Impact of Tobacco Smoking on the Prevalence of Tuberculosis Infection: A Mathematical Study

Abstract

Tobacco smoking is a social problem. It is a well-established fact that the smoking of tobacco products can cause severe health problems. It is also observed that the smokers are having higher risk of getting tuberculosis infection than the non-smokers. In this article a non-linear mathematical model is proposed and analyzed to demonstrate the impact of tobacco smoking on transmission dynamics of the tuberculosis. The basic reproduction number $R_0$ of the model is computed and the stabilities of different equilibria of the model are studied in-detail. The model system exhibits backward bifurcation for some specific set of parameters which suggests that mere reducing the basic reproduction number corresponding to TB below one is not enough to establish the stability of TB-free equilibrium point. So efforts are needed to reduce this basic reproduction number much below one to have TB-free equilibrium to be stable. It is also observed that the smoking habits can influence the stability of co-existence equilibrium and it can lead to persistence of TB in the population even though the basic reproduction number corresponding to TB is much less than one. Furthermore, this model is extended to the optimal control problem and the optimal control model is analyzed using the Pontryagin’s Maximum Principle and is solved numerically using MATLAB$^{*TM}$. Finally, numerical simulations are performed to analyze the effect of optimal control on the infected population. We observe that the optimal control model gives better result as compared to the model without optimal control as it reduces the number of infectives significantly within desired interval of time.