International Journal of Nonlinear Analysis and Applications

Analysis of Cholera model with treatment noncompliance

Document Type : Research Paper

Authors

1 Department Of Computer Science And Mathematics, Mountain Top University, Nigeria

2 Department of Computer Science, Lead City University, Ibadan, Nigeria

Abstract

A model for transmission dynamics of cholera infection between human host and environment is developed. We incorporate the proportion of infectious individuals who do not comply with treatment into the human population. Stability analysis, as well as simulation of the model, is done. The results from the stability analysis show that the disease-free equilibrium solution is locally asymptotically stable if $R_0 < 1$ while the endemic equilibrium solution is globally asymptotically stable when $R_0 > 1$. The technical tool used for our analysis is the theory of competitive systems, compound matrices and stability of periodic orbits. Finally, we investigate, numerically, the influence of seasonal variation on the control of cholera.

Keywords

[1] JR. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: an epidemic model, The Lancet 377 (2011) 1248-1255.
[2] P. Ana, L Paiao, JS. Christiana and FMT Del_m, A cholera mathematical model with vaccination and the biggest outbreak of world's history, Retrieved from http: \\www:aimspress:com=journal=Math, (2018).
[3] PR. Brayton, ML. Tamplin, & RR. Colwell, Enumeration of vibrio cholerae in Bangladesh waters by fluorescent-antibody direct viable count Appl. Environ.Microbiol. 53 (1987) 2862-2865.
[4] V. Capasso and SL. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European, Mediterranean region, Rev.Epidemiol. Sante. 27 (1979) 121-132.
[5] F. Capone, V. De Cataldis and R. De Luca, Inuence of di_usion on the stability of equilibria in a reaction-diffusion system modeling cholera dynamic, J. Math. Biol. 71 (2008) 1107-1131.
[6] C. Codeco, Endemic and epidemic dynamic of cholera: the role of the aquatic reservoir, BMC Infectious Diseases (2001)
[7] Y. Chayu and W. Jin, A cholera transmission model incorporating the impact of medical resources, Mathematical Biosciences and Engineering 16(5) (2019) 5226-5246.
[8] JK. Hale. Ordinary di_erential equations, John Wiley, 1969.
[9] DM. Hartley, JB. Morris, DL. Smith, Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics? Plos Med. 3(1) (2006) e7.
[10] MW. Hirsch, Systems of di_erential equations that are competitive or cooperative IV. Structural stability in 3-dimensional systems, Journal of Di_erential Equations. 80 (1989) 94-106.
[11] AE. Kamuhanda, O. Shaibu and W. Mary, Mathematical modeling and analysis of the dynamics of cholera, Global Journal of Pure and Applied Mathematics. 14(9) (2018) 1259-1275.
[12] AA., King, E.L. Ionides, M. Pascual & M.J. Bouma, Inapparent infections and cholera dynamics. Emerging Infectious Diseases. 14 (2008) 877-881.
[13] JB. Kaper, JG. Morris Jr., & MM. Levine, Clin. Microbiol. Rev. (1995) 48-86.
[14] JP. Lasalle, The stability of dynamical systems. Philadelphia, PA: SIAM, 1976.
[15] AP. Lemos-Paiao, CJ. Silva and DFM Torres An epidemic model for cholera with optimal control treatment, J. Comput. Appl. Math. 318 (2017) 168-180.
[16] MY. Li and JS. Muldowney, A geometric approach to global stability problems, SIAM J.Math.Anal. 27 (1996) 1070-1083.
[17] MY. Li and JS. Muldowney, Global stability for the SEIR model in epidemiology, Mathematical Biosciences 125 (1995) 155-164.
[18] JS. Mulodowney, Compound matrices and ordinary differential equations, Rocky Mountain Journal of Mathematics 20 (1990) 857-872
[19] Z. Mukandavire, S. Liao, J. Wang, H. Ga_, DL. Smith and JG. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Natl Acad. Sci. 108 (2011) 8767-8772.
[20] RLM. Neilan, E. Schaefer, H. Ga_, KR. Fister and S. Lenhart, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Natl Acad. Sci. 108 (2011) 8767-8772.
[21] CC. Obiora and JD. Kelvin, Analysis and optimal control intervention strategies of a waterborne disease model:a realistic case study, Hindawi Journal of Applied Mathematics 14 (2018)
[22] M. Pascual, MJ. Bouma, AP. Dopson, Cholera and climate: revisiting the quantitative evidence, Microbes and Infection 4(2) (2002) 237-245.
[23] HL. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, mathematical survey and monographs, American Mathematical Society (1995) 231-240
[24] HL. Smith, Systems of di_erential equations which generate an order preserving ow, SIAM Review 30 (1988) 87-113.
[25] RP. Sanches, CP. Ferreira and RA. Kraenkel, The role of immunity and seasonality in cholera epidemics, Bulletin of Mathematical Biology 73(12) (2011) 2916-2931.
[26] J. Sepulveda, H. Gomez-Dantes, M. Bronfman, Cholera in the Americas: an overview. Infection 20(5) (1992) 243-248.
[27] J. Tumwiine, JYT. Mugisha and LS. Luboobi, A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity, Applied Mathematics and Computation 189 (2007) 1953-1965.
[28] JH. Tien and DJ. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology 72(6) (2010) 1506-1533.
[29] World Health Organization web page: www.who.org. (2014)
[30] T. Xiaohong, X. Rui and L. Jiazche, Mathematical analysis of a cholera infection model with vaccination strategy, Applied Mathematics and Computation 361 (2019) 517-535.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 29-43
  • Receive Date: 07 January 2021
  • Revise Date: 14 June 2021
  • Accept Date: 28 June 2021