  Chimeric Antigen Receptor T (CAR-T) cell therapy has been proven to be
successful against different leukaemias and lymphomas. This paper makes an
analytical and numerical study of a mathematical model describing the
competition of CAR-T, leukaemias tumor and B cells. Considering its
significance in sustaining anti-CD19 CAR T-cell stimulation, we integrate a
B-cell source term into the model. Through stability and bifurcation analyses,
we reveal the potential for tumor eradication contingent on the continuous
influx of B-cells, uncovering a transcritical bifurcation at a critical B-cell
input. Additionally, we identify an almost heteroclinic cycle between
equilibrium points, providing a theoretical basis for understanding disease
relapse. Analyzing the oscillatory behavior of the system, we approximate the
time-dependent dynamics of CAR T-cells and leukemic cells, shedding light on
the impact of initial tumor burden on therapeutic outcomes. In conclusion, our
study provides insights into CAR T-cell therapy dynamics for acute
lymphoblastic leukemias, offering a theoretical foundation for clinical
observations and suggesting avenues for future immunotherapy modeling research.
