ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Analysis of Transmission Dynamics of Cholera: An Optimal Control Strategy

Journal of Applied Nonlinear Dynamics 11(2) (2022) 387--400 | DOI:10.5890/JAND.2022.06.009

Matthew O. Adewole$^{1,2}$, Akindele Onifade$^1$, Ahmad Izani Md Ismail$^2$, Taye Faniran$^3$,\\ Farah A. Abdullah$^2$

$^1$ Department of Computer Science and Mathematics, Mountain Top University, Prayer City, Ogun State, Nigeria

$^2$ School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia

$^3$ Department of Computer Science, Lead City University, Lagos-Ibadan Expressway, Ibadan, Oyo State, Nigeria

Abstract

Cholera affects populations living with poor sanitary conditions and has caused enormous morbidity and mortality. A mathematical model is presented for the spread of cholera with focus on three human populations; susceptible human, infected human and recovered human. The infected human population was subdivided into two groups - symptomatic individuals and asymptomatic individuals. We obtain the reproductive number and a sensitivity analysis of model parameters is conducted. The sensitivity analysis reveals key parameters which can be used to propose intervention strategies. Our analysis indicates that a single intervention strategy is insufficient for the eradication of the disease. Optimal control strategy is incorporated to find effective solutions for time-dependent controls for eradicating cholera epidemics. We use numerical simulations to explore various optimal control solutions involving single and multiple controls. Our results show that, as in related previous studies, the costs of controls have a direct effect on the duration and strength of each control in an optimal strategy. It is also established that a combination of multiple intervention strategies attains better results than a single-pronged approach since the strength of each control strategy is limited by available resources and social factors.

References

1.  [1] {Cholera vaccines: WHO position paper}. Wkly Epidemiol Rec. 92(34), 477-500. http://apps.who.int/iris/ bitstream/10665/258763/1/WER9234.pdf?ua=1.
2.  [2] Sun, G., Xie, J., Huang, S., Jin, Z., Li, M., and Liu, L. (2017), Transmission dynamics of cholera: mathematical modeling and control strategies, Commun Nonlinear Sci Numer Simul., 45, 235-244. https://doi.org/ 10.1016/j.cnsns.2016.10.007.
3.  [3] {Scientific Committee on Enteric Infections and Foodborne Diseases}. Epidemiology, prevention and control of cholera in {H}ong {K}ong. Center for Health Protection, Department of Health, Hong Kong Special Administrative Region. 2011 http://www.chp.gov.hk/files/pdf/epidemiology\_prevention\_and\_control\_of\_cholera\_in \_hong\_kong\_r.pdf.
4.  [4] Nelson, E.J., Harris, J.B., Glenn Morris Jr, J., Calderwood, S.B., and Camilli, A. (2009), Cholera transmission: the host, pathogen and bacteriophage dynamic, Nature Reviews (Microbiology), 7, 693-702.
5.  [5] Ali, M., Nelson, A.R., Lopez, A.L., and Sack, D.A. (2015), Updated global burden of cholera in endemic countries, PLoS Negl Trop Dis., 9, 1-13. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4455997/ pdf/pntd.0003832.pdf
6.  [6] {Media Centre}. (2017), Cholera. World Health Organization. http://www.who.int/mediacentre/factsheets/ fs107/en/.
7.  [7] Graves, P.M., Deeks, J.J., Demicheli, V., and Jefferson, T. (2010), Vaccines for preventing cholera: killed whole cell or other subunit vaccines (injected). Cochrane Database of Systematic Reviews, 8. http:https://doi.org/10.1002/14651858.CD000974.pub2.
8.  [8] Sinclair, D., Abba, K., Zaman, K., Qadri, F.. and Graves, P.M. (2011), Oral vaccines for preventing cholera. Cochrane Database of Systematic Reviews. https://doi.org/10.1002/14651858.CD008603.pub2.
9.  [9] Andrews, J.R. and Basu, S. (2011), Transmission dynamics and control of cholera in haiti: an epidemic model, Lancet, 377, 1248-1255.
10.  [10] Kong, J.D., Davis, W., and Wang, H. (2014), Dynamics of a cholera transmission model with immunological threshold and natural phage control in reservoir, Bull Math Biol., 76(8), 2025-2051. https://doi.org/10.1007/s11538-014-9996-9.
11.  [11] Mukandavire, Z., Liao, S., Wang, J., Gaff, H., Smith D.L., and Morris J.G. (2011), Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in zimbabwe, Proc Natl Acad Sci USA, 108(21), 8767-8772. https://www.ncbi.nlm.nih.gov/pubmed/21518855.
12.  [12] Mwasa, A. and Tchuenche, J. (2011), Mathematical analysis of a cholera model with public health interventions, BioSystems, 105, 190-200.
13.  [13] Shuai, Z. and van den Driessche, P. (2011), Global dynamics of cholera models with differential infectivity, Math Biosciences, 234(2), 118-126. https://doi.org/10.1016/j.mbs.2011.09.003.
14.  [14] Tian, J.P. and Wang, J. (2011), Global stability for cholera epidemic models, Math Biosciences, 232(1), 31-41. https://doi.org/10.1016/j.mbs.2011.04.001.
15.  [15] Wang, J. and Modnak, C. (2011), Modeling cholera dynamics with controls, Can Appl Math Q., 19(3), 255-273.
16.  [16] Wang, J. and Liao, S. (2012), A generalized cholera model and epidemic-endemic analysis, J Biol Dynnamics, 6(2), 568-589. https://doi.org/10.1080/17513758.2012.658089.
17.  [17] Shuai, Z., Tien, J.H., and {van den Driessche}, P. (2012), Cholera models with hyperinfectivity and temporary immunity, Bull Math Biology, 74(10), 2423-2445. https://doi.org/10.1007/s11538-012-9759-4.
18.  [18] Posny, D., Wang, J., Mukandavire, Z., and Modnak, C. (2015), Analyzing transmission dynamics of cholera with public health interventions. Math Biosciences, 264, 38-53. https://doi.org/10.1016/j.mbs.2015.03.006.
19.  [19] Beryl, M.O., George, L.O., and Fredrick, N.O. (2016), Mathematical analysis of a cholera transmission model incorporating media coverage, Int J of Pure and Applied Mathematics, 11(2), 219-231. http://www.ijpam.eu.
20.  [20] Fung, I.C.H. (2014), Cholera transmission dynamic models for public health practitioners. Emerging Themes in Epidemiology, 11(1), 1-11. http://www.ete-online.com/content/11/1/1.
21.  [21] {World Health Organization (WHO), webpage: www.who.org}
22.  [22] Codeco, C.T. (2001), Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1(1), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC29087/.
23.  [23] Hartley, D.M., Morris J.G.~Jr and Smith, D.L. (2005), Hyperinfectivity: A critical element in the ability of v. cholerae to cause epidemics? PLOS Medicine, 3(1). https://doi.org/10.1371/journal.pmed.0030007.
24.  [24] Diekmann, O., Heesterbeek, J.A.P. and Metz, J.A.J. (1990), On the definition and the computation of the basic reproduction ratio {$R_0$} in models for infectious diseases in heterogeneous populations, J Math Biology, 28(4) 365-382. https://doi.org/10.1007/BF00178324.
25.  [25] Chitnis, N., Hyman, J.M., and Cushing, J.M. (2008), Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull Math Biology 2008;\hspace{0pt}70(5):1272-1296, https://doi.org/10.1007/s11538-008-9299-0.
26.  [26] {Miller Neilan}, R.L., Schaefer, E., Gaff, H., Fister, K.R., and Lenhart, S. (2010), Modeling optimal intervention strategies for cholera, Bulleting of Mathematical Biology, 72(8), 2004-2018. https://doi.org/10.1007/s11538-010-9521-8.
27.  [27] Geering, H.P. (2007), Optimal control with engineering applications, Springer Berlin Heidelberg New York.
28.  [28] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. (1962) The mathematical theory of optimal processes, Interscience Publishers John Wiley \& Sons, Inc.\, New York-London. Translated from the Russian by K.N. Trirogoff; edited by L. W. Neustadt.