Modeling of psoriasis by considering drug influence: A mathematical approach with memory trace

Psoriasis is an immune-mediated genetic disease, characterized by its manifestation on the skin, the joints, or both. In this paper, our primary aim is to increase awareness about the intricate nature of this multifaceted condition, highlight the potential of therapeutic approaches, and examine the factors affecting the future course of psoriasis. This paper introduces a mathematical model for psoriasis, formulated by fractional order differential equations (FODE) with Caputo sense. The model also includes the drug effect of immune-boosting drugs on psoriasis. It has been shown that the solution of the proposed system exists and is unique, and its positivity and boundedness have been proven. Additionally, the local stability and global stability of the co-existing equilibrium point have been investigated. Numerical solutions have been conducted using the Adams Bashforth PECE method to analyze the influence of fractional order derivatives (FODs) and distinct parameters on population dynamics. The graphics have been acquired using the L1 scheme, incorporating a memory trace (MT) mechanism capable of comprehensively capturing and amalgamating historical system dynamics to visualize the memory trace in detail. One of the results deduced from this paper is that the MT disappears when α equals 1. Upon decreasing the fractional order α from 1, the MT experiences a non-linear augmentation starting from zero. This observed MT emphasizes the distinction between derivatives of fractional and integer orders. Within the proposed model, our findings suggest that introducing the immune booster drug efficiently controls psoriasis. Furthermore, when appropriate clinical patient data are available, the proposed results can be used for a specific psoriasis patient. Our study suggests that treatment with the drug may be a new insight into psoriasis treatment and may be proposed as a treatment policy for future clinical trials.