Mathematical Analysis and Optimal Control of Giving up the Smoking Model

Abstract

In this study, we are going to explore mathematically the dynamics of giving up smoking behavior. For this purpose, we will perform a mathematical analysis of a smoking model and suggest some conditions to control this serious burden on public health. The model under consideration describes the interaction between the potential smokers $\left(P\right)$ , the occasional smokers $\left(L\right)$ , the chain smokers $\left(S\right)$ , the temporarily quit smokers $\left(Q_{\mathcal{T}}\right)$ , and the permanently quit smokers $\left(Q_{\mathcal{P}}\right)$ . Existence, positivity, and boundedness of the proposed problem solutions are proved. Local stability of the equilibria is established by using Routh–Hurwitz conditions. Moreover, the global stability of the same equilibria is fulfilled through using suitable Lyapunov functionals. In order to study the optimal control of our problem, we will take into account a two controls’ strategy. The first control will represent the government prohibition of smoking in public areas which reduces the contact between nonsmokers and smokers, while the second will symbolize the educational campaigns and the increase of cigarette cost which prevents occasional smokers from becoming chain smokers. The existence of the optimal control pair is discussed, and by using Pontryagin minimum principle, these two optimal controls are characterized. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to check the equilibria stability, confirm the theoretical findings, and show the role of optimal strategy in controlling the smoking severity.