Fractional optimal control of an HIV/AIDS epidemic model with random testing and contact tracing

Abstract

In this paper, we present a fractional model for transmission dynamics of HIV/AIDS with random testing and contact tracing and incorporate into the model the control parameters of condom use and antiretroviral therapy (ART) aimed at controlling the spread of HIV/AIDS epidemic. The stability of equilibria of the model is discussed using the stability theorem and using the fractional La-Salle invariance principle for fractional differential equations. The effect of the fractional derivative order $$\alpha $$ ( $$ 0.6\le \alpha <1 $$ ) on the HIV/AIDS epidemic model is investigated. The numerical results show that the derivative order $$\alpha $$ can play the role of precautionary measures against infection transmission, treatment of infection and delay in accepting HIV test. We give a general formulation for a fractional optimal control problem, in which the state and co-state equations are given in terms of the left fractional derivatives. We develop the Forward–Backward sweep method using the Adams-type predictor–corrector method to solve the fractional optimal control of the model. The results show that implementing the ART treatment control of diagnosed HIV-infected people together with contact tracing results in a significant decrease in the number of undiagnosed HIV-infected population and AIDS people. Also, implementing all the control efforts decreases significantly the number of diagnosed and undiagnosed HIV-infected population and AIDS people. In addition, the values of the controls are reduced when the fractional derivative order $$\alpha $$ limits to 1 ( $$ 0< \alpha < 1$$ ).