Abstract
In this research work, a delayed giving up smoking model is initially formulated. The model is obtained by dividing the whole population into measurable partitions of four subclasses as well as taking into account harmonic mean type of incidence rate. For the existence of local stability and Hopf bifurcation, some sufficient conditions are derived by incorporating time delay as a bifurcation parameter. Directly afterward, direction and stability of the Hopf bifurcation are investigated. Moreover, using the optimal control strategy in the form of legislation, we propose optimal strategies to lower down the number of smokers. An optimal control solution is obtained for the control problem. In order to characterize the optimal controls, Pontryagin’s maximum principle with time delay is used as the main tool.
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