Theory in Biosciences

, Volume 135, Issue 4, pp 217–230 | Cite as

Optimal control therapy and vaccination for special HIV-1 model with delay

  • Elham Shamsara
  • Jamal Shamsara
  • Zahra AfsharnezhadEmail author
Original Paper


In this paper, we consider a four dimensional model of the human immunodeficiency virus-1 (HIV-1) with delay, which is an extension of some three dimensional models. We approach the treatment problem by adding two controllers to the system for inhibiting viral production. The optimal controller \(u_{1}\) is considered for vaccine and \(u_{2}\) for the drug. The Pontryagin maximum principle with delay is used to characterize these optimal controls. At the end, numerical results are presented to illustrate the optimal solutions. The validity of the model was confirmed by proper semi-quantitative simulation of some clinical data. The model was used to predict the possible beneficial effects of vaccine and anti-retroviral drug administration in HIV-1 disease.


CTL response HIV-1 infection Pontryagin maximum principle with delay Optimal control 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Elham Shamsara
    • 1
  • Jamal Shamsara
    • 2
  • Zahra Afsharnezhad
    • 1
    Email author
  1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesFerdowsi University of Mashhad (FUM)MashhadIran
  2. 2.Pharmaceutical Research Center, School of PharmacyMashhad University of Medical SciencesMashhadIran

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