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

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