Numerical modelling and stability analysis of fractional smoking model

In this work, the effects and propagation of smoking in society are studied by considering the fractional tobacco smoking model. For this reason, the underlying model is investigated both analytically and numerically. The system has two equilibrium points, namely the tobacco-free and endemic equilibrium points. Furthermore, the stability of the model is observed by applying the Jacobian matrix technique. For numerical study, the non-standard finite difference scheme (NSFD) is hybridized with the Grunwald-Letnikov (GL) approximation for the Caputo differential operator. The key features of the continuous model are examined for the projected GL-NSFD scheme. The numerically simulated graphs are plotted to guarantee the positivity, boundedness, and convergence towards the exact steady states. Since the integer order epidemic model cannot accurately capture the nonlinear real phenomenon. Moreover, they cannot predict the future state exactly as the integer order derivatives involved in the models are local by nature, and they do not have the memory effect or history of the system. On the contrary, the fractional order model can capture all the necessary features of the continuous model. The proposed numerical method preserves the structure of the continuous system, for instance, the positivity, boundedness and convergence toward the exact steady states. It is worth mentioning that the projected numerical scheme is consistent with the continuous system.