A control-based mathematical study on psoriasis dynamics with special emphasis on IL−21 and IFN−γ interaction network

Psoriasis is characterized by the excessive growth of keratinocytes (skin cells), which is initiated by chaotic signaling within the immune system and irregular release of cytokines. Pro-inflammatory cytokines: Interleukin 21 ((Formula presented.)) and Interferon gamma ((Formula presented.)), released by (Formula presented.) cell and activated natural killer cells (NK cells) respectively, play central role in the disease pathogenesis. In this work, we have constructed two sets of nonlinear differential equations. One is representing the growth of three vital immune cells (T helper cells (type I and II) and activated NK cells) along with keratinocyte and the other set is for cytokines' ((Formula presented.) and (Formula presented.)) dynamics. The hazardous effects of these cytokines, preconditions for disease persistence and validation of the stability criteria of endemic equilibrium have been studied analytically. We have also observed the effect of the combined biologic therapy (anti (Formula presented.) and (Formula presented.) inhibitor) by considering an optimal control problem. Analytical and numerical results reveal that the impact of activated NK cells on excessive formation of keratinocytes is mostly regulated by the effects of (Formula presented.) and (Formula presented.).