# American Institute of Mathematical Sciences

2011, 8(3): 733-752. doi: 10.3934/mbe.2011.8.733

## Stability analysis and application of a mathematical cholera model

 1 School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China 2 Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529

Received  July 2010 Revised  September 2010 Published  June 2011

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.
Citation: Shu Liao, Jin Wang. Stability analysis and application of a mathematical cholera model. Mathematical Biosciences & Engineering, 2011, 8 (3) : 733-752. doi: 10.3934/mbe.2011.8.733
##### References:
 [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. doi: 10.1128/IAI.73.10.6674-6679.2005. Google Scholar [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. doi: 10.2307/1403510. Google Scholar [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. Google Scholar [4] C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $R_0$ and its role on global stability,, in, 125 (2002). Google Scholar [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. doi: 10.1137/050638941. Google Scholar [6] C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir,, BMC Infectious Diseases, 1 (2001). Google Scholar [7] K. Dietz, The estimation of the basic reproduction number for infections diseases,, Statistical Methods in Medical Research, 2 (1993), 23. doi: 10.1177/096228029300200103. Google Scholar [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. doi: 10.1007/s002850050099. Google Scholar [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. doi: 10.1371/journal.pmed.0030007. Google Scholar [10] P. Hartman, "Ordinary Differential Equations,", John Wiley, (1980). Google Scholar [11] T. R. Hendrix, The pathophysiology of cholera,, Bulletin of the New York Academy of Medicine, 47 (1971), 1169. Google Scholar [12] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar [13] G. A. Korn and T. M. Korn, "Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for References and Review,", Dover Publications, (2000). Google Scholar [14] B. Li, Periodic orbits of autonomous ordinary differential equations: Theory and applications,, Nonlinear Analysis, 5 (1981), 931. doi: 10.1016/0362-546X(81)90055-9. Google Scholar [15] G. Li and J. Zhen, Global stability of an SEI epidemic model with general contact rate,, Chaos Solitons Fractals, 23 (2005), 997. Google Scholar [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. doi: 10.1016/S0025-5564(99)00030-9. Google Scholar [17] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Mathematical Biosciences, 125 (1995), 155. doi: 10.1016/0025-5564(95)92756-5. Google Scholar [18] A. Lajmanovich and J. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Mathematical Biosciences, 28 (1976), 221. doi: 10.1016/0025-5564(76)90125-5. Google Scholar [19] J. B. Kaper, J. G. Morris and M. M. Levine, Cholera,, Clinical Microbiology Reviews \textbf{8} (1995), 8 (1995), 48. Google Scholar [20] H. K. Khalil, "Nonlinear Systems,", Prentice Hall, (1996). Google Scholar [21] A. A. King, E. L. Lonides, M. Pascual and M. J. Bouma, Inapparent infections and cholera dynamics,, Nature, 454 (2008), 877. doi: 10.1038/nature07084. Google Scholar [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. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar [23] P. R. Mason, Zimbabwe experiences the worst epidemic of cholera in Africa,, Journal of Infection in Developing Countries, 3 (2009), 148. doi: 10.3855/jidc.62. Google Scholar [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. Google Scholar [25] D. S. Merrell, S. M. Butler, F. Qadri, et al., Host-induced epidemic spread of the cholera bacterium,, Nature, 417 (2002), 642. doi: 10.1038/nature00778. Google Scholar [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. doi: 10.1016/S0378-4754(02)00002-2. Google Scholar [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. Google Scholar [28] R. M. Nisbet and W. S. C. Gurney, "Modeling Fluctuating Populations,", John Wiley & Sons, (1982). Google Scholar [29] M. Pascual, M. Bouma and A. Dobson, Cholera and climate: Revisiting the quantiative evidence,, Microbes and Infections, 4 (2002), 237. doi: 10.1016/S1286-4579(01)01533-7. Google Scholar [30] M. Pascual, K. Koelle and A. Dobson, Hyperinfectivity in cholera: A new mechanism for an old epidemiological model?,, PLoS Medicine, 3 (2006), 931. doi: 10.1371/journal.pmed.0030280. Google Scholar [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. doi: 10.1016/S0096-3003(99)00212-X. Google Scholar [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. doi: 10.1016/j.jtbi.2008.01.014. Google Scholar [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. doi: 10.1137/0152030. Google Scholar [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. Google Scholar [35] D. Terman, An introduction to dynamical systems and neuronal dynamics,, in, (2005). Google Scholar [36] V. Tudor and I. Strati, "Smallpox, Cholera,", Tunbridge Wells: Abacus Press, (1977). Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [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. doi: 10.1093/ije/dym071. Google Scholar [39] J. Zhang and Z. Ma, Global dynamics of an SEIR epidemic model with saturating contact rate,, Mathematical Biosciences, 185 (2003), 15. doi: 10.1016/S0025-5564(03)00087-7. Google Scholar [40] Center for Disease Control and Prevention, Available from:, \url{http://www.cdc.gov}., (). Google Scholar [41] The Wikipedia, Available from:, \url{http://en.wikipedia.org}., (). Google Scholar [42] World Health Organization, Available from:, \url{http://www.who.org}., (). Google Scholar

show all references

##### References:
 [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. doi: 10.1128/IAI.73.10.6674-6679.2005. Google Scholar [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. doi: 10.2307/1403510. Google Scholar [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. Google Scholar [4] C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $R_0$ and its role on global stability,, in, 125 (2002). Google Scholar [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. doi: 10.1137/050638941. Google Scholar [6] C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir,, BMC Infectious Diseases, 1 (2001). Google Scholar [7] K. Dietz, The estimation of the basic reproduction number for infections diseases,, Statistical Methods in Medical Research, 2 (1993), 23. doi: 10.1177/096228029300200103. Google Scholar [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. doi: 10.1007/s002850050099. Google Scholar [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. doi: 10.1371/journal.pmed.0030007. Google Scholar [10] P. Hartman, "Ordinary Differential Equations,", John Wiley, (1980). Google Scholar [11] T. R. Hendrix, The pathophysiology of cholera,, Bulletin of the New York Academy of Medicine, 47 (1971), 1169. Google Scholar [12] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar [13] G. A. Korn and T. M. Korn, "Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for References and Review,", Dover Publications, (2000). Google Scholar [14] B. Li, Periodic orbits of autonomous ordinary differential equations: Theory and applications,, Nonlinear Analysis, 5 (1981), 931. doi: 10.1016/0362-546X(81)90055-9. Google Scholar [15] G. Li and J. Zhen, Global stability of an SEI epidemic model with general contact rate,, Chaos Solitons Fractals, 23 (2005), 997. Google Scholar [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. doi: 10.1016/S0025-5564(99)00030-9. Google Scholar [17] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Mathematical Biosciences, 125 (1995), 155. doi: 10.1016/0025-5564(95)92756-5. Google Scholar [18] A. Lajmanovich and J. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Mathematical Biosciences, 28 (1976), 221. doi: 10.1016/0025-5564(76)90125-5. Google Scholar [19] J. B. Kaper, J. G. Morris and M. M. Levine, Cholera,, Clinical Microbiology Reviews \textbf{8} (1995), 8 (1995), 48. Google Scholar [20] H. K. Khalil, "Nonlinear Systems,", Prentice Hall, (1996). Google Scholar [21] A. A. King, E. L. Lonides, M. Pascual and M. J. Bouma, Inapparent infections and cholera dynamics,, Nature, 454 (2008), 877. doi: 10.1038/nature07084. Google Scholar [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. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar [23] P. R. Mason, Zimbabwe experiences the worst epidemic of cholera in Africa,, Journal of Infection in Developing Countries, 3 (2009), 148. doi: 10.3855/jidc.62. Google Scholar [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. Google Scholar [25] D. S. Merrell, S. M. Butler, F. Qadri, et al., Host-induced epidemic spread of the cholera bacterium,, Nature, 417 (2002), 642. doi: 10.1038/nature00778. Google Scholar [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. doi: 10.1016/S0378-4754(02)00002-2. Google Scholar [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. Google Scholar [28] R. M. Nisbet and W. S. C. Gurney, "Modeling Fluctuating Populations,", John Wiley & Sons, (1982). Google Scholar [29] M. Pascual, M. Bouma and A. Dobson, Cholera and climate: Revisiting the quantiative evidence,, Microbes and Infections, 4 (2002), 237. doi: 10.1016/S1286-4579(01)01533-7. Google Scholar [30] M. Pascual, K. Koelle and A. Dobson, Hyperinfectivity in cholera: A new mechanism for an old epidemiological model?,, PLoS Medicine, 3 (2006), 931. doi: 10.1371/journal.pmed.0030280. Google Scholar [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. doi: 10.1016/S0096-3003(99)00212-X. Google Scholar [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. doi: 10.1016/j.jtbi.2008.01.014. Google Scholar [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. doi: 10.1137/0152030. Google Scholar [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. Google Scholar [35] D. Terman, An introduction to dynamical systems and neuronal dynamics,, in, (2005). Google Scholar [36] V. Tudor and I. Strati, "Smallpox, Cholera,", Tunbridge Wells: Abacus Press, (1977). Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [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. doi: 10.1093/ije/dym071. Google Scholar [39] J. Zhang and Z. Ma, Global dynamics of an SEIR epidemic model with saturating contact rate,, Mathematical Biosciences, 185 (2003), 15. doi: 10.1016/S0025-5564(03)00087-7. Google Scholar [40] Center for Disease Control and Prevention, Available from:, \url{http://www.cdc.gov}., (). Google Scholar [41] The Wikipedia, Available from:, \url{http://en.wikipedia.org}., (). Google Scholar [42] World Health Organization, Available from:, \url{http://www.who.org}., (). Google Scholar
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