A study on fractional tumor-immune interaction model related to lung cancer via generalized Laguerre polynomials

Abstract

Background Cancer, a complex and deadly health concern today, is characterized by forming potentially malignant tumors or cancer cells. The dynamic interaction between these cells and their environment is crucial to the disease. Mathematical models can enhance our understanding of these interactions, helping us predict disease progression and treatment strategies. Methods In this study, we develop a fractional tumor-immune interaction model specifically for lung cancer (FTIIM-LC). We present some definitions and significant results related to the Caputo operator. We employ the generalized Laguerre polynomials (GLPs) method to find the optimal solution for the FTIIM-LC model. We then conduct a numerical simulation and compare the results of our method with other techniques and real-world data. Results We propose a FTIIM-LC model in this paper. The approximate solution for the proposed model is derived using a series of expansions in a new set of polynomials, the GLPs. To streamline the process, we integrate Lagrange multipliers, GLPs, and operational matrices of fractional and ordinary derivatives. We conduct a numerical simulation to study the effects of varying fractional orders and achieve the expected theoretical results. Conclusion The findings of this study demonstrate that the optimization methods used can effectively predict and analyze complex phenomena. This innovative approach can also be applied to other nonlinear differential equations, such as the fractional Klein–Gordon equation, fractional diffusion-wave equation, breast cancer model, and fractional optimal control problems.

Generalization of Bernoulli polynomials to find optimal solution of fractional hematopoietic stem cells model

“…This section is dedicated to elucidating the fundamental concepts that are employed throughout the paper. (see [30][31][32] and references therein) Consider X(t) as a differentiable function, and let ζ represent the derivative order. The Caputo fractional derivative is defined as follows:…”

“…Their method leveraged generalized Lerch polynomials. Hassani et al [30] devised a fractional tumor-immune interaction model tailored for lung cancer and utilized the generalized Laguerre polynomials method to identify the optimal solution. Ullah and Baleanu [31] proposed a SICA model in a Caputo-type fractional framework for HIV/AIDS transmission dynamics, studying solution nonnegativity, equilibrium points, and stability.…”