脉冲接种SIQRS传染病模型[J]. 系统科学与数学，2010, 30(1): 43-52.
    XU W J. The SIQRS epidemic model of impulsive vaccination with constant input and saturation incidence rate [J]. J. Sys. Sci. & Math. Scis., 2010, 30(1): 43-52.
    [8]
    唐晓明, 薛亚奎. 具有饱和治疗函数与密度制约的SIS传染病模型的后向分支[J]. 数学的实践与认识, 2010, 40(24): 241-246.
    Tang X M, Xue Y K. Backward Bifurcation of a SIS epidemic model with density dependent birth and death rates and saturated treatment function [J]. Mathematics in Practice and Theory, 2010, 40(24): 241-246.
    [9]
    朱凌峰, 李维德, 章培军. 具有连续和脉冲接种的SIQVS传染病模型[J]. 兰州大学学报, 2011, 47(4): 99-102.
    ZHU L F, LI W D, ZHANG P J. A SIQVS epidemic model with continuous and impulsive vaccination [J]. Journal of Lanzhou University, 2011, 47(4) :99-102.
    [10]
    LIU Q, CHEN Q. Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence. Phys. A 2015,428, 140-153.
    [11]
    张改平，董玉才，许飞，等. 具有垂直传染且总人口在变化的SIRS传染病模型的渐近分析[J]. 数学的实践与认识, 2011, 41(8): 139-143.
    ZHANG G P, DONG Y C, XU F, et al. Asymptotic analysis of an SIRS epidemic model with vertical infection and varying population[J]. Mathematics in Practice and Theory , 2011, 41(8): 139-143.
    [12]
    林子植，董霖，李学鹏. 一类具有常数输入和垂直传染的SIRI传染病模型[J]. 数学的实践与认识, 2011, 41(15): 156-164.
    LIN Z Z, DONG L, LI X P. A kind of SIRI epidemic model with constant immigration and vertical transmission[J]. Mathematics in Practice and Theory, 2011, 41(15): 156-164.
    [13]
    马知恩, 周义仓. 传染病动力学的数学建模与研究[M]. 北京: 科学出版社, 2004.
    MA Z E, ZHOU Y C. Mathematics Modeling and Research of Infectious Disease Dynamics[M]. BeiJing: Science Press, 2004.)
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    [1]
    TENG Z, WANG L. Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate[J]. Physica A, 2016，451: 507-518.
    [2]
    YOSHIDA N, HARA T. Global stability of a delayed SIR epidemic model with density dependent birth and death rates[J]. Computational and Applied Mathematics, 2007, 201(2): 339-347.
    [3]
    ZHAO Y., JIANG, D., MAO X, et al. The threshold of a stochastic SIRS epidemic model in a population with varying size[J]. Discrete Contin. Dyn. Syst., 2015, 20(4): 1277-1295.
    [4]
    ZHANG X, SHI Q, MA S. Dynamic behavior of a stochastic SIQS epidemic model with Lévy jumps[J]. Nonlinear Dynamics, 2018, 93(3): 1481-1493.
    [5]
    庞国萍，陈兰荪. 具饱和传染率的脉冲免疫接种SIRS模型[J]. 系统科学与数学, 2007, 27(4): 563-572.
    PANG G P, CHEN L S. The SIRS epidemic model with saturated contact rate and pulse vaccination [J]. J. Sys. Sci. & Math. Scis., 2007, 27(4): 563-572.
    [6]
    LIU Q, CHEN Q. Dynamics of a stochastic SIR epidemic model with saturated incidence[J]. Appl. Math. Comput. 2016，282: 155-166.
    [7]
    徐为坚. 具常数输入率及饱和发生率的脉冲接种SIQRS传染病模型[J]. 系统科学与数学，2010, 30(1): 43-52.
    XU W J. The SIQRS epidemic model of impulsive vaccination with constant input and saturation incidence rate [J]. J. Sys. Sci. & Math. Scis., 2010, 30(1): 43-52.
    [8]
    唐晓明, 薛亚奎. 具有饱和治疗函数与密度制约的SIS传染病模型的后向分支[J]. 数学的实践与认识, 2010, 40(24): 241-246.
    Tang X M, Xue Y K. Backward Bifurcation of a SIS epidemic model with density dependent birth and death rates and saturated treatment function [J]. Mathematics in Practice and Theory, 2010, 40(24): 241-246.
    [9]
    朱凌峰, 李维德, 章培军. 具有连续和脉冲接种的SIQVS传染病模型[J]. 兰州大学学报, 2011, 47(4): 99-102.
    ZHU L F, LI W D, ZHANG P J. A SIQVS epidemic model with continuous and impulsive vaccination [J]. Journal of Lanzhou University, 2011, 47(4) :99-102.
    [10]
    LIU Q, CHEN Q. Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence. Phys. A 2015,428, 140-153.
    [11]
    张改平，董玉才，许飞，等. 具有垂直传染且总人口在变化的SIRS传染病模型的渐近分析[J]. 数学的实践与认识, 2011, 41(8): 139-143.
    ZHANG G P, DONG Y C, XU F, et al. Asymptotic analysis of an SIRS epidemic model with vertical infection and varying population[J]. Mathematics in Practice and Theory , 2011, 41(8): 139-143.
    [12]
    林子植，董霖，李学鹏. 一类具有常数输入和垂直传染的SIRI传染病模型[J]. 数学的实践与认识, 2011, 41(15): 156-164.
    LIN Z Z, DONG L, LI X P. A kind of SIRI epidemic model with constant immigration and vertical transmission[J]. Mathematics in Practice and Theory, 2011, 41(15): 156-164.
    [13]
    马知恩, 周义仓. 传染病动力学的数学建模与研究[M]. 北京: 科学出版社, 2004.
    MA Z E, ZHOU Y C. Mathematics Modeling and Research of Infectious Disease Dynamics[M]. BeiJing: Science Press, 2004.)
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