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Stability analysis of SEIR model with general contact rate

  • 1.

    Public Curriculum Department, Anhui Xinhua University, Hefei 230088, China

  • 2.

    School of Mathematics and Statistics, Xian Jiaotong University, Xian 710049, China

  • 3.

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

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.09.006
  • Received Date: 19 March 2015
  • Accepted Date: 24 July 2015
  • Rev Recd Date: 24 July 2015
  • Publish Date: 30 September 2015
    Abstract

  • Abstract

    A type of SEIR epidemic model with different general contact rates β1(N), β2(N) and β3(N), having infective force in all the latent, infected and immune periods, was studied. And the threshold, basic reproductive number R0 which determines whether a disease is extinct or not, was obtained. By using the Liapunov function method, it was proved that the disease-free equilibrium E0 is globally asymptotically stable and the disease eventually goes away if R0<1. It was also proved that in the case where R0>1, E0 is unstable and the unique endemic equilibrium E* is locally asymptotically stable by Hurwitz criterion theory. It is shown that when disease-induced death rate and elimination rate are zero, the unique endemic equilibrium E* is globally asymptotically stable and the disease persists.

    Abstract

    A type of SEIR epidemic model with different general contact rates β1(N), β2(N) and β3(N), having infective force in all the latent, infected and immune periods, was studied. And the threshold, basic reproductive number R0 which determines whether a disease is extinct or not, was obtained. By using the Liapunov function method, it was proved that the disease-free equilibrium E0 is globally asymptotically stable and the disease eventually goes away if R0<1. It was also proved that in the case where R0>1, E0 is unstable and the unique endemic equilibrium E* is locally asymptotically stable by Hurwitz criterion theory. It is shown that when disease-induced death rate and elimination rate are zero, the unique endemic equilibrium E* is globally asymptotically stable and the disease persists.
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    • References(16)
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    刘丽君，魏来．丙型肝炎病毒的流行病学[J]．传染病信息，2007，20(5)：261-264．
    [2]
    方春红，梁虹，刘美琳．1850例尖锐湿疣形态与分布特点的临床分析[J]．中国麻风皮肤病杂志，2002，18(2)：138-139．
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    Fan M, Li M Y, Wang K. Global stability of an SEIS epidemic model with recruitment and a varying total population size[J]. Mathematical Biosciences, 2001, 170(2): 199-208.
    [4]
    Xu Wenxiong, Zhang Tailei. Global stability for the model with quarantine in epidemiology[J]. Journal of Xian Jiaotong University, 2005，39(2)：210-213．
    徐文雄，张太雷．具有隔离仓室流行病传播数学模型的全局稳定性[J]．西安交通大学学报，2005，39(2)：210-213．
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    Liu W M, van den Driessche P. Epidemiological models with varying total population size and dose-dependent latent period[J]. Mathematical Biosciences, 1995, 128(1-2): 57-69.
    [6]
    Li M Y, Graef J R, Wang L C, et al. Global dynamics of an SEIR model with varying total population size[J]. Mathematics Biosciences, 1999, 160(2): 191-213.
    [7]
    Xu Wenxiong, Zhang Tailei, Xu Zongben. Global stability for a non-linear high dimensional autonomous differential system SEIQR model mi epidemiology[J]. Chinese Journal of Engineering Mathematics, 2007，24(1)：79-86．
    徐文雄，张太雷，徐宗本．非线性高维自治微分系统SEIQR流行病模型全局稳定性[J]．工程数学学报，2007，24(1)：79-86.
    [8]
    Sun Chengjun, Lin Yiping, Tang Shoupeng. Global stability for an special SEIR epidemic model with nonlinear incidence rates[J]. Chaos Solitons and Fractals, 2007, 33(1): 290-297.
    [9]
    Hale J K. Ordinary Differential Equations[M]. New York: Wiley-Interscience, 1969: 296-297.
    [10]
    Thieme H R. Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations[J]. J Math Biol, 1992, 30: 755-760.
    [11]
    Li M Y, Muldowney J S. Global stability for the SEIR model in epidemiology[J]. Mathematics Biosciences, 1995, 125(2): 155-164.
    [12]
    马知恩，周义仓．常微分方程定性与稳定性方法[M]．北京：科教出版社，2001: 147-150．
    [13]
    Muldowney J S. Compound matrices and ordinary differential equations[J]. Rocky Mt J Math, 1990, 20: 857-872.
    [14]
    Hofbauer J, So J. Uniform persistence and repellors for maps[J]. Proc Amer Math Soc, 1989, 107: 1 137-1 142.
    [15]
    Smith H L. Systems of ordinary differential equations which generate an order preserving flow[J]. SIAM Rev, 1988, 30: 87-113.
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    Herzog G, Redheffer R. Nonautonomous SEIRS and Thron models for epidemiology and cell biology[J]. Nonlinear Anal: Real World Applications, 2004, 5: 33-44.
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    [1]
    刘丽君，魏来．丙型肝炎病毒的流行病学[J]．传染病信息，2007，20(5)：261-264．
    [2]
    方春红，梁虹，刘美琳．1850例尖锐湿疣形态与分布特点的临床分析[J]．中国麻风皮肤病杂志，2002，18(2)：138-139．
    [3]
    Fan M, Li M Y, Wang K. Global stability of an SEIS epidemic model with recruitment and a varying total population size[J]. Mathematical Biosciences, 2001, 170(2): 199-208.
    [4]
    Xu Wenxiong, Zhang Tailei. Global stability for the model with quarantine in epidemiology[J]. Journal of Xian Jiaotong University, 2005，39(2)：210-213．
    徐文雄，张太雷．具有隔离仓室流行病传播数学模型的全局稳定性[J]．西安交通大学学报，2005，39(2)：210-213．
    [5]
    Liu W M, van den Driessche P. Epidemiological models with varying total population size and dose-dependent latent period[J]. Mathematical Biosciences, 1995, 128(1-2): 57-69.
    [6]
    Li M Y, Graef J R, Wang L C, et al. Global dynamics of an SEIR model with varying total population size[J]. Mathematics Biosciences, 1999, 160(2): 191-213.
    [7]
    Xu Wenxiong, Zhang Tailei, Xu Zongben. Global stability for a non-linear high dimensional autonomous differential system SEIQR model mi epidemiology[J]. Chinese Journal of Engineering Mathematics, 2007，24(1)：79-86．
    徐文雄，张太雷，徐宗本．非线性高维自治微分系统SEIQR流行病模型全局稳定性[J]．工程数学学报，2007，24(1)：79-86.
    [8]
    Sun Chengjun, Lin Yiping, Tang Shoupeng. Global stability for an special SEIR epidemic model with nonlinear incidence rates[J]. Chaos Solitons and Fractals, 2007, 33(1): 290-297.
    [9]
    Hale J K. Ordinary Differential Equations[M]. New York: Wiley-Interscience, 1969: 296-297.
    [10]
    Thieme H R. Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations[J]. J Math Biol, 1992, 30: 755-760.
    [11]
    Li M Y, Muldowney J S. Global stability for the SEIR model in epidemiology[J]. Mathematics Biosciences, 1995, 125(2): 155-164.
    [12]
    马知恩，周义仓．常微分方程定性与稳定性方法[M]．北京：科教出版社，2001: 147-150．
    [13]
    Muldowney J S. Compound matrices and ordinary differential equations[J]. Rocky Mt J Math, 1990, 20: 857-872.
    [14]
    Hofbauer J, So J. Uniform persistence and repellors for maps[J]. Proc Amer Math Soc, 1989, 107: 1 137-1 142.
    [15]
    Smith H L. Systems of ordinary differential equations which generate an order preserving flow[J]. SIAM Rev, 1988, 30: 87-113.
    [16]
    Herzog G, Redheffer R. Nonautonomous SEIRS and Thron models for epidemiology and cell biology[J]. Nonlinear Anal: Real World Applications, 2004, 5: 33-44.
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