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Journal of Applied Mathematics
Volume 2018, Article ID 3420528, 8 pages
https://doi.org/10.1155/2018/3420528
Research Article

A Stochastic TB Model for a Crowded Environment

Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa

Correspondence should be addressed to Peter Witbooi; az.ca.cwu@ioobtiwp

Received 19 February 2018; Accepted 12 May 2018; Published 13 June 2018

Academic Editor: Zhidong Teng

Copyright © 2018 Sibaliwe Maku Vyambwera and Peter Witbooi. 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

We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation model, and we impose stochastic perturbation. We prove the existence and uniqueness of positive solutions of a stochastic model. We introduce an invariant generalizing the basic reproduction number and prove the stability of the disease-free equilibrium when it is below unity or slightly higher than unity and the perturbation is small. Our main theorem implies that the stochastic perturbation enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to illustrate the analytical findings and the utility of the model.