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Original Articles

On the global stability of a generalized cholera epidemiological model

, &
Pages 1088-1104 | Received 16 Jan 2012, Accepted 31 Aug 2012, Published online: 30 Oct 2012

References

  • Alam , A. , LaRocque , R. C. , Harris , J. B. , Vanderspurt , C. , Ryan , E. T. , Qadri , F. and Calderwood , S. B. 2005 . Hyperinfectivity of human-passaged Vibrio cholerae can be modeled by growth in the infant mouse . Infect. Immun. , 73 : 6674 – 6679 .
  • Berman , A. and Plemmons , R. J. 1994 . Nonnegative Matrices in the Mathematical Sciences , Philadelphia : SIAM .
  • Buonomo , B. and Lacitignola , D. 2008 . On the use of the geometric approach to global stability for three dimensional ODE systems: A bilinear case . J. Math. Anal. Appl. , 348 : 255 – 266 .
  • Butler , G. J. and Waltman , P. 1986 . Persistence in dynamical systems . Proc. Amer. Math. Soc. , 96 : 425 – 430 .
  • Codeço , C. T. 2001 . Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir . BMC Infect. Dis. , 1 ( 1 )
  • Coppel , W. A. 1965 . Stability and Asymptotical Behavior of Differential Equations , Boston, MA : Heath Mathematical Monographs, D. C. Heath .
  • Cross , G. W. 1978 . Three types of matrix stability . Linear Algebra Appl. , 20 : 253 – 263 .
  • van den Driessche , P. and Watmough , J. 2002 . Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission . Math. Biosci. , 180 : 29 – 48 .
  • Fan , M. , Li , M. Y. and Wang , K. 2001 . Global stability of an SEIS epidemic model with recruitment and a varying total population size . Math. Biosci. , 170 : 199 – 208 .
  • Faruque , S. , Islam , M. , Ahmad , Q. , Faruque , A. S.G. , Sack , D. , Nair , G. and Mekalanos , J. 2005 . Self-limiting nature of seasonal cholera epidemics: Role of host-mediated amplification of phage . Proc. Nat. Acad. Sci. , 102 : 6119 – 6124 .
  • Freedman , H. I. , Ruan , S. and Tang , M. 1994 . Uniform persistence and flows near a closed positively invariant set . J. Dyn. Differential Equations , 6 : 583 – 600 .
  • Goh , K. , Teo , S. , Lam , S. and Ling , M. 1990 . Person-to-person transmission of cholera in a psychiatric hospital . J. Inf. , 20 : 193 – 200 .
  • Hartley , D. M. , Morris , J. G. and Smith , D. L. 2006 . Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics? . PLoS Med. , 3 : 0063 – 0069 .
  • Hartman , P. 1980 . Ordinary Differential Equations , New York : John Wiley .
  • Hethcote , H. W. 1976 . Qualitative analysis of communicable disease models . Math. Biosci. , 28 : 335 – 356 .
  • Hethcote , H. W. 2000 . The mathematics of infectious diseases . SIAM Rev. , 42 : 599 – 653 .
  • Jensen , M. , Faruque , S. M. , Mekalanos , J. J. and Levin , B. 2006 . Modeling the role of bacteriophage in the control of cholera outbreaks . Proc. Nat. Acad. Sci. , 103 : 4652 – 4657 .
  • Joh , R. I. , Wang , H. , Weiss , H. and Weitz , J. S. 2009 . Dynamics of indirectly transmitted infectious diseases with immunological threshold . Bull. Math. Biol. , 71 : 845 – 862 .
  • Khalil , H. K. 1996 . textit>Nonlinear Systems , Prentice-Hall, Englewood Cliffs, NJ
  • King , A. A. , Lonides , E. L. , Pascual , M. and Bouma , M. J. 2008 . Inapparent infections and cholera dynamics . Nature , 454 : 877 – 881 .
  • Lajmanovich , A. and Yorke , J. 1976 . A deterministic model for gonorrhea in a nonhomogeneous population . Math. Biosci. , 28 : 221 – 236 .
  • Li , G. and Jin , Z. 2005 . Global stability of an SEI epidemic model with general contact rate . Chaos, Solitons Fractals , 23 : 997 – 1004 .
  • Li , M. Y. , Graef , J. R. , Wang , L. and Karsai , J. 1999 . Global dynamics of a SEIR model with varying total population size . Math. Biosci. , 160 : 191 – 213 .
  • Li , M. Y. and Muldowney , J. S. 1995 . Global stability for the SEIR model in epidemiology . Math. Biosci. , 125 : 155 – 164 .
  • Li , M. Y. and Muldowney , J. S. 1996 . A geometirc approach to global-stability problems . SIAM J. Math. Anal. , 27 : 1070 – 1083 .
  • Li , M. Y. and Muldowney , J. S. 2000 . Dynamics of differential equations on invariant manifolds . J. Differential Equations , 168 : 295 – 320 .
  • Li , M. Y. , Muldowney , J. S. and Driessche , P. V.D. 1999 . Global stability of SEIRS models in epidemiology . Can. Appl. Math. Q. , 7 : 409 – 425 .
  • Li , M. Y. , Smith , H. L. and Wang , L. 2001 . Global dynamics of an SEIR epidemic model with vertical transmission . SIAM J. Math. Anal. , 62 : 58 – 69 .
  • Li , M. Y. and Wang , L. 2002 . Global stability in some SEIR epidemic models , : 295 – 311 . IMA Volumes in Mathematics and Its Application Vol. 126, Springer-Verlag, Berlin
  • Liao , S. and Wang , J. 2011 . Stability analysis and application of a mathematical cholera model . Math. Biosci. Eng. , 8 : 733 – 752 .
  • Ma , S. and Xia , Y. 2008 . “ Mathematical Understanding of Infectious Disease Dynamics ” . Edited by: Ma , S. and Xia , Y. Singapore : Institute for Mathematical Sciences, National University of Singapore . Lecture Notes Series Vol. 16
  • Mena-Lorca , J. and Hethcote , H. W. 1992 . Dynamic models of infectious diseases as regulator of population sizes . J. Math. Biol. , 30 : 693 – 716 .
  • Merrell , D. S. , Butler , S. M. , Qadri , F. , Dolganov , N. A. , Alam , A. , Cohen , M. B. , Calderwood , S. B. , Schoolnik , G. K. and Camilli , A. 2002 . Host-induced epidemic spread of the cholera bacterium . Nature , 417 : 642 – 645 .
  • Mukandavire , Z. , Liao , S. , Wang , J. , Gaff , H. , Smith , D. L. and Morris , J. G. 2011 . Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe . Proc. Nat. Acad. Sci. , 108 : 8767 – 8772 .
  • Muldowney , J. S. 1990 . Compound matrices and ordinary differential equations . Rocky Mountain J. Math. , 20 : 857 – 872 .
  • Nelson , E. J. , Harris , J. B. , Morris , J. G. , Calderwood , S. B. and Camilli , A. 2009 . Cholera transmission: The host, pathogen and bacteriophage dynamics . Nat. Rev.: Microbiol. , 7 : 693 – 702 .
  • Pascual , M. , Rodo , X. , Ellner , S. P. , Colwell , R. and Bouma , M. J. 2000 . Cholera dynamics and El Nino-Southern oscillation . Science , 289 : 1766 – 1769 .
  • Pascual , M. , Bouma , M. and Dobson , A. 2002 . Cholera and climate: Revisiting the quantiative evidence . Microbes Infections , 4 : 237 – 245 .
  • Shuai , Z. and van den Driessche , P. 2011 . Global dynamics of cholera models with differential infectivity . Math. Biosci. , 234 : 118 – 126 .
  • Smith , H. L. 1995 . “ Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems ” . Providence, RI : American Mathematical Society .
  • Smith , H. L. and Zhu , H. R. 1994 . Stable periodic orbits for a class of three dimensional competitive systems . J. Differential Equations , 110 : 143 – 156 .
  • Smith , R. A. 1987 . Orbital stability for ordinary differential equations . J. Differential Equations , 69 : 265 – 287 .
  • Tian , J. P. and Wang , J. 2011 . Global stability for cholera epidemic models . Math. Biosci. , 232 : 31 – 41 .
  • Tien , J. H. and Earn , D. J.D. 2010 . Multiple transmission pathways and disease dynamics in a waterborne pathogen model . Bull. Math. Biol. , 72 : 1502 – 1533 .
  • Tudor , V. and Strati , I. 1977 . Smallpox, Cholera , Tunbridge Wells : Abacus Press .
  • Vynnycky , E. , Trindall , A. and Mangtani , P. 2007 . Estimates of the reproduction numbers of Spanish influenza using morbidity data . Int. J. Epidemiol. , 36 : 881 – 889 .
  • Wang , J. and Liao , S. 2012 . A generalized cholera model and epidemic/endemic analysis . J. Biol. Dyn. , 6 568–589
  • World Health Organization, Available at web page: www.who.org.
  • Zhang , J. and Ma , Z. 2003 . Global dynamics of an SEIR epidemic model with saturating contact rate . Math. Biosci. , 185 : 15 – 32 .