Optimal Control Problems for a Mathematical Model of the Treatment of Psoriasis
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.
Keywordspsoriasis nonlinear controlled system optimal control Pontryagin maximum principle indicator function
Unable to display preview. Download preview PDF.
- 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
- 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
- 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
- 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
- 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.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.E. B. Lee and L. Marcus, Foundations of Optimal Control Theory [Russian translation], Nauka, Moscow (1972).Google Scholar
- 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
- 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].