Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model

  • Received: 10 May 2016 Accepted: 19 September 2016 Published: 01 April 2017
  • MSC : Primary: 35B65, 35K57; Secondary: 47H20

  • We study the global stability issue of the reaction-convection-diffusion cholera epidemic PDE model and show that the basic reproduction number serves as a threshold parameter that predicts whether cholera will persist or become globally extinct. Specifically, when the basic reproduction number is beneath one, we show that the disease-free-equilibrium is globally attractive. On the other hand, when the basic reproduction number exceeds one, if the infectious hosts or the concentration of bacteria in the contaminated water are not initially identically zero, we prove the uniform persistence result and that there exists at least one positive steady state.

    Citation: Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model[J]. Mathematical Biosciences and Engineering, 2017, 14(2): 559-579. doi: 10.3934/mbe.2017033

    Related Papers:

  • We study the global stability issue of the reaction-convection-diffusion cholera epidemic PDE model and show that the basic reproduction number serves as a threshold parameter that predicts whether cholera will persist or become globally extinct. Specifically, when the basic reproduction number is beneath one, we show that the disease-free-equilibrium is globally attractive. On the other hand, when the basic reproduction number exceeds one, if the infectious hosts or the concentration of bacteria in the contaminated water are not initially identically zero, we prove the uniform persistence result and that there exists at least one positive steady state.


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    [1] [ E. Bertuzzo,R. Casagrandi,M. Gatto,I. Rodriguez-Iturbe,A. Rinaldo, On spatially explicit models of cholera epidemics, Journal of the Royal Society Interface, 7 (2010): 321-333.
    [2] [ E. Bertuzzo,L. Mari,L. Righetto,M. Gatto,R. Casagrandi,M. Blokesch,I. Rodriguez-Iturbe,A. Rinaldo, Prediction of the spatial evolution and effects of control measures for the unfolding Haiti cholera outbreak, Geophys. Res. Lett., 38 (2011): 1-5.
    [3] [ V. Capasso,S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. Epidemiol. Sante, 27 (1979): 121-132.
    [4] [ A. Carpenter, Behavior in the time of cholera: Evidence from the 2008-2009 cholera outbreak in Zimbabwe, in Social Computing, Behavioral-Cultural Modeling and Prediction, Springer, 8393 (2014), 237-244.
    [5] [ D. L. Chao,M. E. Halloran,I. M. Longini Jr., Vaccination strategies for epidemic cholera in Haiti with implications for the developing world, Proc. Natl. Acad. Sci. USA, 108 (2011): 7081-7085.
    [6] [ S. F. Dowell,C. R. Braden, Implications of the introduction of cholera to Haiti, Emerg. Infect. Dis., 17 (2011): 1299-1300.
    [7] [ M. C. Eisenberg,Z. Shuai,J. H. Tien,P. van den Driessche, A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013): 105-112.
    [8] [ L. Evans, Partial Differential Equations American Mathematics Society, Providence, Rhode Island, 1998.
    [9] [ H. I. Freedman,X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Differential Equations, 137 (1997): 340-362.
    [10] [ J. K. Hale, Asymptotic Behavior of Dissipative Systems Mathematical surveys and monographs, American Mathematics Society, Providence, Rhode Island, 1988.
    [11] [ D. M. Hartley,J. G. Morris,D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Med., 3 (2006): e7.
    [12] [ S.-B. Hsu,F.-B. Wang,X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013): 265-297.
    [13] [ Y. Lou,X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011): 543-568.
    [14] [ P. Magal,X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005): 251-275.
    [15] [ R. Martin,H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990): 1-44.
    [16] [ Z. Mukandavire,S. Liao,J. Wang,H. Gaff,D. L. Smith,J. G. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Natl. Acad. Sci. USA, 108 (2011): 8767-8772.
    [17] [ R. L. M. Neilan,E. Schaefer,H. Gaff,K. R. Fister,S. Lenhart, Modeling optimal intervention strategies for cholera, B. Math. Biol., 72 (2010): 2004-2018.
    [18] [ R. Piarroux,R. Barrais,B. Faucher,R. Haus,M. Piarroux,J. Gaudart,R. Magloire,D. Raoult, Understanding the cholera epidemic, Haiti, Emerg. Infect. Dis., 17 (2011): 1161-1168.
    [19] [ A. Rinaldo,E. Bertuzzo,L. Mari,L. Righetto,M. Blokesch,M. Gatto,R. Casagrandi,M. Murray,S. M. Vesenbeckh,I. Rodriguez-Iturbe, Reassessment of the 2010-2011 Haiti cholera outbreak and rainfall-driven multiseason projections, Proc. Natl. Acad. Sci. USA, 109 (2012): 6602-6607.
    [20] [ Z. Shuai,P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2011): 118-126.
    [21] [ H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Math. Surveys Monogr. 41 American Mathematical Society, Providence, Rhode Island, 1995.
    [22] [ H. L. Smith,X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001): 6169-6179.
    [23] [ H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992): 755-763.
    [24] [ H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009): 188-211.
    [25] [ J. P. Tian,J. Wang, Global stability for cholera epidemic models, Math. Biosci., 232 (2011): 31-41.
    [26] [ J. H. Tien,D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, B. Math. Biol., 72 (2010): 1506-1533.
    [27] [ J. H. Tien,Z. Shuai,M. C. Eisenberg,P. van den Driessche, Disease invasion on community net-works with environmental pathogen movement, J. Math. Biology, 70 (2015): 1065-1092.
    [28] [ N. K. Vaidya,F.-B. Wang,X. Zou, Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012): 2829-2848.
    [29] [ J. Wang,S. Liao, A generalized cholera model and epidemic-endemic analysis, J. Biol. Dyn., 6 (2012): 568-589.
    [30] [ J. Wang,C. Modnak, Modeling cholera dynamics with controls, Canad. Appl. Math. Quart., 19 (2011): 255-273.
    [31] [ X. Wang,D. Posny,J. Wang, A reaction-convection-diffusion model for cholera spatial dynamics, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016): 2785-2809.
    [32] [ X. Wang,J. Wang, Analysis of cholera epidemics with bacterial growth and spatial movement, J. Biol. Dyn., 9 (2015): 233-261.
    [33] [ W. Wang,X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011): 147-168.
    [34] [ W. Wang,X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012): 1652-1673.
    [35] [ WHO Cholera outbreak, South Sudan Disease Outbreak News, 2014. Available from: http://www.who.int/csr/don/2014_05_30/en/.
    [36] [ J. Wu, null, Theory and Applications of Partial Functional Differential Equations, , Springer, New York, 1996.
    [37] [ K. Yamazaki,X. Wang, Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016): 1297-1316.
    [38] [ X.-Q. Zhao, Dynamical Systems in Population Biology Springer-Verlag, New York, Inc., New York, 2003.
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