In this paper,we present a general formulation for a fractional optimal control problem (FOCP),in which the state and co-state equations are given in terms of the left fractional derivatives.We develop the forward-backward sweep method (FBSM)using the Adamstype predictor-corrector method to solve the FOCP.We present a fractional model for transmission dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS)with treatment and incorporate three control efforts (effective use of condoms,ART treatment and behavioral change control)into the model aimed at controlling the spread of HIV/AIDS epidemic.The necessary conditions for fractional optimal control of the disease are derived and analyzed.The numerical results show that implementing all the control efforts increases the life time and the quality of life those living with HIV and decreases significantly the number of HIV-infected and AIDS people.Also,the maximum levels of the controls and the value of objective functional decrease when the derivative order a limits to 1(0.7≤a <1).In addition,the effect of the fractional derivative order a (0.7≤a <1)on the spread of HIV/AIDS epidemic and the treatment of HIV-infected population is investigated.The results show that the derivative order a can play the role of using ART treatment in the model.
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