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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
II. MATHEMATICAL MODELING
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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.

Keywords

psoriasis nonlinear controlled system optimal control Pontryagin maximum principle indicator function 

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References

  1. 1.
    F. O. Nestle, D. H. Kaplan, and J. Barker, “Psoriasis,” New Engl. J. Med., 361, No. 5, 496–509 (2009).CrossRefGoogle Scholar
  2. 2.
    A. B. Kimball, C. Jacobson, S. Weiss, M. G. Vreeland, and Y. Wu, “The psychosocial burden of psoriasis,” Am. J. Clin. Dermatol., 6, No. 6, 383–392 (2005).CrossRefGoogle Scholar
  3. 3.
    S. L. Mehlis and K. B. Gordon, “The immunology of psoriasis and biologic immunotherapy,” J. Am. Acad. Dermatol., 49, No. 2, 44–50 (2003).CrossRefGoogle Scholar
  4. 4.
    J. E. Gudjonsson, A. Johnston, H. Sigmundsdottir, and H. Valdimarsson, “Immunopathogenic mechanisms in psoriasis,” Clin. Exp. Immunol., 135, No. 1, 1–8 (2004).CrossRefGoogle Scholar
  5. 5.
    A. A. Kubanova, A. A. Kubanov, J. F. Nicolas, L. Puig, J. Prinz, O. R. Katunina, and L. F. Znamenskaya, “Immune mechanisms in psoriasis: New biotherapy strategies,” Vestn. Dermatol. Venerol., 1, 35–47 (2010).Google Scholar
  6. 6.
    M. A. Lowes, M. Suarez-Farinas, and J. G. Krueger, “Immunology of psoriasis,” Ann. Rev. Immunol., 32, 227–255 (2014).CrossRefGoogle Scholar
  7. 7.
    H. B. Oza, R. Pandey, D. Roper, Y. Al-Nuaimi, S. K. Spurgeon, and M. Goodfellow, “Modelling and finite-time stability analysis of psoriasis pathogenesis,” Int. J. Control, 90, No. 8, 1664–1677 (2017).MathSciNetCrossRefGoogle Scholar
  8. 8.
    H. Zhang, W. Hou, L. Henrot, S. Schnebert, M. Dumas, C. Heusele, and J. Yang, “Modelling epidermis homoeostasis and psoriasis pathogenesis,” Journal of Royal Society Interface, 12, 1–22 (2015).CrossRefGoogle Scholar
  9. 9.
    N. J. Savill, “Mathematical models of hierarchically structured cell populations under equilibrium with application to the epidermis,” Cell Proliferat., 36, No. 1, 1–26 (2003).CrossRefGoogle Scholar
  10. 10.
    G. Niels and N. Karsten, “Simulating psoriasis by altering transit amplifying cells,” Bioinformatics, 23, No. 11, 1309–1312 (2007).CrossRefGoogle Scholar
  11. 11.
    M. V. Laptev and N. K. Nikulin, “Numerical modeling of mutual synchronization of auto-oscillations of epidermal proliferative activity in lesions of psoriasis skin,” Biophysics, 54, 519–524 (2009).CrossRefGoogle Scholar
  12. 12.
    N. V. Valeyev, C. Hundhausen, Y. Umezawa, N. V. Kotov, G. Williams, A. Clop, C. Ainali, G. Ouzounis, S. Tsoka, F. O. Nestle, “A systems model for immune cell interactions unravels the mechanism of inflammation in human skin,” PLoS Comput. Biology, 6, No. e10011024, 1–22 (2010).Google Scholar
  13. 13.
    A. Gandolfi, M. Iannelli, and G. Marinoschi, “An age-structured model of epidermis growth,” J. Math. Biol., 62, No. 1, 111–141 (2011).MathSciNetCrossRefGoogle Scholar
  14. 14.
    B. Chattopadhyay and N. Hui, “Immunopathogenesis in psoriasis through a density-type mathematical model,” WSEAS Trans. on Math., 11, 440–450 (2012).Google Scholar
  15. 15.
    P. K. Roy and A. Datta, “Negative feedback control may regulate cytokines effect during growth of keratinocytes in the chronic plaque of psoriasis: a mathematical study,” Int. J. Appl. Math., 25, No. 2, 233–254 (2012).MathSciNetzbMATHGoogle Scholar
  16. 16.
    X. Cao, A. Datta, F. Al Basir, and P. K. Roy, “Fractional-order model of the disease psoriasis: a control based mathematical approach,” J. Syst. Sci. Complex., 29, 1565–1584 (2016).MathSciNetCrossRefGoogle Scholar
  17. 17.
    A. Datta and P. K. Roy, “T-cell proliferation on immunopathogenic mechanism of psoriasis: a control based theoretical approach,” Control Cybern., 42, No. 3, 365–386 (2013).MathSciNetzbMATHGoogle Scholar
  18. 18.
    P. K. Roy and A. Datta, “Impact of cytokine release in psoriasis: a control based mathematical approach,” J. Non. Evolution Equat. and Appl., 2013, No. 3, 23–42 (2013).MathSciNetzbMATHGoogle Scholar
  19. 19.
    A. Datta, X.-Z. Li, and P. K. Roy, “Drug therapy between T-cells and DCs reduces the excess production of keratinocytes: causal effect of psoriasis,” Math. Sci. Intern. Res. J., 3, No. 1, 452–456 (2014).Google Scholar
  20. 20.
    E. Grigorieva and E. Khailov, “Optimal strategies for psoriasis treatment,” MDPI Math. and Comp. Analysis, 23, 1–30 (2018).MathSciNetGoogle Scholar
  21. 21.
    E. Grigorieva and E. Khailov, “Chattering and its approximation in control of psoriasis treatment,” Discrete Cont. Dyn.-B, 24, No. 5, 2251–2280 (2019).MathSciNetzbMATHGoogle Scholar
  22. 22.
    P. K. Roy, J. Bhadra, and B. Chattopadhyay, “Mathematical modeling on immunopathogenesis in chronic plaque of psoriasis: a theoretical study,” Lecture Notes in Eng. and Comp. Sci., 1, 550–555 (2010).Google Scholar
  23. 23.
    A. Datta, D. K. Kesh, and P. K. Roy, “Effect of CD4+T-cells and CD8+T-cells on psoriasis: a mathematical study, “Imhotep Math. Proc., 3, No. 1, 1–11 (2016).Google Scholar
  24. 24.
    E. B. Lee and L. Marcus, Foundations of Optimal Control Theory [Russian translation], Nauka, Moscow (1972).Google Scholar
  25. 25.
    L. S. Pontry;agin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1961).Google Scholar
  26. 26.
    H. Schattler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies. An Application of Geometric Methods, Springer, New York (2015).CrossRefGoogle Scholar
  27. 27.
    F. Bonnans, P. Martinon, D. Giorgi, V. Grelard, S. Maindrault, O. Tissot, and J. Liu, BOCOP 2.0.5 – User Guide (February 8, 2017) [http://bocop.org].

Copyright information

© 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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