# 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), 6674-6679. doi: 10.1128/IAI.73.10.6674-6679.2005. [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. doi: 10.2307/1403510. [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. doi: 10.1137/050638941. [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. doi: 10.1177/096228029300200103. [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. doi: 10.1007/s002850050099. [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. doi: 10.1371/journal.pmed.0030007. [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. doi: 10.1137/S0036144500371907. [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. doi: 10.1016/0362-546X(81)90055-9. [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. doi: 10.1016/S0025-5564(99)00030-9. [17] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Mathematical Biosciences, 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5. [18] A. Lajmanovich and J. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5. [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. doi: 10.1038/nature07084. [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. doi: 10.1016/j.jtbi.2008.04.011. [23] P. R. Mason, Zimbabwe experiences the worst epidemic of cholera in Africa, Journal of Infection in Developing Countries, 3 (2009), 148-151. doi: 10.3855/jidc.62. [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. doi: 10.1038/nature00778. [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. doi: 10.1016/S0378-4754(02)00002-2. [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. doi: 10.1016/S1286-4579(01)01533-7. [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. doi: 10.1371/journal.pmed.0030280. [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. doi: 10.1016/S0096-3003(99)00212-X. [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. doi: 10.1016/j.jtbi.2008.01.014. [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. doi: 10.1137/0152030. [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. doi: 10.1016/S0025-5564(02)00108-6. [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. doi: 10.1093/ije/dym071. [39] J. Zhang and Z. Ma, Global dynamics of an SEIR epidemic model with saturating contact rate, Mathematical Biosciences, 185 (2003), 15-32. doi: 10.1016/S0025-5564(03)00087-7. [40] Center for Disease Control and Prevention, Available from:, \url{http://www.cdc.gov}., (). [41] The Wikipedia, Available from:, \url{http://en.wikipedia.org}., (). [42] World Health Organization, Available from:, \url{http://www.who.org}., ().

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), 6674-6679. doi: 10.1128/IAI.73.10.6674-6679.2005. [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. doi: 10.2307/1403510. [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. doi: 10.1137/050638941. [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. doi: 10.1177/096228029300200103. [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. doi: 10.1007/s002850050099. [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. doi: 10.1371/journal.pmed.0030007. [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. doi: 10.1137/S0036144500371907. [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. doi: 10.1016/0362-546X(81)90055-9. [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. doi: 10.1016/S0025-5564(99)00030-9. [17] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Mathematical Biosciences, 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5. [18] A. Lajmanovich and J. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5. [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. doi: 10.1038/nature07084. [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. doi: 10.1016/j.jtbi.2008.04.011. [23] P. R. Mason, Zimbabwe experiences the worst epidemic of cholera in Africa, Journal of Infection in Developing Countries, 3 (2009), 148-151. doi: 10.3855/jidc.62. [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. doi: 10.1038/nature00778. [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. doi: 10.1016/S0378-4754(02)00002-2. [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. doi: 10.1016/S1286-4579(01)01533-7. [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. doi: 10.1371/journal.pmed.0030280. [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. doi: 10.1016/S0096-3003(99)00212-X. [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. doi: 10.1016/j.jtbi.2008.01.014. [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. doi: 10.1137/0152030. [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. doi: 10.1016/S0025-5564(02)00108-6. [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. doi: 10.1093/ije/dym071. [39] J. Zhang and Z. Ma, Global dynamics of an SEIR epidemic model with saturating contact rate, Mathematical Biosciences, 185 (2003), 15-32. doi: 10.1016/S0025-5564(03)00087-7. [40] Center for Disease Control and Prevention, Available from:, \url{http://www.cdc.gov}., (). [41] The Wikipedia, Available from:, \url{http://en.wikipedia.org}., (). [42] World Health Organization, Available from:, \url{http://www.who.org}., ().
 [1] Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641 [2] Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701 [3] Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033 [4] Peter J. Witbooi, Grant E. Muller, Marshall B. Ongansie, Ibrahim H. I. Ahmed, Kazeem O. Okosun. A stochastic population model of cholera disease. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 441-456. doi: 10.3934/dcdss.2021116 [5] Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479 [6] Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic and Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707 [7] Fred Brauer, Zhisheng Shuai, P. van den Driessche. Dynamics of an age-of-infection cholera model. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1335-1349. doi: 10.3934/mbe.2013.10.1335 [8] Jinliang Wang, Ran Zhang, Toshikazu Kuniya. A note on dynamics of an age-of-infection cholera model. Mathematical Biosciences & Engineering, 2016, 13 (1) : 227-247. doi: 10.3934/mbe.2016.13.227 [9] Feng-Bin Wang, Xueying Wang. A general multipatch cholera model in periodic environments. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1647-1670. doi: 10.3934/dcdsb.2021105 [10] Faker Ben Belgacem. Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters. Inverse Problems and Imaging, 2012, 6 (2) : 163-181. doi: 10.3934/ipi.2012.6.163 [11] Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 [12] Hongyu He, Naohiro Kato. Equilibrium submanifold for a biological system. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1429-1441. doi: 10.3934/dcdss.2011.4.1429 [13] Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124 [14] Xueying Wang, Drew Posny, Jin Wang. A reaction-convection-diffusion model for cholera spatial dynamics. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2785-2809. doi: 10.3934/dcdsb.2016073 [15] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4867-4885. doi: 10.3934/dcdsb.2020316 [16] Wei Yang, Jinliang Wang. Analysis of a diffusive cholera model incorporating latency and bacterial hyperinfectivity. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3937-3957. doi: 10.3934/cpaa.2021138 [17] Leonid Shaikhet. Stability of a positive equilibrium state for a stochastically perturbed mathematical model of glassy-winged sharpshooter population. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1167-1174. doi: 10.3934/mbe.2014.11.1167 [18] Weiyi Zhang, Ling Zhou. Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022062 [19] Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations and Control Theory, 2022, 11 (1) : 283-300. doi: 10.3934/eect.2021003 [20] Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure and Applied Analysis, 2006, 5 (3) : 515-528. doi: 10.3934/cpaa.2006.5.515

2018 Impact Factor: 1.313