Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures

  • Received: 01 July 2013 Accepted: 29 June 2018 Published: 01 September 2014
  • MSC : Primary: 34D20, 34D23; Secondary: 92D30.

  • In this paper, we formulate an SIR epidemic model with hybrid of multigroup and patch structures, which can be regarded as a model for the geographical spread of infectious diseases or a multi-group model with perturbation. We show that if a threshold value, which corresponds to the well-known basic reproduction number $R_0$, is less than or equal to unity, then the disease-free equilibrium of the model is globally asymptotically stable. We also show that if the threshold value is greater than unity, then the model is uniformly persistent and has an endemic equilibrium. Moreover, using a Lyapunov functional technique, we obtain a sufficient condition under which the endemic equilibrium is globally asymptotically stable. The sufficient condition is satisfied if the transmission coefficients in the same groups are large or the per capita recovery rates are small.

    Citation: Toshikazu Kuniya, Yoshiaki Muroya, Yoichi Enatsu. Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures[J]. Mathematical Biosciences and Engineering, 2014, 11(6): 1375-1393. doi: 10.3934/mbe.2014.11.1375

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  • In this paper, we formulate an SIR epidemic model with hybrid of multigroup and patch structures, which can be regarded as a model for the geographical spread of infectious diseases or a multi-group model with perturbation. We show that if a threshold value, which corresponds to the well-known basic reproduction number $R_0$, is less than or equal to unity, then the disease-free equilibrium of the model is globally asymptotically stable. We also show that if the threshold value is greater than unity, then the model is uniformly persistent and has an endemic equilibrium. Moreover, using a Lyapunov functional technique, we obtain a sufficient condition under which the endemic equilibrium is globally asymptotically stable. The sufficient condition is satisfied if the transmission coefficients in the same groups are large or the per capita recovery rates are small.


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    [1] Oxford University, Oxford, 1991.
    [2] in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, Y. Zhou and J. Wu), Higher Education Press, 2009, 65-122.
    [3] in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1956, 81-109.
    [4] Academic Press, New York, 1979.
    [5] Appl. Math. Comput., 218 (2011), 4391-4400.
    [6] 1st edition, John Wiley and Sons, Chichester, 2000.
    [7] Math. Biosci., 80 (1986), 19-22.
    [8] J. Dynam. Diff. Equat., 6 (1994), 583-600.
    [9] Canadian Appl. Math. Quart., 14 (2006), 259-284.
    [10] Proc. Amer. Math. Soc., 136 (2008), 2793-2802.
    [11] in Bioterrorism (eds. H. T. Banks and C. Castillo-Chavez), SIAM, 2003, 211-236.
    [12] J. Math. Anal. Appl., 308 (2005), 343-364.
    [13] Math. Med. Biol., 21 (2004), 75-83.
    [14] Discrete Cont. Dyn. Sys. Series B, 19 (2014), 1105-1118.
    [15] SIAM, Philadelphia, 1976.
    [16] Math. Biosci., 160 (1999), 191-213.
    [17] Canadian Appl. Math. Quart., 17 (2009), 175-187.
    [18] J. Diff. Equat., 284 (2010), 1-20.
    [19] J. Math. Anal. Appl., 361 (2010), 38-47.
    [20] Acta Mathematica Scientia, 33 (2013), 341-361.
    [21] Nonlinear Anal. RWA, 13 (2012), 1581-1592.
    [22] Cambridge University Press, Cambridge, 1995.
    [23] Comput. Math. Appl., 60 (2010), 2286-2291.
    [24] Math. Biosci., 180 (2002), 29-48.
    [25] Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962.
    [26] Math. Biosci., 190 (2004), 97-112.
    [27] Nonlinear Anal. RWA, 11 (2010), 995-1004.
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