Utilizing and investigating the hybrid fractional operators to analyze lung cancer treatment and precautions

Compartmental models, in particular, are essential to epidemiology because they provide a basic mathematical framework for comprehending such systems and can be used to anticipate the evolution of diseases, illustrate epidemic consequences and facilitate public health interventions. For low-immune people, a mathematical model is created utilizing IL₂ and anti-PD-L₁ inhibitors, suggesting that boosting the immune system with antibody cells can enhance it. Using a hybrid fractional-order derivative, the model is further transformed into a fractional-order system. The purpose of this work is to analyze fractional models to have a better understanding of how the order of the fractional derivative influences the spread of lung cancer. A new system, TCDIL2Z, is examined for stability, boundedness, positivity and uniqueness. The system's global stability is also investigated using Lyapunov's first derivative functions. Fractional-order differential equations are solved using the Laplace Adomian Decomposition method, and tables and graphs are included to improve the precision of the numerical findings. Simulations are conducted to identify control situations after detection and treatment. According to the study, the fractional-order lung cancer model can better visualize the dynamics of the disease since it shows a memory effect, in contrast to the classical model. Additionally, it shows that fractional-order derivations have greater reliability than classical order in the explanation of bodily approaches. This research will aid in understanding the spread of the disease and developing control strategies based on justified outcomes.