Post-Treatment Stability Analysis of a Tumor–Immune Model for Non-Small-Cell Lung Cancer under Surgery and Chemotherapy

In this work, we study the post-treatment dynamics of non-small-cell lung cancer through a nonlinear tumor–immune interaction model incorporating the residual effects of surgery and chemotherapy. Tumor cells, cytotoxic CD8\(^+\) T lymphocytes, and dendritic cells are described by a system of ordinary differential equations in which therapeutic interventions are embedded as persistent biological effects acting on both tumor and immune populations. This modeling approach focuses on long-term outcomes following treatment rather than the treatment process itself. The model is shown to be well posed through positivity and boundedness of solutions, ensuring biological feasibility. Three biologically meaningful equilibrium states are identified, corresponding to tumor eradication, immune failure with tumor persistence, and controlled chronic disease. Explicit local stability conditions for these equilibria are derived using linearization and the Routh–Hurwitz criteria, revealing the roles of tumor growth, immune effectiveness, and treatment-induced immune toxicity in shaping post-treatment outcomes. Numerical illustrations support the analytical results and demonstrate robust convergence toward the predicted equilibrium states. A sensitivity analysis highlights the trade-off between chemotherapy-induced tumor killing and immune suppression, emphasizing the importance of immune preservation for favorable long-term tumor control. Overall, the proposed framework provides a mathematically rigorous and biologically interpretable description of post-treatment cancer dynamics and offers qualitative insight into remission, relapse, and disease control following therapy.