Optimal control problem in correlation between smoking and epidemic of respiratory diseases

Smoking appears to be a risk factor that may increase the number of different pulmonary infections. This link is likely to be mediated by smoking adverse effects on the respiratory defenses. A mathematical model to describe correlation between the number of smokers and its effect on the number of infected people suffer from respiratory disease like influenza is constructed in this paper. Promotion of healthy life is accounted in the model as an optimal control problem to reduce the number of smokers. In this work, the transition rates from smokers to non-smokers and from non-smokers to smokers are regarded as the control variables. Assuming the control variables are constant, equilibrium points of the model can be obtained analytically. The basic reproductive ratio as the endemic threshold is taken from the spectral radius of the next-generation matrix. Using numerical simulation, we show that the healthy life promotion can reduce the number of infected person significantly by reducing the number of smokers. Furthermore, different initial conditions to show different situations in the field are also simulated. It is shown that a large effort to increase the transition rate from smokers to non-smokers and to reduce the transition from non-smokers to smokers should be applied in the endemic reduction scenario.