Global stability for cholera epidemic models

Mathematical Biosciences

Volumn 232, Issue 1, 2011, Pages 31-41

  Tian, Jianjun Paul ^a   Wang, Jin ^b  

^a Dept of Economics and Thomas Jefferson Program in Public Policy at the College of William and Mary   (United States)

^b OLD DOMINION UNIVERSITY   (United States)

Author keywords

Cholera model; Dynamical systems; Global stability

Indexed keywords

AUTONOMOUS SYSTEMS; BUILDING BLOCKES; CHOLERA DYNAMICS; CHOLERA MODEL; CLINICAL STUDY; COMPREHENSIVE STUDIES; EPIDEMIC MODELS; GEOMETRIC APPROACHES; GLOBAL STABILITY; GLOBAL STABILITY ANALYSIS; GRAM-NEGATIVE BACTERIA; HUMAN POPULATION; INFECTIOUS DISEASE; MONOTONE DYNAMICAL SYSTEMS; MULTIPLE TRANSMISSION; NON-LINEAR; POINCARE; THREE DIFFERENT TECHNIQUES; VIBRIO CHOLERAE;

BACTERIA; DYNAMICAL SYSTEMS; DYNAMICS; LYAPUNOV FUNCTIONS; MATHEMATICAL MODELS;

EPIDEMIOLOGY;

BACTERIUM; CHOLERA; DISEASE TRANSMISSION; EPIDEMIOLOGY; INFECTIOUS DISEASE;

ARTICLE; BACTERIAL TRANSMISSION; CHOLERA; ENVIRONMENTAL FACTOR; EPIDEMIC; EPIDEMIOLOGICAL DATA; GEOMETRY; GLOBAL STABILITY; HUMAN; INCIDENCE; MATHEMATICAL MODEL; NONHUMAN; VIBRIO CHOLERAE; WATER CONTAMINATION;

BASIC REPRODUCTION NUMBER; CHOLERA; DISEASE OUTBREAKS; HUMANS; INCIDENCE; MODELS, BIOLOGICAL; VIBRIO CHOLERAE; WATER MICROBIOLOGY;

NEGIBACTERIA; VIBRIO CHOLERAE;

EID: 79957653823     PISSN: 00255564     EISSN: 18793134     Source Type: Journal    
DOI: 10.1016/j.mbs.2011.04.001     Document Type: Article

References (60)

1. Alam A., Larocque R.C., Harris J.B., et al. Hyperinfectivity of human-passaged Vibrio cholerae can be modeled by growth in the infant mouse. Infection and Immunity 2005, 73:6674.
2. Buonomo B., Lacitignola D. On the use of the geometric approach to global stability for three dimensional ODE systems: a bilinear case. Journal of Mathematical Analysis and Applications 2008, 348:255.
3. Butler G.J., Waltman P. Persistence in dynamical systems. Proceedings of the American Mathematical Society 1986, 96:425.
4. Capasso V., Paveri-Fontana S.L. A mathematical model for the 1973 cholera epidemic in the european mediterranean region. Revue dépidémoligié et de santé Publiqué 1979, 27:121.
5. Chitnis N., Cushing J.M., Hyman J.M. Bifurcation analysis of a mathematical model for malaria transmission. SIAM Journal on Applied Mathematics 2006, 67:24.
6. Codeço C.T. Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. BMC Infectious Diseases 2001, 1:1.
7. Coppel W.A. Stability and asymptotical behavior of differential equations. Heath Mathematical Monographs 1965, D.C. Heath and Company, Boston.
8. Cross G.W. Three types of matrix stability. Linear Algebra and its Applications 1978, 20:253.
9. Fan M., Li M.Y., Wang K. Global stability of an SEIS epidemic model with recruitment and a varying total population size. Mathematical Boiscience 2001, 170:199.
10. Faruque S., Islam M., Ahmad Q., Faruque A.S.G., Sack D., Nair G., Mekalanos J. Self-limiting nature of seasonal cholera epidemics: role of host-mediated amplification of phage. Proceedings of the National Academy of Sciences 2005, 102:6119.
11. Ghosh M., Chandra P., Sinha P., Shukla J.B. Modeling the spread of carrier-dependent infectious diseases with environmental effect. Applied Mathematics and Computation 2004, 152:385.
12. Glendinning P. Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. Cambridge Texts in Applied Mathematics 1994, Cambridge University.
13. Goh K., Teo S., Lam S., Ling M. Person-to-person transmission of cholera in a psychiatric hospital. Journal of Infection 1990, 20:193.
14. Hartley D.M., Morris J.G., Smith D.L. Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics?. PLoS Medicine 2006, 3:0063.
15. Hartman P. Ordinary Differential Equations 1980, John Wiley, New York.
16. Hendrix T.R. The pathophysiology of cholera. Bulletin of the New York Academy of Medicine 1971, 47:1169.
17. Hethcote H.W. Qualitative analysis of communicable disease models. Mathematical Biosciences 1976, 28:335.
18. Hethcote H.W. The mathematics of infectious diseases. SIAM Review 2000, 42:599.
19. Hirsch M.W. System of differential equations that are competitive or coorperative. VI: a local Cr closing lemma for 3-dimensional systems. Ergodic Theory and Dynamical Systems 1991, 11:443.
20. Jensen M., Faruque S.M., Mekalanos J.J., Levin B. Modeling the role of bacteriophage in the control of cholera outbreaks. Proceedings of the National Academy of Sciences 2006, 103:4652.
21. Khalil H.K. Nonlinear Systems 1996, Prentice Hall, NJ.
22. King A.A., Lonides E.L., Pascual M., Bouma M.J. Inapparent infections and cholera dynamics. Nature 2008, 454:877.
23. Lajmanovich A., Yorke J. A deterministic model for gonorrhea in a nonhomogeneous population. Mathematical Biosciences 1976, 28:221.
24. Li G., Wang W., Jin Z. Global stability of an SEIR epidemic model with constant immigration. Chaos, Solitons and Fractals 2006, 30:1012.
25. Li G., Jin Z. Global stability of an SEI epidemic model with general contact rate. Chaos, Solitons and Fractals 2005, 23:997.
26. Li M.Y., Graef J.R., Wang L., Karsai J. Global dynamics of a SEIR model with varying total population size. Mathematical Biosciences 1999, 160:191.
27. Li M.Y., Muldowney J.S. On Bendixson's criterion. Journal of Differential Equations 1994, 106:27.
28. Li M.Y., Muldowney J.S. Global stability for the SEIR model in epidemiology. Mathematical Biosciences 1995, 125:155.
29. Li M.Y., Muldowney J.S. A geometirc approach to global-stability problems. SIAM Journal on Mathematical Analysis 1996, 27:1070.
30. Li M.Y., Muldowney J.S., Driessche P.V.D. Global stability of SEIRS models in epidemiology. Canadian Applied Mathematics Quarterly 1999, 7:409.
31. Li M.Y., Smith H.L., Wang L. Global dynamics of an SEIR epidemic model with vertical transmission. SIAM Journal on Mathematical Analysis 2001, 62:58.
32. Li M.Y., Wang L. Global stability in some SEIR epidemic models. IMA Volumes in Mathematics and its Application, Spring-Verlag 2002, 126:295.
33. S. Ma, Y. Xia (Eds.), Mathematical Understanding of Infectious Disease Dynamics, Lecture Notes Series, vol. 16, Institute for Mathematical Sciences, National University of Singapore, 2008.
34. Mena-Lorca J., Hethcote H.W. Dynamic models of infectious diseases as regulator of population sizes. Journal of Mathematical Biology 1992, 30:693.
35. Merrell D.S., Butler S.M., Qadri F., et al. Host-induced epidemic spread of the cholera bacterium. Nature 2002, 417:642.
36. Z. Mukandavire, S. Liao, J. Wang, H. Gaff, Estimating the reproductive numbers for the 2008-2009 cholera outbreak in Zimbabwe, submitted, 2010.
37. Muldowney J.S. Compound matrices and ordinary differential equations. Rocky Mountain Journal of Mathematics 1990, 20:857.
38. Nelson E.J., Harris J.B., Morris J.G., Calderwood S.B., Camilli A. Cholera transmission: the host, pathogen and bacteriophage dynamics. Nature Reviews: Microbiology 2009, 7:693.
39. Pascual M., Rodo X., Ellner S.P., Colwell R., Bouma M.J. Cholera dynamics and El Nio-Southern oscillation. Science 2000, 289:1766.
40. Pascual M., Koelle K., Dobson A.P. Hyperinfectivity in cholera: a new mechanism for an old epidemiological model?. PLoS Medicine 2006, 3:931.
41. Pascual M., Bouma M., Dobson A. Cholera and climate: revisiting the quantitative evidence. Microbes and Infections 2002, 4:237.
42. Pourabbas E., d'Onofrio A., Rafanelli M. A method to estimate the incidence of communicable diseases under seasonal fluctuations with application to cholera. Applied Mathematics and Computation 2001, 118:161.
43. Pugh C.C. The closing lemma. American Journal of Mathematics 1967, 89:956.
44. Pugh C.C., Robinson C. The C1 closing lemma including Hamiltonians. Ergodic Theory and Dynamical Systems 1983, 3:261.
45. Redheffer R. Volterra multipliers I. SIAM Journal on Algebraic and Discrete Methods 1985, 6:592.
46. Redheffer R. Volterra multipliers II. SIAM Journal on Algebraic and Discrete Methods 1985, 6:612.
47. Rinaldi F. Global stability results for epidemic models with latent period. IMA Journal of Mathematics Applied in Medicine & Biology 1990, 7:69.
48. Simon C.P., Jacquez J.A. Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations. SIAM Journal on Applied Mathematics 1992, 52:541.
49. Smith H.L. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems 1995, American Mathematical Society, Providence.
50. Smith H.L., Zhu H.R. Stable periodic orbits for a class of three dimensional competitive systems. Journal of Differential Equations 1994, 110:143.
51. Smith R.A. Orbital stability for ordinary differential equations. Journal of Differential Equations 1987, 69:265.
52. Tien J.H., Earn D.J.D. Multiple transmission pathways and disease dynamics in a waterborne pathogen model. Bulletin of Mathematical Biology 2010, 72:1502.
53. Tudor V., Strati I. Smallpox, cholera, Tunbridge Wells 1977, Abacus.
54. van den Driessche P., Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences 2002, 180:29.
55. Vynnycky E., Trindall A., Mangtani P. Estimates of the reproduction numbers of spanish influenza using morbidity data. International Journal of Epidemiology 2007, 36:881.
56. J. Wang, S. Liao, A generalized cholera model and epidemic/endemic analysis, submitted, 2010.
57. Zhang J., Ma Z. Global dynamics of an SEIR epidemic model with saturating contact rate. Mathematical Biosciences 2003, 185:15.
58. Center for Disease Control and Prevention web page: http://www.cdc.gov.
59. The Wikipedia: en.wikipedia.org.
60. World Health Organization web page: http://www.who.org.