Optimal Control Problems for a Mathematical Model of the Treatment of Psoriasis

  • N. L. GrigorenkoEmail author
  • É. V. Grigorieva
  • P. K. Roi
  • E. N. Khailov

We consider a mathematical model of the treatment of psoriasis on a finite time interval. The model consists of three nonlinear differential equations describing the interrelationships between the concentrations of T-lymphocytes, keratinocytes, and dendritic cells. The model incorporates two bounded timedependent control functions, one describing the suppression of the interaction between T-lymphocytes and keratinocytes and the other the suppression of the interaction between T-lymphocytes and dendritic cells by medication. For this model, we minimize the weighted sum of the total keratinocyte concentration and the total cost of treatment. This weighted sum is expressed as an integral over the sum of the squared controls. Pontryagin’s maximum principle is applied to find the properties of the optimal controls in this problem. The specific controls are determined for various parameter values in the BOCOP-2.0.5 program environment. The numerical results are discussed.


psoriasis nonlinear controlled system optimal control Pontryagin maximum principle indicator function 


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • N. L. Grigorenko
    • 1
    Email author
  • É. V. Grigorieva
    • 1
  • P. K. Roi
    • 2
  • E. N. Khailov
    • 1
  1. 1.Faculty of Computational Mathematics and CyberneticsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Centre for Mathematical Biology and Ecology, Department of MathematicsJadavpur UniversityKolkataIndia

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