MATHEMATICAL MODELLING AND ANALYSIS OF CHOLERA DISEASE DYNAMICS WITH CONTROL
Abstract
A deterministic mathematical model of cholera infection incorporating health education campaign, vaccination of susceptible humans, treatment of infected human and water sanitation is developed. It is shown that the solution of the model uniquely exist, it is positive and bounded in a certain region. The disease-free equilibrium (DFE) state of the model was determined and used to compute the basic reproduction number as a threshold for effective disease management. The result from stability analysis for the disease-free equilibrium state (DFEs) shows that it is locally as well as globally asymptotically stable whenever the basic reproduction number is less than unity (). The results obtained from the sensitivity index of show that the control parameters of public health education campaign, vaccination of susceptible individuals, treatment of infected humans and water sanitation are crucial parameters to cholera management. Numerical simulations show that, expanded and improved vaccination among other interventions is crucial in decreasing cholera burden. Furthermore, from the numerical simulations and results it is recommended that a combination of mass and consistent public health education campaigns, expanded vaccination coverage, prompt treatment of infected individuals, with water sanitation, is vital to public health strategies in eradicating cholera infection and deaths in the shortest possible time
References
Ali, M., Nelson, A. R. & Sack, D. (2015). Updated global burden of cholera in endemin countries. PLOS Negl Trop Dis 9(6), e0003832. https://doi:10.1371/journal.pntd.0003832.
Azman, A. S., Rudolph, K. E., Cummings, D. A. & Lesser, J. (2013). The incubation period of cholera: a-systematic review,†Journal of Infection, 66(5), 432–438.
Birkhof, G. & Rota, G.(1982). Ordinary differential equations (8th Edition) Willey, New Jersey.
Castillo-Chavez, C., Feng, C. & Huang, W. (2002). On the computation of and its role on global stability. J. Math. Biol., 35,1-22.
Chitnis, N., Cushing, J. M. & Hyman, J. M. (2006), Bifurcation analysis of a mathematical model for malaria transmission, SIAM Journal on Applied Mathematics, 67(1), 24–45.
Chirambo, R., Mufunda, J., Songolo, P., Kachimba, J. & Vmalika, B. (2016). Epidemiology of the 2016 cholera outbreak of chibombo district of central zambia. Medical Journal of Zambia, 43(2), 61–63.
Codeco, C. (2001). Endemic and epidemic dynamics of cholera: the role of the acquatic reservoir. BMC Infectious Diseases, 1, 1.
Diekmann, J. A., Heesterbeek, J. A. P. & Metz, J.A. J. (1990). ‘On the definition and the computation of the basic reproduction ratio Ro in models for infectious diseases in heterogeneous populations’, Journal of mathematical Biology, 28, 365-382.
Diekmann, O., Heesterbeek, J. A. P. & Roberts, M. G. (2010). ‘The construction of next-generation matrix for compartmental epidemic models’. Journal of the Royal Society Interface, l7, 873–885.
Einarsd’ottir, J., Gunnlaugsson, G. (2001). Health education and cholera in rural Guinea-Bissau. Int. J Infect Dis., 5, 133–138.
Falaye, A., Akarawak, E. E., Cole, A. T., Evans, O., Oluyori, D., Adeyemi, E., Falaye, R. & Adama, N. (2018). A mathematical model for capturing cholera spread and containment options. International Journal of Mathematical Sciences and Computing(IJMSC), 4(1), 15-40.
http://doi.org/10.5815/ijmsc.2018.01.02.
Grinshaw, R. (1990). Nonlinear ordinary differential equations. Blackwell Scientific Publications.
Hargreaves, J. R., Bonell, C. P., Boler, T., Boccia, D., Birdthistle, I., Flecher, A., Pronyk, P. M. & Glynn, J. R. (2008). Systematic review exploring time trends in the association between educational attainment and risk of HIV infection in Sub-Sahara Africa. AIDS, 22, 403–414.
Hartley, D. M., Morris, J. G. Jnr & Smith, D. L. (2006), “Hyperinfectivity: A critical element in the ability of v. cholera to cause epidemic? Plos Medicine, 3(1), 63–69.
Legros, D. (2019). Global cholera epidemiology: Opportunities to reduce the burden of cholera by 2030. The Journal of Infectious Diseases, 219(3), 509. https://doi.org/10.1093/infdis/jiy619
Misra, A., Sharma, A. & Shukla, J. (2011). Modelling and analysis of the effect of awareness programs by media in the spread of infectious diseases. Mathematical and Computer Modelling. 53, 5-6.
Mukandavire, Z., Liao, S., Wang, J., Gaff, H., Smith, D. L.& Morris, J. G. (2011). Estimating the reproductive numbers for the 2008 – 2009 cholera outbreaks in Zimbabwe. Proc Nat’l Acad Sci 108, 8767–8772.
Shrivastava, S. R., Shrivastava, P. S. & Ramasamy, J. (2019). Worldwide implementation of the global roadmap to end cholera by 2030. Int J health Allied Sci., 8, 75–76.
Tarh, J. E. (2019). An overview of cholera epidemiology: A focus on Africa: with a keen interest on Nigeria. International Journal of Tropical Disease & Health, 40(3), 1–17.
Wang, J. & Modrick, C. (2011). Modelling cholera dynamics with controls. Canadian Applied Mathematics Quartely, 19, 255-273.
Wang, X. (2004). A simple proof of Descarte’s rule of signs. American Mathematical Monthly, 111(6), 525-526.
World Health Organisation. (2019, 17 January). Cholera fact sheets.
https://www.who.int/news-room/fact-sheets/detail/Cholera
Wu, J., Dhingra, R., Gambhir, M., Remais, J. V., (2013), Sensitivity analysis of infectious disease models: methods, advances and their application. J. R Soc Interface 10:201221018. http://dx.doi.org/10.1098/rsif.2012.1018.
Copyright (c) 2020 FUDMA JOURNAL OF SCIENCES
This work is licensed under a Creative Commons Attribution 4.0 International License.
FUDMA Journal of Sciences
