Research article

Bifurcation analysis of an age-structured SIRI epidemic model

  • Received: 03 July 2020 Accepted: 15 September 2020 Published: 21 October 2020
  • In this paper, an SIRI epidemic model with age of infection and the proliferation of susceptible individuals with logistic growth is investigated. By using the theory of integral semigroup and Hopf bifurcation theory for semilinear equations with non-dense domain, it is shown that if the threshold parameter is greater than unity, sufficient condition is derived for the occurrence of the Hopf bifurcation. Numerical simulations are carried out to illustrate the theoretical results.

    Citation: Xiaohong Tian, Rui Xu, Ning Bai, Jiazhe Lin. Bifurcation analysis of an age-structured SIRI epidemic model[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7130-7150. doi: 10.3934/mbe.2020366

    Related Papers:

  • In this paper, an SIRI epidemic model with age of infection and the proliferation of susceptible individuals with logistic growth is investigated. By using the theory of integral semigroup and Hopf bifurcation theory for semilinear equations with non-dense domain, it is shown that if the threshold parameter is greater than unity, sufficient condition is derived for the occurrence of the Hopf bifurcation. Numerical simulations are carried out to illustrate the theoretical results.
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    [1] E. Beretta, Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33 (1995), 250-260.
    [2] H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.
    [3] A. Kaddar, Stability analysis in a delayed SIR epidemic model with a saturated incidence rate, Nonlinear Anal. Model. Control, 15 (2010), 299-306.
    [4] Q. Liu, D. Jiang, Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation, Appl. Math. Lett., 73 (2017), 8-15.
    [5] W. Ma, M. Song, Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), 1141-1145.
    [6] P. Magal, O. Seydi, G. F. Webb, Final size of an epidemic for a two-group SIR model, SIAM J. Appl. Math., 76 (2016), 2042-2059.
    [7] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59.
    [8] D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136-139.
    [9] A. M. Spagnuolo, M. Shillor, L. Kingsland, A. Thatcher, M. Toeniskoetter, B. Wood, A logistic delay differential equation model for chagas disease with interrupted spraying schedules, J. Biol. Dyn., 6 (2012), 377-394.
    [10] L. Gao, H. W. Hethcote, Disease transmission models with density-dependent demographics, J. Math. Biol., 30 (1992), 717-731.
    [11] J. Li, Z. Teng, G. Wang, L. Zhang, C. Hu, Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment, Chaos Soliton. Fract., 99 (2017), 63-71.
    [12] S. P. Rajasekar, M. Pitchaimani, Ergodic stationary distribution and extinction of a stochastic SIRS epidemic model with logistic growth and nonlinear incidence, Appl. Math. Comput., 377 (2020), 125143.
    [13] N. Yoshida, T. Hara, Global stability of a delayed SIR epidemic model with density dependent birth and death rates, J. Comput. Appl. Math., 201 (2007), 339-347.
    [14] J. Zhang, J. Li, Z. Ma, Global analysis of SIR epidemic models with population size dependent contact rate, J. Eng. Math., 21 (2004), 259-267.
    [15] J. Zhou, H. W. Hethcote, Population size dependent incidence in models for diseases without immunity, J. Math. Biol., 32 (1994), 809-834.
    [16] L. Zhu, H. Hu, A stochastic SIR epidemic model with density dependent birth rate, Adv. Difference Equ., 330 (2015), 1-12.
    [17] X. Rui, X. Tian, F. Zhang, Global dynamics of a tuberculosis transmission model with age of infection and incomplete treatment, Adv. Difference Equ., 242 (2017), 1-34.
    [18] J. Yang, Z. Qiu, Z. Li, Global stability of an age-structured cholera model, Math. Biosci. Eng., 11 (2014), 641-665.
    [19] Z. Feng, W. Huang, C. C. Chavez, Global behavior of a multi-group SIS epidemic model with age structure, J. Differ. Equations, 218 (2005), 292-324.
    [20] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434.
    [21] P. Magal, C. C. McCluskey, G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.
    [22] X. Duan, J. Yin, X. Li, Global Hopf bifurcation of an SIRS epidemic model with age-dependent recovery, Chaos Soliton. Fract., 104 (2017), 613-624.
    [23] Y. Chen, S. Zou, J. Yang, Global analysis of an SIR epidemic model with infection age and saturated incidence, Nonlinear Anal. Real World Appl., 30 (2016), 16-31.
    [24] P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differ. Equ., 65 (2001), 1-35.
    [25] P. Magal, S. Ruan, On semilinear Cauchyproblems with non-dense domain, Adv. Differ. Equations, 14 (2009), 1041-1084.
    [26] Z. Liu, P. Magal, S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.
    [27] Z. Wang, Z. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model, J. Math. Anal. Appl., 385 (2012), 1134-1150.
    [28] Y. Song, S. Yuan, Bifurcation analysis in a predator-prey system with time delay, Nonlinear Anal., 7 (2006), 265-284.
    [29] S. M. Blower, A. R. Mclean, T. C. Porco, P. M. Small, P. C. Hopewell, M. A. Sanchez, et al., The intrinsic transmission dynamics of tuberculosis epidemics, Nat. Med., 1 (1995), 815-821.
    [30] C. C. Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404.
    [31] H. Cao, D. Yan, S. Zhang, X. Wang, Analysis of dynamics of recurrent epidemics: periodic or non-periodic, B. Math. Biol., 81 (2019), 4889-4907.
    [32] P. van den Driessche, L. Wang, X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4(2) (2007), 205-219.
    [33] A. Hoare, D. G. Regan, D. P. Wilson, Sampling and sensitivity analyses tools (SaSAT) for computational modelling, Theor. Biol. Med. Model., 5 (2008), 4.
    [34] S. Marino, I. B. Hogue, C. J. Ray, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196.
    [35] A. Ducrot, Z. Liu, P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518.

    © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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