Mathematical Biosciences and Engineering (MBE)

Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model
Pages: 559 - 579, Issue 2, April 2017

doi:10.3934/mbe.2017033      Abstract        References        Full text (530.9K)           Related Articles

Kazuo Yamazaki - Department of Mathematics, University of Rochester, Rochester, NY 14627, United States (email)
Xueying Wang - Washington State University, Department of Mathematics and Statistics, Pullman, WA 99164-3113, United States (email)

1 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.
2 E. Bertuzzo, L. Mari, L. Righetto, M. Gatto, R. Casagrandi, M. Blokesch, I. Rodriguez-Iturbe and 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 and 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 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.
6 S. F. Dowell and 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 and 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 and 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 and 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 and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297.       
13 Y. Lou and 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 and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.       
15 R. Martin and 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 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.
17 R. L. M. Neilan, E. Schaefer, H. Gaff, K. R. Fister and 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 and 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 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.
20 Z. Shuai and 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 and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.       
23 H. R. Thieme, Convergence results and a PoincarĂ©-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 and J. Wang, Global stability for cholera epidemic models, Math. Biosci., 232 (2011), 31-41.       
26 J. H. Tien and 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 and 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 and 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 and S. Liao, A generalized cholera model and epidemic-endemic analysis, J. Biol. Dyn., 6 (2012), 568-589.       
30 J. Wang and C. Modnak, Modeling cholera dynamics with controls, Canad. Appl. Math. Quart., 19 (2011), 255-273.       
31 X. Wang, D. Posny and J. Wang, A reaction-convection-diffusion model for cholera spatial dynamics, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2785-2809.
32 X. Wang and J. Wang, Analysis of cholera epidemics with bacterial growth and spatial movement, J. Biol. Dyn., 9 (2015), 233-261.       
33 W. Wang and 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 and 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, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996.       
37 K. Yamazaki and 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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