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Qiang Hou. GLOBAL ANALYSIS OF A MULTI-GROUP ANIMAL EPIDEMIC MODEL WITH INDIRECT INFECTION AND TIME DELAY[J]. Journal of Applied Analysis & Computation, 2016, 6(4): 1023-1040. doi: 10.11948/2016066
Citation: Qiang Hou. GLOBAL ANALYSIS OF A MULTI-GROUP ANIMAL EPIDEMIC MODEL WITH INDIRECT INFECTION AND TIME DELAY[J]. Journal of Applied Analysis & Computation, 2016, 6(4): 1023-1040. doi: 10.11948/2016066
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GLOBAL ANALYSIS OF A MULTI-GROUP ANIMAL EPIDEMIC MODEL WITH INDIRECT INFECTION AND TIME DELAY

  • Department of Mathematics, North University of China, No.3 Xueyuan Road, Taiyuan 030051, Shanxi Province, China

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    Abstract
  • The transmission mechanism of some animal diseases is complex because of the multiple transmission pathways and multiple-group interactions, which lead to the limited understanding of the dynamics of these diseases transmission. In this paper, a delay multi-group dynamic model is proposed in which time delay is caused by the latency of infection. Under the biologically motivated assumptions, the basic reproduction number R0 is derived and then the global stability of the disease-free equilibrium and the endemic equilibrium is analyzed by Lyapunov functionals and a graph-theoretic approach as for time delay. The results show the global properties of equilibria only depend on the basic reproductive number R0:the disease-free equilibrium is globally asymptotically stable if R0 ≤ 1; if R0>1, the endemic equilibrium exists and is globally asymptotically stable, which implies time delay span has no effect on the stability of equilibria. Finally, some specific examples are taken to illustrate the utilization of the results and then numerical simulations are used for further discussion. The numerical results show time delay model may experience periodic oscillation behaviors, implying that the spread of animal diseases depends largely on the prevention and control strategies of all sub-populations.
    MSC: 34D20;37N25
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  • [1] R. Breban, J.M. Drake, D.E. Stallknecht and P. Rohani, The role of environmental transmission in recurrent avian influenza epidemics, PLoS Comput. Biol., 5(2009)(10), e1000346. doi:10.1371/journal.pcbi.1000346.

    Google Scholar

    [2] C.J. Briggs and H.C.J. Godfray, The dynamics of insect-pathogen interactions in stage-structured populations, Am. Nat., 145(1995), 845-887.

    Google Scholar

    [3] C.T. Codeço, Endemic and epidemic dynamics of cholera:the role of the aquatic reservoir, BMC Infect. Dis., 1(2001)(1), doi:10.1186/1471-2334-1-1.

    Google Scholar

    [4] O. Diekmann, J.A.P. Heesterbeek and J.A.J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28(1990), 365-382.

    Google Scholar

    [5] Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model, Nonlinear Anal. RWA., 13(2012), 2120-2133.

    Google Scholar

    [6] N.M. Ferguson, C.A. Donnelly and R.M. Anderson, The foot-and-mouth epidemic in great Britain:pattern of spread and impact of interventions, Science, 292(2001), 1155-1160.

    Google Scholar

    [7] H. Guo, M.Y. Li and Z.S Shuai, Global stability of the endemic equilibrium of multi-group SIR epidemic models, Canad. Appl. Math. Quart., 14(2006)(3), 259-284.

    Google Scholar

    [8] H. Guo, M.Y. Li and Z.S. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136(2008), 2793-2802.

    Google Scholar

    [9] J.K. Hale, P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20(1989), 388-395.

    Google Scholar

    [10] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63(2011), 125-139.

    Google Scholar

    [11] V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci., 111(1992), 1-71.

    Google Scholar

    [12] M.J. Keeling and C.A. Gilligan, Meta-population dynamics of bubonic plague, Nature, 407(2000), 903-905.

    Google Scholar

    [13] M.J. Keeling, M.E.J. Woolhouse, R.M. May, G. Davies and B.T. Grenfell, Modeling vaccination strategies against foot-and-mouth disease, Nature, 421(2003)(9), 136-142.

    Google Scholar

    [14] R.J. Knell, M. Begon and D.J. Thompson, Transmission dynamics of Bacillus thuringiensis infecting Plodia interpunctella:a test of the mass action assumption with an insect pathogen, Proc. R. Soc. London B Biol. Sci., 263(1996), 75-81.

    Google Scholar

    [15] M.Y. Li, Z.S. Shuai and C.C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361(2010), 38-47.

    Google Scholar

    [16] M.T. Li, G.Q. Sun, Y.F. Wu, J. Zhang and Z. Jin, Transmission dynamics of a multi-group brucellosis model with mixed cross infection in public farm, Appl. Math. Comput., 237(2014), 582-594.

    Google Scholar

    [17] W.M. Liu and S.A.Levin, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23(1986), 187-204.

    Google Scholar

    [18] S. Roy, T.F. McElwain and Y. Wan, A network control theory approach to modeling and optimal control of zoonoses:case study of brucellosis transmission in sub-saharan Africa, PLoS Negl. Trop. Dis., 5(2011)(10), e1259. doi:10.1371/journal.pntd. 0001259.

    Google Scholar

    [19] G. Röst, J.H. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5(2008), 389-402.

    Google Scholar

    [20] G. Pappas, P. Papadimitriou, N. Akritidis, L. Christou and E.V. Tsianos, The new global map of human brucellosis, Lancet Infect. Dis., 6(2006), 91-99.

    Google Scholar

    [21] P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180(2002), 29-48.

    Google Scholar

    [22] R. Xu and Z.E. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. RWA., 10(2009), 3175-3189.

    Google Scholar

    [23] Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. RWA., 11(2010), 995-1004.

    Google Scholar

    [24] J. Zhang, Z. Jin, G.Q. Sun, T. Zhou and S.G. Ruan, Analysis of rabies in China:transmission dynamics and control, PLoS ONE, 6(2011)(7), e20891. doi:10.1371/journal. pone.0020891.

    Google Scholar

    [25] X. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.

    Google Scholar

    [26] J. Zinsstag, S. Dürra, M. Penny, R. Mindekem, F. Roth, S. Menendez Gonzalez, S. Naissengar and J. Hattendorf, Transmission dynamics and economics of rabies control in dogs and humans in an African city, Proc. Natl. Acad. USA., 106(2009), 14996-15001.

    Google Scholar

    [27] J. Zinsstag, F. Roth, D. Orkhon, G. Chimed-Ochir, M. Nansalmaa, J. Kolar and P. Vounatsou, A model of animal-human brucellosis transmission in Mongolia, Prev. Vet. Med., 69(2005), 77-95.

    Google Scholar

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