Protection of children from vaccine preventable diseases, such as measles is among primary goal for health worker. Measles is a highly contagious disease that can spread in a population depending on the number of peoples susceptible or infected and also depending on their dynamics in the community. The model monitors the temporal dynamics of a childhood disease in the presence of preventive vaccine. We presented a nonlinear time fractional model of measles in order to understand the outbreaks of this epidemic disease. The Caputo fractional derivative operator of order ∈ (0,1] is employed to obtain the system of fractional differential equations. The numerical solution of the time fractional model has been procured by employing Laplace Adomian decomposition method (LADM), qualitative and sensitivity analysis of the model was performed. Qualitative results shows that the model has endemic equilibrium which locally asymptotically stable for 0 > 1 and otherwise unstable. The convergence analysis and nonnegative solutions are verified for the proposed scheme. Simulation of different epidemiological classes at the effect of fractional parameter revealed that most individuals undergoing treatment join the recovered class. This method proves to be very efficient techniques for solving epidemic model to control infectious disease.

Smoking is a large problem in the entire world. Despite overwhelming facts about smoking, it is still a very bad habit which is widely spread and accepted socially. Among smokers, often the desire to quit smoking arises. A large number of smokers attempt to quit, but only a few of them are successful. In this research, the nonstandard finite difference scheme is applied on system which is dynamically consistent, easy to implement and show a good agreement to control the bad impact of smoking for long period of time and to eradicate a death killer factor in the world spread by smoking. We have discussed the qualitative behavior of the model and numerical simulations are carried out to support the analytical results.