Asymptotic behavior of Tuberculosis between smokers and non-smokers

Abstract

Tuberculosis (TB) is an ailment caused by Mycobacterium tuberculosis. So, by the continuous spread of TB, this study presents a new mathematical model of TB that investigates the impact of smoking and contact rate on the transmission dynamics of the disease in a population. The basic reproduction number R 0 , of the model, is computed by employing the next generation matrix approach and the dynamical behavior of the model is explored in detail. Mathematical analysis reveals that the disease-free equilibrium solution is globally asymptotically stable when the associated basic reproduction number is less than unity. It is further shown that the model has a unique endemic equilibrium point, which is proved to exist when R 0 > 1 . The global asymptotic stability of the unique endemic equilibrium, when the associated basic reproduction number exceeds unity, is investigated through numerical simulations. Sensitivity analysis is carried out to identify key parameters that have the greatest influence on the transmission dynamics of TB in the population. The sensitivity results show that the top four parameters of the model, that have the most influence on R 0 of the model are the recruitment rate of smokers into the population, contact rate, progression rate to latent TB stage, and recovery rate of infectious individuals, with other key parameters influencing the outcome of the other output responses. Numerical experiments are performed to support the analytical findings. Numerical simulation of the model shows that, if the contact rate α = 0 . 001 , then R 0 is estimated to be 0.4611, which shows that TB dies out of the population. It further shows that, if the contact rate α = 0 . 005 , R 0 is estimated to be 2.3058, which indicates that TB establishes itself in the population. We observed from the above results that the smoking habit should be discouraged in society to reduce the prevalence of TB in the population.