Investigation of singular regimens in a controlled model of psoriasis treatment

Abstract

Psoriasis is characterized by an overgrowth of keratinocytes (skin cells) resulted from chaotic signaling in the immune system and irregular release of cytokines. Anti-inflammatory cytokines, such as IL−21 and IFN −γ released by the T h 1 -cells and activated killer cells (NK-cells) play a central role in pathogenesis of this disease. So, in this paper, we propose two systems of nonlinear differential equations: one system describes the growth of immune cells (T -helper cells of type I and II, as well as activated NK-cells) along with keratinocytes; another system sets the dynamics of cytokines (IL − 21 and IFN − γ). Since these systems use different time scales, we transform them into one system of differential equations, including the immune T h 1 -and T h 2 -cells, activated NKcells, and epidermal keratinocytes. Within this description of the dynamics of psoriasis, we study the effect of combined bio-therapy, including the action of the IL − 21 inhibitor together with anti-IFN − γ therapy. To do this, we introduce two bounded control functions into the system and formulate on a given time interval, the problem of minimizing the total cost of the applied immune therapy and its impact on the proliferation of activated NK-cells and keratinocytes at the end of the treatment time. Analysis of such a problem is carried out using the Pontryagin maximum principle. As a result of this analysis, the properties of optimal controls and their possible types are established. It is shown that each such control is either a bang-bang function over the entire time interval, or in addition to non-singular bang-bang sections, it can have a singular regimen. Possible types of singular regimens are studied; for them, the necessary optimality conditions are checked, the singular regimens's formulas are found, so as the ways the singular regimens concatenate with non-singular (bang-bang) sections. Some numerically computed optimal controls are given alone with a discussion of the numerical difficulties in detection of singular regimens.