Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Transmission dynamics of cholera with hyperinfectious and hypoinfectious vibrios: mathematical modelling and control strategies

1 Institute of Applied Mathematics, Army Engineering University, Shijiazhuang 050003, Hebei, P.R. China
2 Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, P.R. China

Special Issues: Transmission dynamics in infectious diseases

Cholera is a common infectious disease caused by Vibrio cholerae, which has different infectivity. In this paper, we propose a cholera model with hyperinfectious and hypoinfectious vibrios, in which both human-to-human and environment-to-human transmissions are considered. By analyzing the characteristic equations, the local stability of disease-free and endemic equilibria is established. By using Lyapunov functionals and LaSalle’s invariance principle, it is verified that the global threshold dynamics of the model can be completely determined by the basic reproduction number. Numerical simulations are carried out to illustrate the corresponding theoretical results and describe the cholera outbreak in Haiti. The study of optimal control helps us seek cost-effective solutions of time-dependent control strategies against cholera outbreaks, which shows that control strategies, such as vaccination and sanitation, should be taken at the very beginning of the outbreak and become less necessary after a certain period.
  Article Metrics

Keywords Cholera model; hyperinfectious vibrios; human-to-human transmission; environment-to-human transmission; global dynamics; optimal control

Citation: Jiazhe Lin, Rui Xu, Xiaohong Tian. Transmission dynamics of cholera with hyperinfectious and hypoinfectious vibrios: mathematical modelling and control strategies. Mathematical Biosciences and Engineering, 2019, 16(5): 4339-4358. doi: 10.3934/mbe.2019216


  • 1. Cholera Fact Sheets, World Health Organization, August 2017. Available from: https://www.who.int/en/news-room/fact-sheets/detail/cholera.
  • 2. J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol., 72 (2010), 1506–1533.
  • 3. X. Zhou and J. Cui, Threshold dynamics for a cholera epidemic model with periodic transmission rate, Appl. Math. Model., 37 (2013), 3093–3101.
  • 4. D. Posny, J. Wang, Z. Mukandavire, et al., Analyzing transmission dynamics of cholera with public health interventions, Math. Biosci., 264 (2015), 38–53.
  • 5. Y. Wang and J. Cao, Global stability of general cholera models with nonlinear incidence and removal rates, J. Frankl. Inst., 352 (2015), 2464–2485.
  • 6. J. P. Tian and J. Wang, Global stability for cholera epidemic models, Math. Biosci., 232 (2011), 31–41.
  • 7. D. M. Hartley, J. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics? PLoS Med., 3 (2006), 63–69.
  • 8. R. L. M. Neilan, E. Schaefer, H. Gaff, et al., Modeling optimal intervention strategies for cholera, Bull. Math. Biol., 72 (2010), 2004–2018.
  • 9. Z. Mukandavire, S. Liao, J. Wang, et al., Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe, PNAS, 108 (2011), 8767–8772.
  • 10. T. K. Sengupta, R. K. Nandy, S. Mukhopadhyay, et al., Characterization of a 20-kDa pilus protein expressed by a diarrheogenic strain of non-O1/non-O139 vibrio cholerae, FEMS Microbiol. Lett., 160 (1998), 183–189.
  • 11. C. T. Codec ¸o, Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infect. Dis., 1 (2001), 1–14.
  • 12. C. Modnak, A model of cholera transmission with hyperinfectivity and its optimal vaccination control, Int. J. Biomath., 10 (2017), 1750084.
  • 13. A.P.Lemos-Paião, C.J.SilvaandD.F.M.Torres, A cholera mathematical model with vaccination and the biggest outbreak of world's history, AIMS Mathematics, 3 (2018), 448–463.
  • 14. J. K. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, 1st edition, Springer-Verlag, New York, 1993.
  • 15. P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48.
  • 16. O. Sharomi and T. Malik, Optimal control in epidemiology, Ann. Oper. Res., 251 (2017), 55–71.
  • 17. J. K. K. Asamoah, F. T. Oduro, E. Bonyah, et al., Modelling of rabies transmission dynamics using optimal control analysis, J. Appl. Math., 12 (2017), 1–23.
  • 18. L. Cesari, Optimization-Theory and Applications. Problems with ordinary differential equations, in: Applications of Mathematics, vol. 17, Springer-Verlag, New York, 1983.
  • 19. W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, 1st edition, Springer-Verlag, New York, 1975.
  • 20. L. Pontryagin, V. Boltyanskii, R. Gramkrelidze, et al., The Mathematical Theory of Optimal Processes, John Wiley & Sons, 1962.
  • 21. C. J. Silva and D. F. M. Torres, Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Math. Biosci., 244 (2013), 154–164.
  • 22. M. McAsey, L. Mou and W. Han, Convergence of the forward-backward sweep method in optimal control, Comput. Optim. Appl., 53 (2012), 207–226.
  • 23. O. Zakary, A. Larrache, M. Rachik, et al., Effect of awareness programs and travel-blocking operations in the control of HIV/AIDS outbreaks: a multi-domains SIR model, Adv. Differ. Equ., 2016 (2016), 169.
  • 24. C. Modnak, J. Wang and Z. Mukandavire, Simulating optimal vaccination times during cholera outbreaks, Int. J. Biomath., 7 (2014), 1450014.
  • 25. J. Wang and C. Modnak, Modeling cholera dynamics with controls, Canadian Appl. Math. Quart., 19 (2011), 255–273.
  • 26. D.S.Merrell, S.M.Butler, F.Qadri, et al., Host-induced epidemic spread of the cholera bacterium, Nature, 417 (2002), 642–645.
  • 27. S. Lenhart and J. T. Workman, Optimal control applied to biological models, Chapman & Hall/CRC Press, Boca Raton, 2007.
  • 28. Global Task Force on Cholera Control, Cholera Country Profile: Haiti, World Health Organization, May 2011. Available from: https://www.who.int/cholera/countries/HaitiCountryProfileMay2011.pdf.
  • 29. A. P. Lemos-Paião, C. J. Silva and D. F.M. Torres, An epidemic model for cholera with optimal control treatment, J. Comput. Appl. Math., 318 (2017), 168–180.


Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved