American Institute of Mathematical Sciences

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

Stability analysis and application of a mathematical cholera model

Shu Liao 1, and  Jin Wang 2,

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.
Keywords: cholera model., Stability, equilibrium, dynamical system.
Mathematics Subject Classification: Primary: 34D20, 92D3.
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: http://www.cdc.gov.

[41]

The Wikipedia, Available from: http://en.wikipedia.org.

[42]

World Health Organization, Available from: 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: http://www.cdc.gov.

[41]

The Wikipedia, Available from: http://en.wikipedia.org.

[42]

World Health Organization, Available from: http://www.who.org.

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