Mathematical modeling and control of lung cancer with IL2 cytokine and anti-PD-L1 inhibitor effects for low immune individuals.

Mathematical formulations are crucial in understanding the dynamics of disease spread within a community. The aim of this work is to examine that the Lung Cancer detection and treatment by introducing IL2 and anti-PD-L1 inhibitor for low immune individuals. Mathematical model is developed with the created hypothesis to increase immune system by antibody cell's and Fractal-Fractional operator (FFO) is used to turn the model into a fractional order model. A newly developed system TCDIL2Z is examined both qualitatively and quantitatively in order to determine its stable position. The boundedness, positivity and uniqueness of the developed system are examined to ensure reliable bounded findings, which are essential properties of epidemic models. The global derivative is demonstrated to verify the positivity with linear growth and Lipschitz conditions are employed to identify the rate of effects in each sub-compartment. The system is investigated for global stability using Lyapunov first derivative functions to assess the overall impact of IL2 and anti-PD-L1 inhibitor for low immune individuals. Fractal fractional operator is used to derive reliable solution using Mittag-Leffler kernel. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease and fractional represents the fractional ordered derivative operator. We use combine operators to see real behavior of spread as well as control of lung cancer with different dimensions and continuous monitoring. Simulations are conducted to observe the symptomatic and asymptomatic effects of Lung Cancer disease to verify the relationship of IL2, anti-PD-L1 inhibitor and immune system. Also identify the real situation of the control for lung cancer disease after detection and treatment by introducing IL2 cytokine and anti-PD-L1 inhibitor which helps to generate anti-cancer cells of the patients. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies from our justified outcomes.

A general class of derivative free optimal root finding methods based on rational interpolation.

We construct a new general class of derivative free n-point iterative methods of optimal order of convergence 2 (n-1) using rational interpolant. The special cases of this class are obtained. These methods do not need Newton's iterate in the first step of their iterative schemes. Numerical computations are presented to show that the new methods are efficient and can be seen as better alternates.