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Open AccessOpen Access JUSTC Research Articles: Mathematics

Global dynamics of an SEIQR model with saturation incidence rate and hybrid strategies

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

Cite this:
https://doi.org/10.52396/JUST-2020-1046
  • Received Date: 02 April 2020
  • Rev Recd Date: 10 September 2020
  • Publish Date: 28 February 2021
    Abstract

  • Abstract

    An SEIQR epidemic model with the saturation incidence rate and hybrid strategies was proposed, and the stability of the model was analyzed theoretically and numerically. Firstly, the basic reproduction number R0 was derived, which determines whether the disease was extinct or not. Secondly, through LaSalle's invariance principle, it was proved that the disease-free equilibrium is globally asymptotically stable and the disease generally dies out when R0<1. By Routh-Hurwitz criterion theory, it was proved that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R0>1. Thirdly, according to the periodic orbit stability theory and the second additive compound matrix, it was proved that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R0>1. Finally, some numerical simulations were carried out to illustrate the results.

    Abstract

    An SEIQR epidemic model with the saturation incidence rate and hybrid strategies was proposed, and the stability of the model was analyzed theoretically and numerically. Firstly, the basic reproduction number R0 was derived, which determines whether the disease was extinct or not. Secondly, through LaSalle's invariance principle, it was proved that the disease-free equilibrium is globally asymptotically stable and the disease generally dies out when R0<1. By Routh-Hurwitz criterion theory, it was proved that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R0>1. Thirdly, according to the periodic orbit stability theory and the second additive compound matrix, it was proved that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R0>1. Finally, some numerical simulations were carried out to illustrate the results.
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    • References(21)
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    [2]
    Ma Yanli. Global dynamics of an SEIR model with infectious force in latent and recovered period and standard incidence rate. International Journal of Applied Physics and Mathematics, 2017, 7(1): 1-11.
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    Liu L. A delayed SIR model with general nonlinear incidence rate. Advances in Difference Equations, 2015, 329: 1-10.
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    Li T, Xue Y. Global stability analysis of a delayed SEIQR epidemic model with quarantine and latent. Applied Mathematics, 2013, 4: 109-117.
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    Bai Z. Global dynamics of a SEIR model with information dependent vaccination and periodically varying transmission rate. Mathematical Methods in the Applied Sciences, 2015, 38(11): 2403-2410.
    [14]
    Shen M, Xiao Y. Global stability of a multi-group SVEIR epidemiological model with the vaccination age and infection age. Acta Applicandae Mathematicae, 2016,144: 137-157.
    [15]
    Liu D, Wang B, Guo S. Stability analysis of a novel epidemics model with vaccination and nonlinear infectious rate. Applied Mathematics and Computation, 2013, 221: 786-801.
    [16]
    Chauhan S, Misra O P, Dhar J. Stability analysis of SIR model with vaccination. American Journal of Computational and Applied Mathematics, 2014, 4: 17-23.
    [17]
    Ma Yanli, Liu Jiabao, Li Haixia. Global dynamics of an SIQR model with vaccination and elimination hybrid strategies. Mathematics, 2018, 6(12): 328-339.
    [18]
    Zhang Z. Stability properties in an SIR model with asynchronous pulse vaccination and pulse elimination. Journal of Shanxi Normal University(Natural Science Edition), 2012, 2: 8-11.
    [19]
    Ma Zhi'en, Zhou Yicang, Wang Wendi, et al.Mathematical Modeling and Research on Dynamics of Infectious Diseases. Beijing: Science Press,2004.(in Chinese)
    [20]
    Li Dongmei, Liu Weihua, Zheng Zhongtao. Qualitative analysis of epidemic models with vaccination and isolation. Journal of Harbin University of Science and Technology, 2012, 17(2): 122-126. (in Chinese)
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    Wang Caiyun, Ji Xiaoming, Jia Jianwen. Global analysis of an SEIQR model with the exposed individuals quarantined. Journal of Shanxi Normal University (Natural Science Edition), 2015, 29(4): 11-14.(in Chinese).
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    [1]
    Yuan Y, Bélair J. Threshold dynamics in an SEIRS model with latency and temporary immunity. Journal of Mathematical Biology, 2014, 69: 875-904.
    [2]
    Ma Yanli. Global dynamics of an SEIR model with infectious force in latent and recovered period and standard incidence rate. International Journal of Applied Physics and Mathematics, 2017, 7(1): 1-11.
    [3]
    Sun C, Hsieh Y H. Global analysis of an SEIR model with varying population size and vaccination. Applied Mathematical Modelling, 2010, 34(10):2685-2697.
    [4]
    Ma Yanli, Zhang Zhonghua.Asymptotical analysis of SEIR model with infectious force in latent and immune periods. Journal of University of Science and Technology of China, 2016, 46(2): 95-103.(in Chinese)
    [5]
    Xu R, Ma Z. Global stability of a delayed SEIRS epidemic model with saturation incidence rate. Nonlinear Dynamics, 2010, 61: 229-239.
    [6]
    Xu R. Global dynamics of an SEIS epidemic model with saturation incidence and latent period. Applied Mathematics and Computation, 2012, 218(15): 7927-7938.
    [7]
    Liu L. A delayed SIR model with general nonlinear incidence rate. Advances in Difference Equations, 2015, 329: 1-10.
    [8]
    Li T, Xue Y. Global stability analysis of a delayed SEIQR epidemic model with quarantine and latent. Applied Mathematics, 2013, 4: 109-117.
    [9]
    Silva C M. A nonautonomous epidemic model with general incidence and isolation. Mathematical Methods in the Applied Sciences, 2014, 37(13): 1974-1991.
    [10]
    Ma Yanli, Zhang Zhonghua, Liu Jiabao, et al. An SIQR mode with impulsive vaccination and impulsive elimination. Journal of University of Science and Technology of China, 2018, 48(2): 111-117. (in Chinese)
    [11]
    Tan X X, Li S J, Dai Q W, et al. An epidemic model with isolated intervention based on cellular automata. Advanced Materials Research, 2014, 926: 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. Biosystems, 2015,129(3): 50-65.
    [13]
    Bai Z. Global dynamics of a SEIR model with information dependent vaccination and periodically varying transmission rate. Mathematical Methods in the Applied Sciences, 2015, 38(11): 2403-2410.
    [14]
    Shen M, Xiao Y. Global stability of a multi-group SVEIR epidemiological model with the vaccination age and infection age. Acta Applicandae Mathematicae, 2016,144: 137-157.
    [15]
    Liu D, Wang B, Guo S. Stability analysis of a novel epidemics model with vaccination and nonlinear infectious rate. Applied Mathematics and Computation, 2013, 221: 786-801.
    [16]
    Chauhan S, Misra O P, Dhar J. Stability analysis of SIR model with vaccination. American Journal of Computational and Applied Mathematics, 2014, 4: 17-23.
    [17]
    Ma Yanli, Liu Jiabao, Li Haixia. Global dynamics of an SIQR model with vaccination and elimination hybrid strategies. Mathematics, 2018, 6(12): 328-339.
    [18]
    Zhang Z. Stability properties in an SIR model with asynchronous pulse vaccination and pulse elimination. Journal of Shanxi Normal University(Natural Science Edition), 2012, 2: 8-11.
    [19]
    Ma Zhi'en, Zhou Yicang, Wang Wendi, et al.Mathematical Modeling and Research on Dynamics of Infectious Diseases. Beijing: Science Press,2004.(in Chinese)
    [20]
    Li Dongmei, Liu Weihua, Zheng Zhongtao. Qualitative analysis of epidemic models with vaccination and isolation. Journal of Harbin University of Science and Technology, 2012, 17(2): 122-126. (in Chinese)
    [21]
    Wang Caiyun, Ji Xiaoming, Jia Jianwen. Global analysis of an SEIQR model with the exposed individuals quarantined. Journal of Shanxi Normal University (Natural Science Edition), 2015, 29(4): 11-14.(in Chinese).
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