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The aim of this work is to design a stochastic framework to solve the fractional-order differential model based on the breast cancer progression during the immune-chemotherapeutic treatment phase, including certain control parameters such as anti-cancer medications, ketogenic diet and immune boosters. The developed model considers tumor density progression throughout chemotherapy treatment, as well as an immune response during normal cell–tumor cell interaction. This study’s subject seems to be to demonstrate the implications and significance of the fractional-order breast cancer mathematical model. The goal of these studies is to improve accuracy in the breast cancer model by employing fractional derivatives. This study also includes an integer, nonlinear mathematical system with immune-chemotherapeutic treatment impacts. The mathematical system divides the fractional-order breast cancer mathematical model among four manifestations: normal cell population (N), tumor cells (T), immune response class (I), and estrogen compartment (E), i.e., (NTIE). The fractional-order NTIE mathematical system is still not published previously, nor has it ever been addressed employing the stochastic solvers’ strength. To solve a fractional-order NTIE mathematical system, stochastic solvers based on the Levenberg–Marquardt backpropagation scheme (LMBS) and neural networks (NNs), namely, LMBNNs, are been constructed. To solve the fractional-order NTIE mathematical model, three cases with varying values for this same fractional order have been supplied. The statistics used to offer the numerical solutions of the fractional-order NTIE mathematical model are divided as follows: 75% in training, 15% in testing, and 10% in the authorization. The acquired numerical findings were compared using the reference solutions to determine the accuracy of the LMBNNs using Adams–Bashforth–Moulton. The numerical performances employing error histograms (EHs), state transitions (STs), regression, correlation, including mean square error (MSE) have been further supplied to authenticate overall capability, competence, validity, consistency, as well as exactness of such LMBNNs.

fractal and fractional

Numerical Investigations of the Fractional-Order Mathematical Model Underlying Immune-Chemotherapeutic Treatment for Breast Cancer Using the Neural Networks