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2013, 2013(special): 747-757. doi: 10.3934/proc.2013.2013.747

Analyzing the infection dynamics and control strategies of cholera

Jianjun Paul Tian 1, , Shu Liao 2, and  Jin Wang 3,

1. 

Mathematics Department, College of William and Mary, Williamsburg, VA 23187, United States
2. 

School of Mathematics and Statistics, Chongqing Technology and Business University, China
3. 

Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529

Received  July 2012 Published  November 2013

We conduct a rigorous analysis for the differential equation-based cholera model proposed in [3]. Unlike traditional infectious disease SIR-type models, this model explicitly includes cholerae bacteria from the environments, and the incidence rate is a dose-dependent Michaelis-Menten type functional response. By extending the theory of monotone dynamical systems, we prove that the endemic equilibrium, when it exists, of the model is globally asymptotically stable, implying the persistence of the disease in the absence of interventions. We then modify the model by incorporating various control strategies, and study the subsequent dynamics. We find that with strong control measures, the basic reproduction number will be reduced below 1 so that the disease-free equilibrium is globally asymptotically stable. With weak controls, instead, epidemicity still occurs and a unique and globally stable endemic equilibrium state exists, though at a lower infection level and with a reduced disease outbreak growth rate. The analytical predictions are confirmed by numerical results.
Keywords: disease control., Dose-dependent infection, cholera model, global stability, dynamical systems.
Mathematics Subject Classification: Primary: 34D20, 92D3.
Citation: Jianjun Paul Tian, Shu Liao, Jin Wang. Analyzing the infection dynamics and control strategies of cholera. Conference Publications, 2013, 2013 (special) : 747-757. doi: 10.3934/proc.2013.2013.747
References:
[1]

R.M. Anderson and R.M. May, Infectious diseases of humans, Oxford University Press, 1991. Google Scholar

[2]

G. J. Butler and P. Waltman, Persistence in dynamical systems, Journal of Differential Equations 63: 255-263, 1986.  Google Scholar

[3]

C.T. Codeço, Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infectious Diseases, 1:1, 2001. Google Scholar

[4]

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: 0063-0069, 2006. Google Scholar

[5]

P. Hartman, Ordinary differential equations, John Wiley, New York, 1980.  Google Scholar

[6]

A.A. King, E.L. Lonides, M. Pascual and M.J. Bouma, Inapparent infections and cholera dynamics, Nature, 454: 877-881, 2008. Google Scholar

[7]

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. Google Scholar

[8]

M.M. Levine, D.R. Nalin, M.B. Rennels, et al., Genetic susceptibility to cholera, Annals of Human Biology, 6: 369-374, 1979. Google Scholar

[9]

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: 191-213, 1999.  Google Scholar

[10]

M.Y. Li and J.S. Muldowney, Global stability for the SEIR model in epidemiology, Mathematical Biosciences, 125: 155-164, 1995.  Google Scholar

[11]

Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D.L. Smith and J.G. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proceedings of the National Academy of Sciences, 108: 8767-8772, 2011. Google Scholar

[12]

J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain Journal of Mathematics, 20: 857-872, 1990.  Google Scholar

[13]

M.A. Savageau, Michaelis-Menten mechanism reconsidered: implications of fractal kinetics, Journal of Theoretical Biology, 176: 115-124, 1995. Google Scholar

[14]

H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, 1995.  Google Scholar

[15]

H.L. Smith and H.R. Zhu, Stable periodic orbits for a class of three dimensional competitive systems, Journal of Differential Equations, 110: 143-156, 1994.  Google Scholar

[16]

J.P. Tian and J. Wang, Global stability for cholera epidemic models, Mathematical Biosciences, 232: 31-41, 2011.  Google Scholar

[17]

J.H. Tien and D.J.D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology, 72: 1502-1533, 2010.  Google Scholar

[18]

J. Wang and S. Liao, A generalized cholera model and epidemic-endemic analysis, Journal of Biological Dynamics, 6: 568-589, 2012.  Google Scholar

[19]

J. Zhang and Z. Ma, Global dynamics of an SEIR epidemic model with saturating contact rate, Mathematical Biosciences, 185: 15-32, 2003.  Google Scholar

show all references

References:
[1]

R.M. Anderson and R.M. May, Infectious diseases of humans, Oxford University Press, 1991. Google Scholar

[2]

G. J. Butler and P. Waltman, Persistence in dynamical systems, Journal of Differential Equations 63: 255-263, 1986.  Google Scholar

[3]

C.T. Codeço, Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infectious Diseases, 1:1, 2001. Google Scholar

[4]

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: 0063-0069, 2006. Google Scholar

[5]

P. Hartman, Ordinary differential equations, John Wiley, New York, 1980.  Google Scholar

[6]

A.A. King, E.L. Lonides, M. Pascual and M.J. Bouma, Inapparent infections and cholera dynamics, Nature, 454: 877-881, 2008. Google Scholar

[7]

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. Google Scholar

[8]

M.M. Levine, D.R. Nalin, M.B. Rennels, et al., Genetic susceptibility to cholera, Annals of Human Biology, 6: 369-374, 1979. Google Scholar

[9]

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: 191-213, 1999.  Google Scholar

[10]

M.Y. Li and J.S. Muldowney, Global stability for the SEIR model in epidemiology, Mathematical Biosciences, 125: 155-164, 1995.  Google Scholar

[11]

Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D.L. Smith and J.G. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proceedings of the National Academy of Sciences, 108: 8767-8772, 2011. Google Scholar

[12]

J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain Journal of Mathematics, 20: 857-872, 1990.  Google Scholar

[13]

M.A. Savageau, Michaelis-Menten mechanism reconsidered: implications of fractal kinetics, Journal of Theoretical Biology, 176: 115-124, 1995. Google Scholar

[14]

H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, 1995.  Google Scholar

[15]

H.L. Smith and H.R. Zhu, Stable periodic orbits for a class of three dimensional competitive systems, Journal of Differential Equations, 110: 143-156, 1994.  Google Scholar

[16]

J.P. Tian and J. Wang, Global stability for cholera epidemic models, Mathematical Biosciences, 232: 31-41, 2011.  Google Scholar

[17]

J.H. Tien and D.J.D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology, 72: 1502-1533, 2010.  Google Scholar

[18]

J. Wang and S. Liao, A generalized cholera model and epidemic-endemic analysis, Journal of Biological Dynamics, 6: 568-589, 2012.  Google Scholar

[19]

J. Zhang and Z. Ma, Global dynamics of an SEIR epidemic model with saturating contact rate, Mathematical Biosciences, 185: 15-32, 2003.  Google Scholar

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