Abstract: A nonlinear tumor-immune interaction model is used to show the effect chemotherapy has on the qualitative aspects of cancer cell proliferation. The model uses three coupled ordinary differential equations for tumor cells (T), effector immune cells (E) and chemotherapy (C). Tumor growth is modeled using a logistic equation; immune mediated killing of tumor cells (via a saturated response); stimulation of immune activity via tumor antigens; immunological exhaustion (i.e., reduction in immune activity due to chronic antigen presentation); cytotoxic effects of chemotherapy directly on tumors and indirect effects on the immune system leading to suppression. The mathematical nature of this system is studied in terms of its fundamental characteristics (positivity, boundedness and existence of an invariant set which satisfies biological constraints) so as to establish that the system will be well posed. In addition to establishing the existence of tumor free and co-existence steady-states, a linearization about each steady-state using the Jacobian matrix and application of the Routh-Hurwitz criteria are used to examine the local stability of the steady-states. Using the chemotherapy input rate as a control parameter to induce bifurcations within the system it is demonstrated that this model can exhibit transcritical and Hopf bifurcations. These types of bifurcations are shown to explain transitions in tumor burden (persistent or eliminated) in relation to levels of chemotherapy use, transitions between an immune controlled state of co-existence, oscillatory behavior related to remission-relapse patterns and complete tumor eradication. Numerical simulations were performed to confirm the results obtained through analytical techniques and to determine a critical chemotherapy dose level at which persistent tumor burden changes to tumor elimination. Finally, sensitivity analyses were conducted to demonstrate that treatment efficacy was dependent upon several factors, including intensity of chemotherapy, efficiency of the immune response against tumors, aggressiveness of the tumor population and toxic effects of chemotherapy.

Keywords: Tumor–immune interaction, chemotherapy, bifurcation analysis, local stability, tumor-free equilibrium, coexistence equilibrium, Hopf bifurcation, transcritical bifurcation, mathematical oncology, numerical simulation.

Downloads: | DOI: 10.17148/IARJSET.2026.13465