A Stochastic TB Model for a Crowded Environment

Hindawi 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 Sibaliwe Maku Vyambwera and Peter Witbooi 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; 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. 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. 1. Introduction Tuberculosis (TB) continues to be a major global health problem that is responsible for 1.5 million deaths worldwide each year [1]. TB is most prevalent in communities with socioeconomical problems but is not confined to such. The authors in [2, 3] associate TB infection with poverty and underdevelopment of some countries. It has been observed globally that one of the major factors driving TB infection is overcrowding. TB mostly occurs in poorest countries that are not developed and particularly where a population is overcrowded and in countries that are influenced by war. Conflict is the most common cause of large population displacement, which often results in relocation to temporary settlements such as camps. Factors including malnutrition and overcrowding in camp settings further increase the exposure to TB infection in these populations. Following up on a paper of Ssematimba et al. [3] regarding internally displaced people’s camps in Uganda, Buonomo and Lacitignola [2] proposed a model that considers the dynamics of TB in concentration camps with a case study in Uganda. Another type of crowded environment which provides favourable conditions for TB to flourish is prisons and more so if the prison is full beyond its capacity. There are more than 10 million inmates in prisons all over the world. The United States of America is in the top rank with about 2.2 million inmates while South Africa is in rank 11 [4]. South African prison has approximately 160000 inmates in custody, of which 120000 are sentenced individuals while the rest are awaiting trial. This means that a large number of inmates are kept in remand population and some of them might not be found guilty at the end of the process, after having been exposed to high risk of TB infection. Mathematical models have been used to model TB by considering the size of the area and how size and density affect the extent to which TB can invade a certain population [2, 3, 5–7]. Quite obviously, considering the manner in which TB is aerially transmitted from one person to another, the prison situation provides favourable conditions for TB to flourish. TB is an infectious disease caused by bacillus Mycobacterium tuberculosis that most often affects the lungs (pulmonary TB) and can affect other parts as well such as brain, kidneys, and spine (extrapulmonary TB) [8, 9]. The TB infection can take place when an infected individual releases some droplet nuclei which can remain airborne in any indoor area for up to four hours. The tubercle bacillus can persist in a dark area for several hours but it is exceptionally sensitive to sunshine. The risk of infection increases as the length of prison stay increases and the sentenced offenders are more likely to get TB infection as compared to the awaiting trial inmates. Against this background the paper [10] offers a model for the population dynamics of TB in a prison or prison system. 2 In particular, it computes the parameters relevant to South Africa for the given model, using publicly available data. The current paper considers a stochastic form of the model in [10]. It is well understood that stochastic differential equation (sde) attempts to reflect the effect of random disturbances in or on a system. A second reason for studying sde models is that it is good to know that a given model carries some resilience against small disturbances. In this case we consider the transmission parameters to be stochastically perturbed, similarly to [11]. Stochastic pertubation has been studied by Yang and Mao [12]; they considered a multigroup SEIR epidemic model. In most cases, it has been observed in [12, 13] that introducing a stochastic perturbation into an unstable disease-free equilibrium model system of ordinary differential equation may lead to a system being stable in sde. Stochastic differential equation models for various diseases have been studied and similar work has been done in [11, 12, 14–16]. Our paper focuses on the analysis of TB in prisons as prisons have been recognized as institutions with very high TB burden as compared to a general population [17]. For a deterministic model of similar type, in [10] we computed parameter values pertaining to South Africa. For the stochastic model in this paper the focus is on mathematical analysis. In Section 2, the model is introduced, based on the paper of Buonomo and Lacitignola [2]. The existence and uniqueness of the solution to the stochastic models is investigated by using the Lyapunov method in Section 3. Stability of the disease-free equilibrium for stochastic models is shown in Section 4. We show our results by means of numerical simulations and conclude in Section 5. 2. The Model We introduce a stochastic compartmental model which is based on the deterministic model in the paper of Buonomo and Lacitignola [2]. We divide the population, which is of size 𝑁(𝑡) at time 𝑡, into four compartments, namely, the class 𝑆(𝑡) of susceptible individuals 𝑆(𝑡), the class 𝐸(𝑡) of individuals infected with TB who are not infectious, the class 𝐼(𝑡) of individuals infected with active TB who are infectious, and the class 𝑇(𝑡) of individuals under treatment. It is important to note that in general populations removal of individuals out of the system is only by death. In this model, as in [10], the removal is by death or by discha (...truncated)