YANG Ya-li, LI Jian-quan, LIU Wan-meng, TANG San-yi. Global Stability of a Vector-Borne Epidemic Model With Distributed Delay and Nonlinear Incidence[J]. Applied Mathematics and Mechanics, 2013, 34(12): 1291-1299. doi: 10.3879/j.issn.1000-0887.2013.12.008
Citation: YANG Ya-li, LI Jian-quan, LIU Wan-meng, TANG San-yi. Global Stability of a Vector-Borne Epidemic Model With Distributed Delay and Nonlinear Incidence[J]. Applied Mathematics and Mechanics, 2013, 34(12): 1291-1299. doi: 10.3879/j.issn.1000-0887.2013.12.008

Global Stability of a Vector-Borne Epidemic Model With Distributed Delay and Nonlinear Incidence

doi: 10.3879/j.issn.1000-0887.2013.12.008
Funds:  The National Natural Science Foundation of China(11071256;11171267;11301320;11371369);China Postdoctoral Science Foundation(2013M532016)
  • Received Date: 2013-08-14
  • Rev Recd Date: 2013-12-04
  • Publish Date: 2013-12-16
  • An SIR vector-borne epidemic model with distributed delay and nonlinear incidence was established, the basic reproduction number determining the uniform persistence of the disease was found. When the basic reproduction number was not greater than 1, the disease died out finally; when the basic reproduction number was greater than 1, the model had a unique endemic equilibrium, and the disease uniformly persisted in the population. By constructing Lyapunov functional, it was proved that, under certain conditions, the endemic equilibrium was globally stable in the feasible region only when it existed. In addition, the non-uniqueness of the suitable Lyapunov functionals was shown for proving the global stability of the endemic equilibrium.
  • loading
  • [1]
    Ma Z E, Li J. Dynamical Modeling and Analysis of Epidemics[M]. World Scientific, 2009.
    [2]
    Capasso V, Serio G. A generalization of the Kermack-McKendrick deterministic epidemic model[J]. Mathematical Biosciences,1978, 42(1/2): 43-61.
    [3]
    Liu W M, Levin S A, Iwasa Y. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models[J]. Journal of Mathematical Biology, 1986, 23(2): 187-204.
    [4]
    Gao S J, Chen L S, Nieto J J, Torres A. Analysis of a delayed epidemic model with pulse vaccination and saturation incidence[J]. Vaccine, 2006, 24(35/36): 6037-6045.
    [5]
    Korobeinikov A, Maini P K.  Nonlinear incidence and stability of infectious disease models[J]. Mathematical Medicine and Biology, 2005, 22(2): 113-128.
    [6]
    Cooke K L. Stability analysis for a vector disease model[J]. Rocky Mountain Journal of Mathematics, 1979, 9(1): 31-42.
    [7]
    Xu R, Ma Z E. Stability of a delayed SIRS epidemic model with a nonlinear incidence rate[J]. Chaos Solutions and Fractals,2009, 41(5): 2319-2325.
    [8]
    Huang G, Takeuchi Y, Ma W B , Wei D J. Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate[J]. Bulletin of Mathematical Biology, 2010, 72(5): 1192-1207.
    [9]
    Beretta E, Takeuchi Y. Convergence results in SIR epidemic models with varying population size[J]. Nonlinear Analysis: Theory, Method and Applications, 1997, 28(12): 1909-1921.
    [10]
    Takeuchi Y, Ma W B,  Beretta E. Global asymptotic properties of a delay SIR epidemic model with finite incubation times[J]. Nonlinear Analysis: Theory, Methods and Applications, 2000, 42(6): 931-947.
    [11]
    Nakata Y, Enatsu Y, Muroya Y. On the global stability of an SIRS epidemic model with distributed delays[J]. Discrete and Continuous Dynamical Systems, 2011(Supp): 1119-1128.
    [12]
    Enatsu Y, Nakata Y, Muroya Y.  Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model[J]. Nonlinear Analysis: Real World Applications, 2012, 13(5): 2120-2133.
    [13]
    Kuang Y. Delay Differential Equations With Applications in Population Dynamics [M]. San Diego: Academic Press, 1993.
    [14]
    Li J Q, Song X C, Gao F Y. Global stability of a viral infection model with delays and two types of target cells[J]. Journal of Applied Analysis and Computation, 2012, 2(3): 281-292.
  • 加载中

Catalog

    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

    Article Metrics

    Article views (1557) PDF downloads(1153) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return