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

  • H. KheiriEmail author
  • M. Jafari
Original Research


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 (\( 0< \alpha < 1\)).


Fractional optimal control Diseases modeling Numerical simulation HIV/AIDS dynamics 

Mathematics Subject Classification

26A33 37B25 37M05 49K15 


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Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesUniversity of TabrizTabrizIran

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