LAUSR

In the present work, we have proposed a new mathematical model of alcohol abuse with delay, saturated incidence function and logistic recruitment. The model is made up of the following four population classes: occasional drinkers, heavy drinkers, drinkers during treatment and drinkers who are temporarily recovered. In particular, we incorporate time delay because the non consumer population will take a period of time to become an alcohol consumer. We have studied the optimal control problem by considering a saturated control function and an objective function of type $$L^{1}$$ . The delay is incorporated in our model to make it more realistic and to describe the latency period. The existence of the optimal control is also proved. Pontryagin’s maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation.