This paper investigates the global stability analysis of two-strain epidemic model with two general incidence rates. The problem is modelled by a system of six nonlinear ordinary differential equations describing the evolution of susceptible, exposed, infected and removed individuals. The wellposedness of the suggested model is established in terms of existence, positivity and boundedness of solutions. Four equilibrium points are given, namely the disease-free equilibrium, the endemic equilibrium with respect to strain 1, the endemic equilibrium with respect to strain 2, and the last endemic equilibrium with respect to both strains. By constructing suitable Lyapunov functional, the global stability of the disease-free equilibrium is proved depending on the basic reproduction number R 0. Furthermore, using other appropriate Lyapunov functionals, the global stability results of the endemic equilibria are established depending on the strain 1 reproduction number R 1 0 and the strain 2 reproduction number R 2 0. Numerical simulations are performed in order to confirm the different theoretical results. It was observed that the model with a generalized incidence functions encompasses a large number of models with classical incidence functions and it gives a significant wide view about the equilibria stability. Numerical comparison between the model results and COVID-19

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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 P , the occasional smokers L , the chain smokers S , the temporarily quit smokers Q T , and the permanently quit smokers Q P . 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.

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