Hybrid Mathematical-PINN Model for Tumor-Immune Interaction with Holling Type-II Functional Response and Optimal Control

Code accompanying the article “Hybrid Mathematical-PINN Model for Tumor-Immune Interaction with Holling Type-II Functional Response and Optimal Control”. The package reproduces the dynamical-systems results for the three-variable tumor, hunting-cell and resting-cell model with Holling Type-II responses and tumor-induced immune suppression (baseline parameters of Table 2, μd = 0.05). It contains the RK4 reference solution; the exact analysis of the interior equilibria, including the quintic coexistence condition, the saddle-node bifurcation near w2 = 0.4385 and a count of admissible equilibria over the parameter ranges studied; the Sotomayor conditions for the transcritical bifurcation of the hunting-cell-free equilibrium; the Hopf point and first Lyapunov coefficient; and the regenerated bifurcation diagram (Figure 4) with its limit-cycle envelope. A verification folder holds independent re-implementations of a forward–backward sweep for the Pontryagin optimal-control problem and of a state-only physics-informed neural network, with analytic gradients checked against finite differences. [Scripts used for Figures 1–3 and 5–9 and Table 6 are in the folder.] This version updates the earlier deposit to match the revised article. The code is written in Python (NumPy, SciPy, SymPy, mpmath, Matplotlib); the README gives the run order, expected outputs and run times.