# Dynamics of Cholera Pathogen Carriers and Effect of Hygiene Consciousness in Cholera Outbreaks

Document Type : Research Article

Authors

1 Department of Mathematics, Faculty of Physical Sciences, Kebbi State University of Science and Technology, Aliero, Nigeria.

2 Department of Mathematics, Faculty of Physical Sciences, University of Ilorin, Ilorin, Nigeria,

Abstract

We derive a deterministic mathematical model that scrutinizes the dy-namics of cholera pathogen carriers and the hygiene consciousness of in-dividuals, before the illness, during its prevalence, and after the disease’s outbreaks. The dynamics can effectively help in curtailing the disease, but its effects had less coverage in the literature. Boundedness of the solu-tion of the model, its existence, and uniqueness are ascertained. Effects of cholera pathogen carriers and hygiene consciousness of individuals in controlling the disease or allowing its further spread are analyzed. The differential transformation method is used to obtain series solutions of the differential equations that make the system of the model. Simulations of the series solutions of the model are carried out and displayed in graphs. The dynamics of the concerned state variables and parameters in the model are interpreted via the obtained graphs. It is observed that higher hygiene consciousness of individuals can drastically reduce catching cholera disease at onset and further spread of its infections in the population, this in turn, shortens the period of cholera epidemic.

Keywords

Main Subjects

#### References

[1] Adagbada, A.O., Adesida, S.A., Nwaokorie, F.O., Niemogha, M.T. and Coker, A.O. Cholera epidemiology in Nigeria: an overview. Pan Afr. Med. J. 12(1) (2012).
[2] Ahmad, M.Z., Alsarayreh, D., Alsarayreh, A. and Qaralleh, I. Differ-ential transformation method (DTM) for solving SIS and SI epidemic models. Sains Malays. 46 (10) (2017) 2007–2017.
[3] Akinboro, F.S., Alao, S. and Akinpelu, F.O. Numerical solution of SIR model using differential transformation method and variational iteration method. General Mathematics Notes 22(2) (2014) 82–92.
[4] Batiha, K. and Batiha, B. A new algorithm for solving linear ordinary differential equations. World Appl. Sci. J. 15 (12) (2011) 1774–1779.
[5] Chakraborty, A.K., Shahrear, P. and Islam, M.A. Analysis of epidemic model by differential transform method. J. Multidiscip. Eng. Sci. Technol. 4(2) (2017) 6574–6581.
[6] CCodeço, C.T., Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. BMC Infect. Dis. 1 (2001) 1–14.
[7] Edwards, C.H. and Penney, D.E. Differential equations and boundary value problems: computing and modeling. Pearson Educación. 2000.
[8] Edward, S. and Nyerere, N. A mathematical model for the dynamics of cholera with control measures. Comput. Appl. Math. 4(2) (2015) 53–63.
[9] Egbetade, S.A. and Ibrahim, M.O. Modelling the Impact of BCG vac-cines on tuberculosis epidemics. Journal of Mathematical Modelling and Application, 9(1) (2014) 49–55.
[10] Elhia, M., Laaroussi, A., Rachik, M., Rachik, Z. and Labriji, E., Global stability of a susceptible-infected-recovered (SIR) epidemic model with two infectious stages and treatment. Int. J. Sci. Res. 3(5) (2014) 114–121.
[11] Harris JB, LaRoque RC, Qadri F, Ryan ET, Calderwood SB. Seminar, The Lancet 2012.
[12] Jahan, S., Cholera–epidemiology, prevention and control. Significance, Prevention and Control of Food Related Diseases. Croatia: InTech, (2016) 145–157.
[13] Javidi, M. and Ahmad, B., A study of a fractional-order cholera model. Appl. Math. Inf. Sci. 8(5) (2014) p.2195.
[14] Kenmogne, F. Generalizing of differential transform method for solving nonlinear differential equations. J. Appl. Comp. Math. 4 (2015) p.196.
[15] Mafuta, P., Mushanyu, J. and Nhawu, G. Invariant region, endemic equilibria and stability analysis. IOSR J. Math. 10(2) (2014) 118–120.
[16] Madubueze, C.E., Kimbir, A.R., Onah, E.S. and Aboiyar, T., Existence and uniqueness of solution of ebola virus disease model with contact tracing and quarantine as controls. Nigerian J. Math. Appl. 25 (2016) 111–121.
[17] Merrell, D.S., Butler, S.M., Qadri, F., Dolganov, N.A., Alam, A., Cohen, M.B., Calderwood, S.B., Schoolnik, G.K. and Camilli, A., Host-induced epidemic spread of the cholera bacterium. Nature, 417(6889) (2002) 642–645.
[18] Mirzaee, F., Differential transform method for solving linear and non-linear systems of ordinary differential equations. Appl. Math. Sci. 5(70) (2011) 3465–3472.
[19] Najafgholipour, M. and Soodbakhsh, N., Modified differential transform method for solving vibration equations of MDOF systems. Civil Eng. J. 2(4) (2016) 123–139.
[20] Nelson, E.J., Harris, J.B., Glenn Morris Jr, J., Calderwood, S.B. and Camilli, A., Cholera transmission: the host, pathogen and bacteriophage dynamic. Nat. Rev. Microbiol. 7(10) (2009) 693–702.
[21] Ochoche, M.J., Madubueze, E.C. and Akaabo, T.B., A mathematical model on the control of cholera: hygiene consciousness as a strategy. J. Math. Comput. Sci. 5(2) (2015) 172–187.
[22] Panja, P. and Mondal, S.K., A mathematical study on the spread of Cholera. South Asian J. Math. 4(2) (2014) 69–84.
[23] Patil, N. and Khambayat, A., Differential transform method for ordinary differential equations. Res. J. Math. Stat. Sci. 3 (2014) 330–337.
[24] Penrose, K., Castro, M.C.D., Werema, J. and Ryan, E.T., Informal urban settlements and cholera risk in Dar es Salaam, Tanzania. PLoS Negl. Trop. Dis. 4(3) (2010) p.e631.
[25] Todar K. Online Textbook of Bacteriology. www.onlinetextbookofbacteriology.net 2008.
[26] Tuite, A.R., Chan, C.H. and Fisman, D.N., Cholera, canals, and conta-gion: Rediscovering Dr Beck’s report. J. Public Health Policy, 32 (2011) 320–333.
[27] Uwishema, O., Okereke, M., Onyeaka, H., Hasan, M.M., Donatus, D., Martin, Z., Oluwatomisin, L.A., Mhanna, M., Olumide, A.O., Sun, J. and Adanur, I., Threats and outbreaks of cholera in Africa amidst COVID-19 pandemic: a double burden on Africa’s health systems. Trop-ical medicine and health, 49(1) (2021) p.93.
[28] Wang, X. and Wang, J., Analysis of cholera epidemics with bacterial growth and spatial movement. J. Biol. Dyn. 9(sup1) (2015) 233–261.
[29] Wang, J., Mathematical models for cholera dynamics—A review. Mi-croorganisms, 10(12) (2022) p.2358.
[30] Soltanalizadeh, B. and Branch, S., Application of differential transfor-mation method for solving a fourth-order parabolic partial differential equations. Int. J. Pure Appl. Math. 78(3) (2012) 299–308.
[31] Munganga, J.M.W., Mwambakana, J.N., Maritz, R., Batubenge, T.A. and Moremedi, G.M., Introduction of the differential transform method to solve differential equations at undergraduate level. Int. J. Math. Educ. Sci. Technol. 45(5) (2014) 781–794.
[32] Saeed, R.K. and Rahman, B.M., Differential transform method for solv-ing system of delay differential equation. Aust. J. Basic Appl. Sci. 5(4) (2011) 201–206.