A Mathematical Model for the Dynamics of Cholera with Control Measures

Stephen Edward; Nkuba Nyerere

doi:doi:10.11648/j.acm.20150402.14

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A Mathematical Model for the Dynamics of Cholera with Control Measures

Stephen Edward, Nkuba Nyerere

Published in Applied and Computational Mathematics (Volume 4, Issue 2)

Received: 25 February 2015 Accepted: 13 March 2015 Published: 21 March 2015

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Abstract

Cholera, an acute gastro-intestinal infection and a waterborne disease continues to emerge in developing countries and remains an important global health challenge. In this paper, we formulate a mathematical model that captures some essential dynamics of cholera transmission with public health educational campaigns, vaccination, sanitation and treatment as control strategies in limiting the disease. The reproduction numbers with single and combined controls are computed and compared with each other to assess the possible community benefits. Numerical simulation shows that in a unique control strategy, treatment yields the best results followed by education campaign, then sanitation and vaccination being the last. Furthermore, we noted that the control of cholera is very much better when we incorporated more than one strategy, in two controls the results were better than one strategy, and in three control strategies the results were far better than in two control strategies. Further simulations with all four interventions showed the best results among all combinations attained before. We performed sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence.

Published in	Applied and Computational Mathematics (Volume 4, Issue 2)
DOI	10.11648/j.acm.20150402.14
Page(s)	53-63
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Modeling, Sensitivity, Sanitation, Education, Treatment, Vaccination, Epidemiology

References

[1]	World Health Organization web page: www.who.org
[2]	K. Goh, S. Teo, S. Lam, M. Ling. Person-to-person transmission of cholera in a psychiatric hospital. Journal of Infection 20 (1990) 193
[3]	H.W. Hethcote, The mathematics of infectious diseases, SIAM Review 42 (2000) 599.
[4]	Mukandavire Z, Liao S, Wang J, Gaff H, Smith DL, Morris JG, Estimating the reproductive numbers for the2008–2009 cholera outbreaks in Zimbabwe, Proc Natl Acad Sci 108:8767–8772, 2011.
[5]	Tuite AR, Tien J, Eisenberg M, Earn DJ, Ma J, Fisman DN, and Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions, Ann Intern Med 154:593–601, 2010.
[6]	WHO/Cholera Fact 2010.
[7]	J.M, Ochoche. A Mathematical Model for the Dynamics of Cholera with Control Strategy. International Journal of Science and Technology, 2013; 2 (11): 797-803.
[8]	P.T. Tian, S. Liao, J. Wang. Dynamical analysis and control strategies in modeling cholera. A monograph, 2010.
[9]	Hargreaves JR, Bonell CP, Boler T, Boccia D, Birdthistle I, Fletcher A, Pronyk PM, Glynn JR. Systematic review exploring time trends in the association between educational attainment and risk of HIV infection in sub-Saharan Africa. AIDS 22:403–414, 2008.
[10]	Barry M, The tail end of Guinea worm Global eradication without a drug or a vaccine. New England J Med 356:2561–2563, 2007.
[11]	Smith RJ, Cloutier P, Harrison J, Desforges A: A mathematical model for the eradication of Guinea Worm Disease, in Mushayabasa S (ed.). Understanding the Dynamics of Emerging and Re-emerging Infectious Diseases Using Mathematical Models, pp. 157–177, 2012.
[12]	Ramjee G, Gouws E, Andrews A, Myer L, Weber AE. The acceptability of a vaginal microbicide among South African men. Fam Plan Perspect 27:164–170, 2001.
[13]	Einarsd´ottir J, Gunnlaugsson G. Health education and cholera in rural Guinea-Bissau. Int J Infect Dis 5:133–138, 2001.
[14]	Hubley J, Communicating Health: An Action Guide to Health Education and Health Promotion, MacMillan Press, London, 1993.
[15]	Rideal S. Disinfection and Disinfectants (an introduction to the study of).Griffin, London, 1895.
[16]	C. T. Codeço. Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. BMC Infectious Diseases 1 (2001), 1.
[17]	D. M. Hartley, J. G. Morris Jr. and D. L. Smith. Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics? PLoS Medicine 3(2006), 0063–0069.
[18]	D. S. Merrell, S. M. Butler, F. Qadri et al. Host-induced epidemic spread of the cholera bacterium, Nature 417 (2002), 642–645.
[19]	S. Liao and J. Wang. Stability analysis and application of a mathematical cholera model. Math. Biosci. Eng. 8 (2011), 733–752.
[20]	R. I. Joh, H. Wang, H. Weiss and J. S. Weitz. Dynamics of indirectly transmitted infectious diseases with immunological threshold. Bull. Math. Biol. 71 (2009), 845–862.
[21]	Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. L. Smith and J. G. Morris Jr. Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe. Proc. Nat. Acad. Sci. 108 (2011), 8767–8772.
[22]	J. H. Tien and D. J. D. Earn. Multiple transmission pathways and disease dynamics in a waterborne pathogen model. Bull. Math. Biol. 72 (2010), 1502–1533.
[23]	J. Tian and J. Wang. Global stability for cholera epidemic models. Math. Biosci. 232 (2011), 31–41.
[24]	R. L. M. Neil an, E. Schaefer, H. Gaff, K. R. Fister and S. Lenhart, Modelingoptimal intervention strategiesfor cholera, Bull. Math. Biol. 72 (2010), 2004–2018.
[25]	Diekman, O., Heesterbeek, J.A.P and Metz, J.A.P. On the definition and Computation of the basic reproduction ratio R_0 in the model of infectious disease in Heterogeneous populations. Journal of Mathematical Biology. 2(1):265-382,1990
[26]	Van den Driessche, P and Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 2002, 180 (1–2): 29–48.
[27]	Stephen, E., Dmitry, K. and Silas, M. Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination. Applied and Computational Mathematics. Vol. 3, No. 4, 2014, pp. 150-162.doi: 10.11648/j.acm.20140304.16
[28]	Stephen, E., Dmitry, K., and Silas, M. Modeling the Impact of Immunization on the epidemiology of Varicella-Zoster Virus. Mathematical theory and Modeling.Vol.4, No. 8, 2014, pp.46-56.
[29]	N. Nyerere, L. S. Luboobi and Y. Nkansah-Gyekye. Modeling the Effect of Screening and Treatment on the Transmission of Tuberculosis Infections. Mathematical Theory and Modeling. Vol.4 , No. 7,2014,pp.51-62.
[30]	N. Nyerere, L. S. Luboobi and Y. Nkansah-Gyekye. Bifurcation and Stability analysis of the dynamics of Tuberculosis model incorporating, vaccination, Screening and treatment. Communications in Mathematical biology and Neuroscience, Vol. 2014 (2014), Article ID 3.
[31]	J. R. Andrews, S. Basu. Transmission dynamics and control of cholera in Haiti: An epidemic model. Lancet 2011; 377: 1248–55.
[32]	Chitnis, N., Hyman, J.M., and Cusching, J. M. (2008). Determining important Parameters in the spread of malaria through the sensitivity analysis of a mathematical Model. Bulletin of Mathematical Biology 70 (5):1272–12
[33]	Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. L. Smith and J. G. Morris Jr. Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Nat. Acad. Sci. 108 (2011), 8767–8772.
[34]	D. M. Hartley, J. G. Morris Jr. and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?. PLoS Medicine 3(2006), 0063–0069.
[35]	Senelani D. Hove-Musekwa, Farai Nyabadzac, Christinah Chiyaka , Prasenjit Das, Agraj Tripathi, Zindoga Mukandavire. Modelling and analysis of the effects of malnutrition in the spread of Cholera. Mathematical and Computer Modelling 53 (2011)1583–1595.
[36]	World Health Organization web page: www.who.org.
[37]	WHO (2012) http://www.who.int/gho/epidemicdiseases/ cholera/ en/index.
[38]	Mwasa, A., Tchuenche, J. M., Mathematical analysis of a cholera model with public health interventions, Biosystems 105:190–200, 2011.

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Stephen Edward, Nkuba Nyerere. (2015). A Mathematical Model for the Dynamics of Cholera with Control Measures. Applied and Computational Mathematics, 4(2), 53-63. https://doi.org/10.11648/j.acm.20150402.14

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Stephen Edward; Nkuba Nyerere. A Mathematical Model for the Dynamics of Cholera with Control Measures. Appl. Comput. Math. 2015, 4(2), 53-63. doi: 10.11648/j.acm.20150402.14

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Stephen Edward, Nkuba Nyerere. A Mathematical Model for the Dynamics of Cholera with Control Measures. Appl Comput Math. 2015;4(2):53-63. doi: 10.11648/j.acm.20150402.14

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@article{10.11648/j.acm.20150402.14,
  author = {Stephen Edward and Nkuba Nyerere},
  title = {A Mathematical Model for the Dynamics of Cholera with Control Measures},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {2},
  pages = {53-63},
  doi = {10.11648/j.acm.20150402.14},
  url = {https://doi.org/10.11648/j.acm.20150402.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150402.14},
  abstract = {Cholera, an acute gastro-intestinal infection and a waterborne disease continues to emerge in developing countries and remains an important global health challenge. In this paper, we formulate a mathematical model that captures some essential dynamics of cholera transmission with public health educational campaigns, vaccination, sanitation and treatment as control strategies in limiting the disease. The reproduction numbers with single and combined controls are computed and compared with each other to assess the possible community benefits. Numerical simulation shows that in a unique control strategy, treatment yields the best results followed by education campaign, then sanitation and vaccination being the last. Furthermore, we noted that the control of cholera is very much better when we incorporated more than one strategy, in two controls the results were better than one strategy, and in three control strategies the results were far better than in two control strategies. Further simulations with all four interventions showed the best results among all combinations attained before. We performed sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence.},
 year = {2015}
}

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TY  - JOUR
T1  - A Mathematical Model for the Dynamics of Cholera with Control Measures
AU  - Stephen Edward
AU  - Nkuba Nyerere
Y1  - 2015/03/21
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150402.14
DO  - 10.11648/j.acm.20150402.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 53
EP  - 63
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150402.14
AB  - Cholera, an acute gastro-intestinal infection and a waterborne disease continues to emerge in developing countries and remains an important global health challenge. In this paper, we formulate a mathematical model that captures some essential dynamics of cholera transmission with public health educational campaigns, vaccination, sanitation and treatment as control strategies in limiting the disease. The reproduction numbers with single and combined controls are computed and compared with each other to assess the possible community benefits. Numerical simulation shows that in a unique control strategy, treatment yields the best results followed by education campaign, then sanitation and vaccination being the last. Furthermore, we noted that the control of cholera is very much better when we incorporated more than one strategy, in two controls the results were better than one strategy, and in three control strategies the results were far better than in two control strategies. Further simulations with all four interventions showed the best results among all combinations attained before. We performed sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence.
VL  - 4
IS  - 2
ER  -

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Author Information

Stephen Edward

Department of Mathematics, College of Natural and Mathematical Sciences, University of Dodoma, Dodoma, Tanzania
Nkuba Nyerere

Departments of Biometry and Mathematics, Sokoine University Of Agriculture, Morogoro, Tanzania

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Stephen Edward, Nkuba Nyerere. (2015). A Mathematical Model for the Dynamics of Cholera with Control Measures. Applied and Computational Mathematics, 4(2), 53-63. https://doi.org/10.11648/j.acm.20150402.14

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ACS Style

Stephen Edward; Nkuba Nyerere. A Mathematical Model for the Dynamics of Cholera with Control Measures. Appl. Comput. Math. 2015, 4(2), 53-63. doi: 10.11648/j.acm.20150402.14

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AMA Style

Stephen Edward, Nkuba Nyerere. A Mathematical Model for the Dynamics of Cholera with Control Measures. Appl Comput Math. 2015;4(2):53-63. doi: 10.11648/j.acm.20150402.14

Copy | Download

@article{10.11648/j.acm.20150402.14,
  author = {Stephen Edward and Nkuba Nyerere},
  title = {A Mathematical Model for the Dynamics of Cholera with Control Measures},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {2},
  pages = {53-63},
  doi = {10.11648/j.acm.20150402.14},
  url = {https://doi.org/10.11648/j.acm.20150402.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150402.14},
  abstract = {Cholera, an acute gastro-intestinal infection and a waterborne disease continues to emerge in developing countries and remains an important global health challenge. In this paper, we formulate a mathematical model that captures some essential dynamics of cholera transmission with public health educational campaigns, vaccination, sanitation and treatment as control strategies in limiting the disease. The reproduction numbers with single and combined controls are computed and compared with each other to assess the possible community benefits. Numerical simulation shows that in a unique control strategy, treatment yields the best results followed by education campaign, then sanitation and vaccination being the last. Furthermore, we noted that the control of cholera is very much better when we incorporated more than one strategy, in two controls the results were better than one strategy, and in three control strategies the results were far better than in two control strategies. Further simulations with all four interventions showed the best results among all combinations attained before. We performed sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - A Mathematical Model for the Dynamics of Cholera with Control Measures
AU  - Stephen Edward
AU  - Nkuba Nyerere
Y1  - 2015/03/21
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150402.14
DO  - 10.11648/j.acm.20150402.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 53
EP  - 63
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150402.14
AB  - Cholera, an acute gastro-intestinal infection and a waterborne disease continues to emerge in developing countries and remains an important global health challenge. In this paper, we formulate a mathematical model that captures some essential dynamics of cholera transmission with public health educational campaigns, vaccination, sanitation and treatment as control strategies in limiting the disease. The reproduction numbers with single and combined controls are computed and compared with each other to assess the possible community benefits. Numerical simulation shows that in a unique control strategy, treatment yields the best results followed by education campaign, then sanitation and vaccination being the last. Furthermore, we noted that the control of cholera is very much better when we incorporated more than one strategy, in two controls the results were better than one strategy, and in three control strategies the results were far better than in two control strategies. Further simulations with all four interventions showed the best results among all combinations attained before. We performed sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence.
VL  - 4
IS  - 2
ER  -

Copy | Download