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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 |
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] | |
[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] | |
[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}., ().
|
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