Research article

Global stability of an age-structured infection model in vivo with two compartments and two routes


  • Received: 11 January 2022 Revised: 21 June 2022 Accepted: 23 June 2022 Published: 02 August 2022
  • In this paper, for an infection age model with two routes, virus-to-cell and cell-to-cell, and with two compartments, we show that the basic reproduction ratio $ R_0 $ gives the threshold of the stability. If $ R_0 > 1 $, the interior equilibrium is unique and globally stable, and if $ R_0 \le 1 $, the disease free equilibrium is globally stable. Some stability results are obtained in previous research, but, for example, a complete proof of the global stability of the disease equilibrium was not shown. We give the proof for all the cases, and show that we can use a type reproduction number for this model.

    Citation: Tsuyoshi Kajiwara, Toru Sasaki, Yoji Otani. Global stability of an age-structured infection model in vivo with two compartments and two routes[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11047-11070. doi: 10.3934/mbe.2022515

    Related Papers:

  • In this paper, for an infection age model with two routes, virus-to-cell and cell-to-cell, and with two compartments, we show that the basic reproduction ratio $ R_0 $ gives the threshold of the stability. If $ R_0 > 1 $, the interior equilibrium is unique and globally stable, and if $ R_0 \le 1 $, the disease free equilibrium is globally stable. Some stability results are obtained in previous research, but, for example, a complete proof of the global stability of the disease equilibrium was not shown. We give the proof for all the cases, and show that we can use a type reproduction number for this model.



    加载中


    [1] R. Qesmi, J. Wu, J. Wu, J. M. Feffernan, Influence of backward bifurcation in a model of hepatitis B and C viruses, Math. Biosci., 224 (2010), 118–125. https://doi.org/10.1016/j.mbs.2010.01.002 doi: 10.1016/j.mbs.2010.01.002
    [2] R. Qesmi, S. Elsaadan, J. M. Heffernan, J. Wu, A hepatitis B and C virus model with age since infection that exhibits backward bifurcation, SIAM J. Appl. Math., 71 (2011), 1509–1530. https://doi.org/10.1137/10079690X doi: 10.1137/10079690X
    [3] T. Kajiwara, T. Sasaki, Y. Takeuchi, Construction of Lyapunov functions of the models for infectious diseases in vivo: from simple models to complex models, Math. Biosci. Eng., 12 (2015), 117–133. https://doi.org/10.3934/mbe.2015.12.117 doi: 10.3934/mbe.2015.12.117
    [4] W. Hübner, G. P. McNerney, P. Chen, B. M. Dale, R. E. Gordon, F. Y. Chuang, et al., Quantitative 3D video microscopy of HIV transfer across T cell virological synapses, Science, 323 (2009), 1743–1747. https://doi.org/10.1126/science.1167525 doi: 10.1126/science.1167525
    [5] H. Pourbashash, S. S. Pilyugin, P. de Leenheer, C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete Contin. Dyn. Syst. Ser. B, 10 (2014), 3341–3337. https://doi.org/10.3934/dcdsb.2014.19.3341 doi: 10.3934/dcdsb.2014.19.3341
    [6] X. Lai, X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898–917. https://doi.org/10.1137/130930145 doi: 10.1137/130930145
    [7] J. Wang, J. Lang, X. Zou, Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal. Real World Appl., 34 (2017), 75–96. https://doi.org/10.1016/j.nonrwa.2016.08.001 doi: 10.1016/j.nonrwa.2016.08.001
    [8] P. Wu, H. Zhao, Dynamics of an HIV infection model with two infection routes and evolutionary competition between two viral strains, Appl. Math. Model., 84 (2020), 240–264. https://doi.org/10.1016/j.apm.2020.03.040 doi: 10.1016/j.apm.2020.03.040
    [9] C. Y. Cheng, Y. Dong, Y. Takeuchi, An age-structured virus model with two routes of infection in heterogeneous environment, Nonlinear Anal. Real World Appl., 39 (2018), 464–491. https://doi.org/10.1016/j.nonrwa.2017.07.013 doi: 10.1016/j.nonrwa.2017.07.013
    [10] P. Wu, H. Zhao, Mathematical analysis of multi-target cells and multi-strain age-structured model with two HIV infection route, Int. J. Biomath., 14 (2021), 2150057. https://doi.org/10.1142/S1793524521500571 doi: 10.1142/S1793524521500571
    [11] M. G. Roberts, J. A. P. Heesterbeek, A new method for estimating the effort required to control an infectious disease, Proc. Roy. Soc. B., 270 (2003), 1359–1364. https://doi.org/10.1098/rspb.2003.2339 doi: 10.1098/rspb.2003.2339
    [12] H. Smith, H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Soceity, Providence, 2011.
    [13] T. Kajiwara, T. Sasaki, Y. Otani, Global stability of age-structured models for pathogen-immune interaction, J. Appl. Math. Comput., 50 (2019), 631–660. https://doi.org/10.1007/s12190-018-1194-8 doi: 10.1007/s12190-018-1194-8
    [14] J. A. P. Heesterbeek, M. G. Roberts, The type-reproduction number T in models for infectious disease control, Math. Biosci., 206 (2007), 3–10. https://doi.org/10.1016/j.mbs.2004.10.013 doi: 10.1016/j.mbs.2004.10.013
    [15] R. D. Demasse, A. Ducrot, An age-structured within-host model for multistrain malaria infection, SIAM J. Appl. Math., 73 (2013), 572–593. https://doi.org/10.1137/120890351 doi: 10.1137/120890351
    [16] T. Kajiwara, T. Sasaki, Y. Otani, Global stability of age-structured multistrain models for pathogen-immune interaction, J. Appl. Math. Comput., 62 (2020), 239–279. https://doi.org/10.1007/s12190-019-01283-w doi: 10.1007/s12190-019-01283-w
    [17] G. Sell, Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. https://doi.org/10.1007/978-1-4757-5037-9
    [18] T. Kajiwara, T. Sasaki, Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Anal. Real World Appl., 13 (2012), 1802–1826. https://doi.org/10.1016/j.nonrwa.2011.12.011 doi: 10.1016/j.nonrwa.2011.12.011
  • Reader Comments
  • © 2022 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1444) PDF downloads(81) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog