On the verification of existence of backward bifurcation for a mathematical model of cholera dynamics.

A. A. Ayoade; O. J. Peter; F. A. Oguntolu; C. Y. Ishola; S Amadiegwu

A. A. Ayoade Department of Mathematics, University of Ilorin, Ilorin, Kwara State, Nigeria.
O. J. Peter Department of Mathematics, University of Ilorin, Ilorin, Kwara State, Nigeria.
F. A. Oguntolu Department of Mathematics Federal University of Technology, Minna, Nigeria.
C. Y. Ishola Department of Mathematics, National Open University of Nigeria Jabi, Abuja, Nigeria.
S Amadiegwu Department of Mathematics, School of General Studies, Maritime Academy of Nigeria, Oron, Akwa Ibom, State Nigeria.

Keywords: Model, Stability, Center manifold theory, Bifurcation

Abstract

A cholera transmission model, which incorporates preventive measures, is studied qualitatively. The stability results together with the center manifold theory are used to investigate the existence of backward bifurcation for the model. The epidemiological consequence of backward bifurcation is that the disease may still persist in the population even when the classical requirement of the reproductive number R_0 being less than one is satisfied.

References

Isere, A. O., Osemwenkhae, J. E., & Okuonghae, D. Optimal control model for the outbreak of cholera in Nigeria. African Journal of Mathematics and Computer Science Research, 7(2), 24-30.(2014).

. Ochoche, J. M. A mathematical model for the transmission dynamics of cholera with control strategy. International Journal of Science and Technology,2(11), 212-217. (2013).

. Peter, O. J & Ibrahim, M. O. Application of Differential Transform Method in Solving a Typhoid Fever Model. International Journal of Mathematical analysis and Optimization.1(1),250-260.(2017). Retrieved from http://ijmao.unilag.edu.ng/article/view/12

Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. J. On the definition and computation of the basic reproduction ratio R_0 in models for infectious diseases in heterogeneous population. Journal of Mathematical Biology, 28, 365-382. (1990).

Lashari, A. A., Hattaf, K., Zaman, G., & Li, X-z. Backward bifurcation and optimal control of a vector borne disease. Applied Mathematics and Information Sciences,

Hu, Z. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete and Continuous Dynamical System, Series B,15(1), 93-112. (2011).

Wang, J. & Modnak, C. Modeling cholera dynamics with controls.Canadian Applied Mathematics Quarterly, 19(3),255-273. (2011).

.

Buonomo, B., & Lacitignola, D. On the backward bifurcation of a vaccination model with nonlinear incidence. Nonlinear Analysis: Modelling and Control, 16 (1), 30-46.(2011).

Kadaleka, S. Assessing the effects of nutrition and treatment in cholera dynamics: The case of Malawi. M. Sc. Dissertation, University of Dares Salaam. (2011).

Okosun, K. O. & Smith, R. Optimal control analysis of Malaria-Schistosomiasis co-infection dynamics. Mathematical Biosciences and Engineering,14(2), 337-405.(2017).