American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: Editor-in-chief: Mohamed Seddeek
Open Access
Journal Browser
American Journal of Applied Mathematics and Statistics. 2019, 7(1), 1-8
DOI: 10.12691/ajams-7-1-1
Open AccessArticle

Modeling and Analysis of Cholera Dynamics with Vaccination

Nneamaka Judith Ezeagu1, , Houénafa Alain Togbenon1 and Edwin Moyo1

1Department of Mathematics, Pan African University Institute for Basic Sciences, Technology and Innovations (PAUISTI), Nairobi, Kenya

Pub. Date: December 24, 2018

Cite this paper:
Nneamaka Judith Ezeagu, Houénafa Alain Togbenon and Edwin Moyo. Modeling and Analysis of Cholera Dynamics with Vaccination. American Journal of Applied Mathematics and Statistics. 2019; 7(1):1-8. doi: 10.12691/ajams-7-1-1


A mathematical model for the transmission of cholera dynamics with a class of quarantined and vaccination parameter as control strategies is proposed in this paper. It is shown through mathematical analysis that the solution of the model uniquely exist, is positive and bounded in a certain region. The disease-free and endemic equilibrium points of the model are obtained. By using the next generation matrix, the basic reproduction number was computed around the disease-free equilibrium points, and it was shown through the Jacobian matrix that the disease free equilibrium is locally asymptotic stable if Rh<1. Numerical simulation was carried to understand the impact of the incorporated controls as the system evolves over time. Results show that effective quarantine, vaccination and proper sanitation reduce the disease contact rates and thus eliminates the spread of cholera.

cholera vaccination basic reproduction number equilibrium points

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit


[1]  World Health Organization (WHO), “Cholera key facts,” 2018. [Online] Available: [Accessed Oct. 9, 2018]
[2]  Azman, A.S., Rudolph, K.E., Cummings, D.A. and Lessler, J. “The incubation period of cholera: a systematic review,” Journal of Infection, 66(5), 432-438, 2013.
[3]  Yang, X., Chen, L. and Chen, J., “Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models,” Computers & Mathematics with Applications, 32(4), 109-116, 1996.
[4]  Chirambo, R., Mufunda, J., Songolo, P., Kachimba, J. and Vwalika, B., “Epidemiology of the 2016 cholera outbreak of chibombo district central zambia,” Medical Journal of Zambia, 43(2), 61-63, 2016.
[5]  Edward, S. and Nyerere, N. “A mathematical model for the dynamics of cholera with control measures,” Applied and Computational Mathematics, 4(2), 53-63, 2015.
[6]  Ochoche, J.M., “A mathematical model for the transmission dynamics of cholera with control strategy,” International Journal of Science and Technology, 2(11), 797-803, 2013.
[7]  Codeco, C.T., “Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir,” BMC Infectious diseases, 1(1), 1, 2001.
[8]  Cui, J., Wu, Z. and Zhou, X., “Mathematical analysis of a cholera model with vaccination,” Journal of Applied Mathematics, 2014.
[9]  Mukandavire, Z., Liao, S., Wang, J., Gaff, H., Smith, D.L. and Morris, J.G., “Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in zimbabwe,” Proceedings of the National Academy of Sciences, 108(21), 8767-8772, 2011.
[10]  Wang, J. and Modnak, C., “Modeling cholera dynamics with controls,” Canadian applied mathematics quarterly, 19(3), 255-273, 2011.
[11]  Lemos-Paião, A.P., Silva, C.J. and Torres, D.F., “An epidemic model for cholera with optimal control treatment,” Journal of Computational and Applied Mathematics, 318, 168-180, 2017.
[12]  Sun, G.-Q., J.-H., Huang, S.-H., Z., Li., M.-T. And Liu, Li., “Transmission dynamics of cholera: Mathematical modeling and control strategies,” Communications in Nonlinear Science and Numerical Simulation, 45, 235-244, 2017.
[13]  Centers for Disease Control and Prevention (CDC), “Cholera-vibrio cholerae infection,” 2018. [Online] Available: [Accessed Oct. 14, 2018].
[14]  Mwasa, A. and Tchuenche, J.M., “Mathematical Analysis of a cholera model with public health interventions,” Biosystems, 105(3), 190-200, 2011.
[15]  Sanches, R.P., Ferreira, C.P. and Kraenkel, R.A., “The role of immunity and seasonality in cholera epidemics,” Bulletin of mathematical biology, 73(12), 2916-2931, 2011.
[16]  Diekmann, O., Heesterbeek, J.A.P. and Metz, J.A., “On the definition and the computation of the basic reproduction ratio r0 in models for infectious diseases in heterogeneous populations,” Journal of mathematical biology, 28 (4), 365-382, 1990.
[17]  Van den Driessche, P. and Watmough, J., “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical biosciences, 180(1), 29-48, 2002.
[18]  Ezeagu, N.J., Orwa, G.O., and Winckler, M.J., “Transient analysis of a finite capacity m/m/1 queuing system with working breakdowns and recovery policies,” Global Journal of Pure and Applied Mathematics, 14(8), 1049-1065, 2018.
[19]  Togbenon, H.A., Degla, G.A., and Kimathi, M.E., “Stability analysis using nonstandard finite difference method and model simulation for multi-mutation and drug resistance: A case of immune-suppression,” Journal of Mathematical Theory and Modeling, 8(7), 77-96, 2018.
[20]  Togbenon, H.A., Kimathi, M.E., Degla, G.A., “Modeling multimutation and drug resistance: A case of immune-suppresion,” Global Journal of Pure and Applied Mathematics, 14(6), 787-807, 2018.