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Mathematical Biosciences and Engineering

2013, Volume 10, Issue 3: 499-521. doi: 10.3934/mbe.2013.10.499
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A singularly perturbed SIS model with age structure

  • 1.
    School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban
  • 2.
    School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4041
  • 3.
    Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw
  • Received: 01 May 2012 Accepted: 29 June 2018 Published: 01 April 2013

  • MSC : Primary: 34E15, 92D30; Secondary: 34E13.

  • We present a preliminary study of an SIS model with a basic age structure and we focus on a disease with quick turnover, such as influenza or common cold. In such a case the difference between the characteristic demographic and epidemiological times naturally introduces two time scales in the model which makes it singularly perturbed. Using the Tikhonov theorem we prove that for certain classes of initial conditions the nonlinear structured SIS model can be approximated with very good accuracy by lower dimensional linear models.

    Citation: Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure[J]. Mathematical Biosciences and Engineering, 2013, 10(3): 499-521. doi: 10.3934/mbe.2013.10.499

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  • We present a preliminary study of an SIS model with a basic age structure and we focus on a disease with quick turnover, such as influenza or common cold. In such a case the difference between the characteristic demographic and epidemiological times naturally introduces two time scales in the model which makes it singularly perturbed. Using the Tikhonov theorem we prove that for certain classes of initial conditions the nonlinear structured SIS model can be approximated with very good accuracy by lower dimensional linear models.


    [1] Physics of Life Reviews, 8 (2011), 19-20.
    [2] preprint.
    [3] in preparation.
    [4] Physics of Life Reviews, 8 (2011), 1-18.
    [5] Springer-Verlag, New York, 1993.
    [6] Math. Intelligencer, 12 (1990), 57-64.
    [7] in "Mathematical Epidemiology" (eds. F. Brauer, P. van den Driessche and J. Wu), LNM 1945, Springer, Berlin, (2008), 3-18.
    [8] J. Differ. Equ., 31 (1979), 53-98.
    [9] J. Math. Biol., 60 (2010), 347-386.
    [10] J. Math. Anal. Appl., 18 (1967), 129-134.
    [11] in "Dynamical Systems" (ed. R. Johnson), LNM 1609, Springer, Berlin, (1995), 44-118.
    [12] SIAM J. Math. Anal., 33 (2001), 286-314.
    [13] in "Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic" (eds. V. Capasso and M. Lachowicz), LNM 1940, Springer, (2008), 201-68.
    [14] Prob. Engin. Mech., 26 (2011), 54-60.
    [15] SIAM J. Appl. Math., 52 (1992), 1688-1706.
    [16] Springer, New York, 2003.
    [17] http://www.tdi.texas.gov/pubs/videoresource/fscommoncold.pdf.
    [18] Ecol. Model., 6 (1992), 287-308.
    [19] 174 (2011), 109-117.
    [20] SIAM Reviews, 31 (1989), 446-477.
    [21] African Institute of Mathematical Sciences Postgraduate Diploma Essay 2011/12, http://archive.aims.ac.za/2011-12.
    [22] PLoS ONE, 6, e27140.
    [23] Princeton University Press, Princeton, 2003.
    [24] Springer, Berlin, 1985.
    [25] Nauka, Moscow, 1973, in Russian.
    [26] Moscow State University, 1978 (in Russian) (translation: Mathematical Research Center Technical Summary Report 2039, Madison, 1980).
    [27] SIAM Review, 36 (1994), 440-452.
    [28] Springer, New York, 2005.

  • This article has been cited by:

    1. Eddy Kimba Phongi, Jacek Banasiak, 2015, Canard-type solutions in epidemiological models, 1-60133-018-9, 85, 10.3934/proc.2015.0085
    2. Jacek Banasiak, Mirosław Lachowicz, 2014, Chapter 1, 978-3-319-05139-0, 1, 10.1007/978-3-319-05140-6_1
    3. Jacek Banasiak, Mirosław Lachowicz, 2014, Chapter 4, 978-3-319-05139-0, 105, 10.1007/978-3-319-05140-6_4
    4. Jacek Banasiak, Aleksandra Falkiewicz, Milaine S. S. Tchamga, 2018, Chapter 13, 978-3-319-71485-1, 249, 10.1007/978-3-319-71486-8_13
    5. Christian Kuehn, 2015, Chapter 20, 978-3-319-12315-8, 665, 10.1007/978-3-319-12316-5_20
    6. J. Banasiak, M.S. Seuneu Tchamga, Delayed stability switches in singularly perturbed predator–prey models, 2017, 35, 14681218, 312, 10.1016/j.nonrwa.2016.10.013
    7. Rodrigue Yves M'pika Massoukou, Hermane Mambili-Mamboundou, Singular perturbation analysis of a two-site model for an epidemic in age-structured population, 2018, 11, 1793-5245, 1850087, 10.1142/S1793524518500870
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    9. Yunyun Niu, Yulin Chen, Detian Kong, Bo Yuan, Jieqiong Zhang, Jianhua Xiao, Strategy evolution of panic pedestrians in emergent evacuation with assailants based on susceptible-infected-susceptible model, 2021, 570, 00200255, 105, 10.1016/j.ins.2021.04.040
    10. J. Banasiak, S.Y. Tchoumi, Multiscale malaria models and their uniform in-time asymptotic analysis, 2024, 03784754, 10.1016/j.matcom.2024.02.015
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