Stem Cell Based Fractional-Order Dynamical Model of Psoriasis: A Mathematical Study - Kushary, Subhankar; Ghosh, Tushar; Makinde, Oluwole Daniel; Li, Xue-Zhi; Roy, Priti Kumar (Jambura Journal of Biomathematics (JJBM), 2025)
Psoriasis is a chronic  autoimmune skin  disorder driven by  dysregulated immune responses,  where  abnormal interactions between  T  cells  and  dendritic cells  lead to  excessive   inflammatory  cytokine  production.    This triggers the hyper-proliferation of  epidermal keratinocytes while  depleting mesenchymal stem  cells  (MSCs), which play  a crucial role in immune modulation. The progression behavior of psoriasis is not only  influenced by their present  state but also by  the historical evolution of underlying  cellular interactions. Memory stages  and complex interplay  among   immune  components at  different   temporal   scales  significantly  modulate disease expression. Motivated by this, we proposed a mathematical model  of psoriasis to a fractional-order framework in  order  to  incorporate  memory-dependent  effects  and  non-local  characteristics.   This   article  deals  with   a four-dimensional  model  of  psoriasis involving  concentrations of  T  cells,  dendritic cells,  keratinocytes, and mesenchymal stem  cells  (MSCs) in  order  to predict  the  temporal   evolution in  the  considered cell  densities during the  disease  dissemination process.     Using  Caputo, Caputo-Fabrizio, and  Atangana-Baleanu-Caputo operators,   we  analyze  how   memory  influences disease   dynamics.    In-depth  mathematical analysis  of  the solution of  the  fractionalized  model   has  been  thoroughly  investigated.   The  stability of  the  model   is  also examined using generalized Ulam–Hyers stability criteria.  The considered population densities  are numerically evaluated using  various  fractional orders  with   considered  fractional  operators  to  capture  non-local effects. Optimal control  is  implemented on  the  fractionalized system  using the  Forward-Backward Sweep  Method (FBSM), emphasizing the impacts  of two biologics, namely TNF-α inhibitors and IL-23 blockers,  via  considered operators.  Numerical simulations are performed in support of the theoretical analyses, accompanied by detailed discussions from  both mathematical and  biological viewpoints.  Results based on optimal control  effectiveness analysis indicate  that a combined control  strategy,  particularly under  the Caputo-Fabrizio operator,  optimally reduces  keratinocyte density.  Which offers  deeper  insights into  disease  progression and  effective  therapeutic approaches.