Mathematical Biosciences and Engineering (MBE)

Pages: 733 - 752,    Volume: 8 ,   Issue: 3 ,       July 2011  

Stability analysis and application of a mathematical cholera model

doi:10.3934/mbe.2011.8.733          Full text: (565.3K)

Shu Liao - School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China (email)
Jin Wang - Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, United States (email)

Abstract: In this paper, we conduct a dynamical analysis of the deterministic cholera model proposed in [9]. We study the stability of both the disease-free and endemic equilibria so as to explore the complex epidemic and endemic dynamics of the disease. We demonstrate a real-world application of this model by investigating the recent cholera outbreak in Zimbabwe. Meanwhile, we present numerical simulation results to verify the analytical predictions.

Keywords: Stability, equilibrium, dynamical system, cholera model.
Mathematics Subject Classification: Primary: 34D20, 92D30.


[1]  A. Alam, R. C. Larocque, J. B. Harris, et al., Hyperinfectivity of human-passaged Vibrio cholerae can be modeled by growth in the infant mouse, Infection and Immunity, 73 (2005), 6674-6679.
[2]  S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, International Statistical Review, 62 (1994), 229-243.
[3]  V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the european mediterranean region, Revue Dépidémoligié et de Santé Publiqué, 27 (1979), 121-132.
[4]  C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $R_0$ and its role on global stability, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction," IMA, 125, Springer-Verlag, 2002.
[5]  N. Chitnis, J. M. Cushing and J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM Journal on Applied Mathematics, 67 (2006), 24-45.
[6]  C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1, 2001.
[7]  K. Dietz, The estimation of the basic reproduction number for infections diseases, Statistical Methods in Medical Research, 2 (1993), 23-41.
[8]  J. Dushoff, W. Huang and C. Castillo-Chavez, Backwards bifurcation and catastrophe in simple models of fatal diseases, Journal of Mathematical Biology, 36 (1998), 227-248.
[9]  D. M. Hartley, J. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Medicine, 3 (2006), 63-69.
[10]  P. Hartman, "Ordinary Differential Equations," John Wiley, New York, 1980.
[11]  T. R. Hendrix, The pathophysiology of cholera, Bulletin of the New York Academy of Medicine, 47 (1971), 1169-1180.
[12]  H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.
[13]  G. A. Korn and T. M. Korn, "Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for References and Review," Dover Publications, Mineola, NY, 2000.
[14]  B. Li, Periodic orbits of autonomous ordinary differential equations: Theory and applications, Nonlinear Analysis, 5 (1981), 931-958.
[15]  G. Li and J. Zhen, Global stability of an SEI epidemic model with general contact rate, Chaos Solitons Fractals, 23 (2005), 997-1004.
[16]  M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Mathematical Biosciences, 160 (1999), 191-213.
[17]  M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Mathematical Biosciences, 125 (1995), 155-164.
[18]  A. Lajmanovich and J. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28 (1976), 221-236.
[19]  J. B. Kaper, J. G. Morris and M. M. Levine, Cholera, Clinical Microbiology Reviews 8 (1995), 48-86.
[20]  H. K. Khalil, "Nonlinear Systems," Prentice Hall, NJ, 1996.
[21]  A. A. King, E. L. Lonides, M. Pascual and M. J. Bouma, Inapparent infections and cholera dynamics, Nature, 454 (2008), 877-881.
[22]  S. Marino, I. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in system biology, Journal of Theoretical Biology, 254 (2008), 178-196.
[23]  P. R. Mason, Zimbabwe experiences the worst epidemic of cholera in Africa, Journal of Infection in Developing Countries, 3 (2009), 148-151.
[24]  J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases as regulator of population sizes, Journal of Mathematical Biology, 30 (1992), 693-716.
[25]  D. S. Merrell, S. M. Butler, F. Qadri, et al., Host-induced epidemic spread of the cholera bacterium, Nature, 417 (2002), 642-645.
[26]  S. M. Moghadas and A. B. Gumel, Global stability of a two-stage epidemic model with generalized non-linear incidence, Mathematics and Computers in Simulation, 60 (2002), 107-118.
[27]  E. J. Nelson, J. B. Harris, J. G. Morris, S. B. Calderwood and A. Camilli, Cholera transmission: The host, pathogen and bacteriophage dynamics, Nature Reviews: Microbiology, 7 (2009), 693-702.
[28]  R. M. Nisbet and W. S. C. Gurney, "Modeling Fluctuating Populations," John Wiley & Sons, New York, 1982.
[29]  M. Pascual, M. Bouma and A. Dobson, Cholera and climate: Revisiting the quantiative evidence, Microbes and Infections, 4 (2002), 237-245.
[30]  M. Pascual, K. Koelle and A. Dobson, Hyperinfectivity in cholera: A new mechanism for an old epidemiological model?, PLoS Medicine, 3 (2006), 931-933.
[31]  E. Pourabbas, A. d'Onofrio and M. Rafanelli, A method to estimate the incidence of communicable diseases under seasonal fluctuations with application to cholera, Applied Mathematics and Computation, 118 (2001), 161-174.
[32]  T. C. Reluga, J. Medlock and A. S. Perelson, Backward bifurcation and multiple equilibria in epidemic models with structured immunity, Journal of Theoretical Biology, 252 (2008), 155-165.
[33]  C. P. Simon and J. A. Jacquez, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations, SIAM Journal on Applied Mathematics, 52 (1992), 541-576.
[34]  B. H. Singer and D. Kirschner, Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection, Mathematical Biosiences and Engineering, 1 (2004), 91-93.
[35]  D. Terman, An introduction to dynamical systems and neuronal dynamics, in "Tutorials in Mathematical Biosciences I," Springer, Berlin/Heidelberg, 2005.
[36]  V. Tudor and I. Strati, "Smallpox, Cholera," Tunbridge Wells: Abacus Press, 1977.
[37]  P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.
[38]  E. Vynnycky, A. Trindall and P. Mangtani, Estimates of the reproduction numbers of spanish influenza using morbidity data, International Journal of Epidemiology, 36 (2007), 881-889.
[39]  J. Zhang and Z. Ma, Global dynamics of an SEIR epidemic model with saturating contact rate, Mathematical Biosciences, 185 (2003), 15-32.
[40]  Center for Disease Control and Prevention, Available from:
[41]  The Wikipedia, Available from:
[42]  World Health Organization, Available from:

Received:   July   2010;   Accepted:   September  2010;   Published:   June  2011.


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