A semi-analytical approach for solving a fractional-order mathematical model of skin cell damage and repair driven by environmental pollutants

Environmental pollutants such as particulate matter and ozone generate persistent oxidative stress in the skin, promoting inflammation, premature aging, and impaired repair. Classical integer-order models fail to capture the cumulative and delayed nature of these responses. Here, we introduce a fractional-order nonlinear differential equation model that incorporates biological memory to simulate pollutant-induced skin damage and repair dynamics. Using the extended Laplace Decomposition Method, we obtained stable semi-analytical solutions across a wide range of fractional orders. Model simulations show that the fractional order α governs memory strength: higher values reproduce rapid, reversible responses typical of healthy skin, whereas lower values generate prolonged damage accumulation consistent with aged or chronically exposed skin. Two representative scenarios demonstrate how combined changes in α and the repair coefficient β differentiate resilient versus vulnerable phenotypes. The model also reveals a critical damage threshold beyond which injury accelerates, reflecting tipping-point behavior documented in environmental dermatology. This fractional framework provides a biologically grounded and mathematically flexible tool for analyzing cumulative pollutant stress, offering a foundation for future extensions incorporating spatial dynamics, stochasticity, and empirical validation using advanced skin models.