Tuberculosis (TB) associates with HIV/AIDS at any stage of the total infection period. In recent years, HIV and TB have become the leading causes of death due to infectious and long lasting disease. On the basis of the characteristics of disease transmission of HIV and TB, we formulate a mathematical model. The model consists of ten compartments with nonlinear ordinary differential equations. The model is locally asymptotically stable in case of disease free equilibrium point when basic reproduction number is less than unity and locally asymptotically stable in case of endemic equilibrium point when the basic reproduction number is greater than unity. We apply optimal control theory to the co-infection model in the form of Pontryagin’s Maximum Principle by introducing the treatment control and the vaccination control to decrease the transmission of the disease burden. Finally, we perform the numerical simulations of the optimal control model to observe the dynamics of the system in support of analytical findings. J. of Sci. and Tech. Res. 3(1): 103-114, 2021

Bangladesh is one of the largest tobacco users in the world being troubled by smoking related issues. In this paper we consider a compartmental mathematical model of smoking in which the population is divided into five compartments: susceptible, expose, smokers, temporary quitters and permanent quitters described by ordinary differential equations. We study by including the conversion rate from light smoker to permanent quit smokers. The basic reproduction number R0 has been derived and then we found two euilibria of the model one of them is smoking-free and other of them is smoking-present. We establish the positivity, boundedness of the solutions and perform stability analysis of the model. To decrease the smoking propensity in Bangladesh we perform numerical simulation for various estimations of parameters which offer understanding to give up smoking and how they influence the smoker and exposed class. This model gives us legitimate thought regarding the explanations for the spread of smoking in Bangladesh.