# Optimal Strategies of the Psoriasis Treatment by Suppressing the Interaction Between T-Lymphocytes and Dendritic Cells

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## Abstract

This report contains the results devoted to the study of a mathematical model of psoriasis, proposed by P. K. Roy. This model is formulated as a Cauchy problem for the system of three nonlinear differential equations that describe the relationships between the concentrations of T-lymphocytes, keratinocytes and dendritic cells, the interactions of which cause the occurrence of psoriasis. Moreover, in this model we include a bounded scalar control responsible for a dose of a medication that suppresses the interaction of T-lymphocytes and dendritic cells. On the given time interval, for the control mathematical model, a problem of minimizing the concentration of keratinocytes at the end of time interval is considered. To analyze this problem, the Pontryagin maximum principle is applied. The adjoint system and the maximum condition for the optimal control are written. Using the corresponding system of differential equations, the switching function describing the behavior of the optimal control is studied. Such a system of equations allows us to determine the type of the optimal control: whether this control is only of a bang-bang type, or, in addition to the portions of a bang-bang type, it also contains a singular arc. When a singular arc occurs, the report discusses its order, the fulfillment of the necessary optimality condition for it, and its concatenation with portions of a bang-bang type. The obtained analytical results are illustrated by numerical calculations. The corresponding conclusions are made.

## Keywords

Psoriasis Nonlinear system Optimal control Pontryagin maximum principle Switching function Singular arc## Notes

### Acknowledgements

E. N. Khailov was funded by Russian Foundation for Basic Research according to the research project 18-51-45003 \(\mathrm{IND\_a}\).

## References

- 1.Bonnans, F., Martinon, P., Giorgi, D., Grélard, V., Maindrault, S., Tissot, O., Liu, J.: BOCOP 2.0.5—User Guide. http://bocop.org. Accessed 8 Feb 2017
- 2.Datta, A., Roy, P.K.: T-cell proliferation on immunopathogenic mechanism of psoriasis: a control based theoretical approach. Control Cybern.
**2013**, 23–42 (2013)MathSciNetzbMATHGoogle Scholar - 3.Datta, A., Li, X.-Z., Roy, P.K.: Drug therapy between T-cells and DCS reduces the excess production of keratinicytes: ausal effect of psoriasis. Math. Sci. Int. Res. J.
**3**, 452–456 (2014)Google Scholar - 4.Datta, A., Kesh, D.K., Roy, P.K.: Effect of \(CD4^{+}\) T-cells and \(CD8^{+}\) T-cells on psoriasis: a mathematical study. Imhotep Math. Proc.
**3**, 1–11 (2016)Google Scholar - 5.Grigorieva, E., Khailov, E., Deignan, P.: Optimal treatment strategies for control model of psoriasis. In: Proceedings of the SIAM Conference on Control and its Applications (CT17), Pittsburgh, Pennsylvania, USA, pp. 86–93, 10–12 July 2017CrossRefGoogle Scholar
- 6.Gudjonsson, J.E., Johnston, A., Sigmundsdottir, H., Valdimarsson, H.: Immunopathogenic mechanisms in psoriasis. Rev. Clin. Exp. Immunol.
**135**, 1–8 (2004)CrossRefGoogle Scholar - 7.Lee, E.B., Marcus, L.: Foundations of Optimal Control Theory. Wiley, New York (1967)Google Scholar
- 8.Lowes, M.A., Suárez-Fariñas, M., Krueger, J.G.: Immunology of psoriasis. Annu. Rev. Immunol.
**32**, 227–255 (2014)CrossRefGoogle Scholar - 9.Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: Mathematical Theory of Optimal Processes. Wiley, New York (1962)zbMATHGoogle Scholar
- 10.Roy, P.K., Bradra, J., Chattopadhyay, B.: Mathematical modeling on immunopathogenesis in chronic plaque of psoriasis: a theoretical study. Lect. Notes Eng. Comput. Sci.
**1**, 550–555 (2010)Google Scholar - 11.Roy, P.K., Datta, A.: Impact of cytokine release in psoriasis: a control based mathematical approach. J. Nonlinear Evol. Equ. Appl.
**2013**, 23–42 (2013)MathSciNetzbMATHGoogle Scholar - 12.Schättler, H., Ledzewicz, U.: Geometric Optimal Control. Theory, Methods and Examples. Springer, New York, Heidelberg, Dordrecht, London (2012)CrossRefGoogle Scholar
- 13.Schättler, H., Ledzewicz, U.: Optimal Control for Mathematical Models of Cancer Therapies. An Application of Geometric Methods. Springer, New York, Heidelberg, Dordrecht, London (2015)CrossRefGoogle Scholar