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

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.