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Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model

Abstract / Introduction Related Papers Cited by
  • In this paper, we study the initial boundary value problem of a reaction-convection-diffusion epidemic model for cholera dynamics, which was developed in [38], named susceptible-infected-recovered-susceptible-bacteria (SIRS-B) epidemic PDE model. First, a local well-posedness result relying on the theory of cooperative dynamics systems is obtained. Via a priori estimates making use of the special structure of the system and continuation of local theory argument, we show that in fact this problem is globally well-posed. Secondly, we analyze the local asymptotic stability of the solutions based on the basic reproduction number associated with this model.
    Mathematics Subject Classification: Primary: 35B65, 35K57; Secondary: 47H20.

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  • [1]

    J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: an epidemic model, Lancet, 377 (2011), 1248-1255.doi: 10.1016/S0140-6736(11)60273-0.

    [2]

    M. Bani-Yaghoub, R. Gautam, Z. Shuai, P. v. d. Driessche and R. Ivanek, Reproduction numbers for infections with free-living pathogens growing in the environment, J. Biol. Dyn., 6 (2012), 923-940.doi: 10.1080/17513758.2012.693206.

    [3]

    O. Bratteli and P. E. T. Jorgensen, Positive Semigroups of Operators, and Applications, Springer, Dordrecht, Holland, 1984.doi: 10.1007/978-94-009-6484-6.

    [4]

    E. Bertuzzo, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, On spatially explicit models of cholera epidemics, Journal of the Royal Society Interface, 7 (2010), 321-333.doi: 10.1098/rsif.2009.0204.

    [5]

    D. L. Chao, M. E. Halloran and 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.doi: 10.1073/pnas.1102149108.

    [6]

    C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1 (2001), p1.doi: 10.1186/1471-2334-1-1.

    [7]

    W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups, in Differential Equations in Banach Spaces, Lecture Notes in Math. 1223, A. Favini and E. Obrecht eds., Springer-Verlag, Berlin, Heidelberg, 1223 (1986), 61-73.doi: 10.1007/BFb0099183.

    [8]

    S. F. Dowell and C. R. Braden, Implications of the introduction of cholera to Haiti, Emerg. Infect. Dis., 17 (2011), 1299-1300.doi: 10.3201/eid1707.110625.

    [9]

    P. V. D. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.doi: 10.1016/S0025-5564(02)00108-6.

    [10]

    M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013), 105-112.doi: 10.1016/j.mbs.2013.08.003.

    [11]

    G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-state Problems, Second edition, Springer Monographs in Mathematics, Springer, New York, 2011.doi: 10.1007/978-0-387-09620-9.

    [12]

    J. P. Grover, S.-B. Hsu and F.-B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone, Math. Biosci., 222 (2009), 42-52.doi: 10.1016/j.mbs.2009.08.006.

    [13]

    D. M. Hartley, J. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Med., 3 (2006), 63-69.doi: 10.1371/journal.pmed.0030007.

    [14]

    S.-B. Hsu, J. Jiang and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Differential Equations, 248 (2010), 2470-2496.doi: 10.1016/j.jde.2009.12.014.

    [15]

    S.-B. Hsu, F.-B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, J. Dynam. Differential Equations, 23 (2011), 817-842.doi: 10.1007/s10884-011-9224-3.

    [16]

    S.-B. Hsu, F.-B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297.doi: 10.1016/j.jde.2013.04.006.

    [17]

    R. I. Joh, H. Wang, H. Weiss and J. S. Weitz, Dynamics of indirectly transmitted infectious diseases with immunological threshold, Bull. Math. Biol., 71 (2009), 845-862.doi: 10.1007/s11538-008-9384-4.

    [18]

    S. Kakutani, Concrete representation of abstract (M)-spaces (a characterization of the space of continuous functions), Ann. Math., 42 (1941), 994-1024.doi: 10.2307/1968778.

    [19]

    C. Kapp, Zimbabwe's humanitarian crisis worsens, Lancet, 373 (2009), p447.doi: 10.1016/S0140-6736(09)60151-3.

    [20]

    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, 1995.

    [21]

    S. Liao and J. Wang, Stability analysis and application of a mathematical cholera model, Math. Biosci. Eng., 8 (2011), 733-752.doi: 10.3934/mbe.2011.8.733.

    [22]

    R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.doi: 10.1090/S0002-9947-1990-0967316-X.

    [23]

    C. Mugero and A. Hoque, Review of Cholera Epidemic in South Africa with Focus on KwaZulu-Natal Province, Technical Report. KwaZulu-Natal Department of Health, Pietermaritzburg, South America, 2001.

    [24]

    Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. L. Smith and J. G. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Natl. Acad. Sci. USA, 108 (2011), 8767-8772.doi: 10.1073/pnas.1019712108.

    [25]

    A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983.doi: 10.1007/978-1-4612-5561-1.

    [26]

    R. Piarroux, R. Barrais, B. Faucher, R. Haus, M. Piarroux, J. Gaudart, R. Magloire and D. Raoult, Understanding the cholera epidemic, Haiti, Emerg. Infect. Dis., 17 (2011), 1161-1168.doi: 10.3201/eid1707.110059.

    [27]

    L. Righetto, E. Bertuzzo, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, Modeling human movement in a cholera spreading along fluvial systems, Ecohydrol., 4 (2011), 49-55.doi: 10.1002/eco.122.

    [28]

    A. Rinaldo, E. Bertuzzo, L. Mari, L. Righetto, M. Blokesch, M. Gatto, R. Casagrandi, M. Murray, S. M. Vesenbeckh and 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.doi: 10.1073/pnas.1203333109.

    [29]

    D. A. Sack, R. B. Sack and C.-L. Chaignat, Getting serious about cholera, New Engl. J. Med., 355 (2006), 649-651.doi: 10.1056/NEJMp068144.

    [30]

    Z. Shuai, J. H. Tien and P. van den Driessche, Cholera models with hyperinfectivity and temporary immunity, Bull. Math. Biol., 74 (2012), 2423-2445.doi: 10.1007/s11538-012-9759-4.

    [31]

    Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.doi: 10.1137/120876642.

    [32]

    Z. Shuai, J. A. P. Heesterbeek and P. van den Driessche, Extending the type reproduction number to infectious disease control targeting contacts between types, J. Math. Biol., 67 (2013), 1067-1082.doi: 10.1007/s00285-012-0579-9.

    [33]

    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.

    [34]

    J. P. Tian and J. Wang, Global stability for cholera epidemic models, Math. Biosci., 232 (2011), 31-41.doi: 10.1016/j.mbs.2011.04.001.

    [35]

    H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.doi: 10.1137/080732870.

    [36]

    A. R. Tuite, J. H. Tien, M. C. Eisenberg, D. J. D. Earn, J. Ma and D. N. Fisman, Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions, Annals of Internal Medicine, 154 (2011), 593-601.doi: 10.7326/0003-4819-154-9-201105030-00334.

    [37]

    X. Wang and J. Wang, Analysis of cholera epidemics with bacterial growth and spatial movement, J. Biol. Dyn., 9 (2015), 233-261.doi: 10.1080/17513758.2014.974696.

    [38]

    X. Wang, D. Posny and J. Wang, A Reaction-Convection-Diffusion Model for Cholera Spatial Dynamics submitted.

    [39]

    W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.doi: 10.1137/120872942.

    [40]
    [41]
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