Stability, control and discretization for a smoking model

In this paper, the deterministic model of smoking consisting of five classes, has been qualitatively analysed. Explicit formula for the reproduction number has been obtained. Equilibria have been found and their global asymptotic stability has been discussed. Method of matrix theoretic, with the Perron eigenvectors, is used to get the global asymptotic stability of smoking-free equilibrium. It is shown that unique endemic equilibrium is globally asymptotically stable by using graph theoretic approach. To know the important factors, through which the disease spreads rapidly, sensitivity analysis of basic reproduction number and endemic level of smokers has been performed. This sensitivity analysis urged to modify the existing problem by inserting two controls namely prevention and treatment. Optimal control problem has been designed on the basis of sensitivity analysis. The existence of controls has been proved analytically and numerically it is shown that these applied controls significantly reduce the number of smokers. Moreover, a dynamically consistent nonstandard difference scheme is implemented to obtain a 5-dimensional discrete-time smoking model. Necessary and sufficient conditions are obtained for local dynamics of equilibria for discrete-time smoking model. Fourth-order implicit Runge–Kutta method is implemented to see the effectiveness of proposed nonstandard difference scheme. Numerical simulations have been done for the verification of the analytical results.