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Computational and Mathematical Methods in Medicine
Volume 2014, Article ID 912306, 20 pages
http://dx.doi.org/10.1155/2014/912306
Research Article

Dynamics of Mycobacterium and bovine tuberculosis in a Human-Buffalo Population

1Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
2Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2

Received 4 May 2014; Accepted 20 June 2014; Published 2 September 2014

Academic Editor: Issam El Naqa

Copyright © 2014 A. S. Hassan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new model for the transmission dynamics of Mycobacterium tuberculosis and bovine tuberculosis in a community, consisting of humans and African buffalos, is presented. The buffalo-only component of the model exhibits the phenomenon of backward bifurcation, which arises due to the reinfection of exposed and recovered buffalos, when the associated reproduction number is less than unity. This model has a unique endemic equilibrium, which is globally asymptotically stable for a special case, when the reproduction number exceeds unity. Uncertainty and sensitivity analyses, using data relevant to the dynamics of the two diseases in the Kruger National Park, show that the distribution of the associated reproduction number is less than unity (hence, the diseases would not persist in the community). Crucial parameters that influence the dynamics of the two diseases are also identified. Both the buffalo-only and the buffalo-human model exhibit the same qualitative dynamics with respect to the local and global asymptotic stability of their respective disease-free equilibrium, as well as with respect to the backward bifurcation phenomenon. Numerical simulations of the buffalo-human model show that the cumulative number of Mycobacterium tuberculosis cases in humans (buffalos) decreases with increasing number of bovine tuberculosis infections in humans (buffalo).