Abstract
A two-dimensional tumor-immune model with the time delay of the adaptive
immune response is considered in this paper. The model accounts for the
interaction between cytotoxic T lymphocytes (CTLs) and cancer cells on
the surface of a solid tumor with a necrotic kernel. The system has
three equilibria. Both zero and maximum tumor volume equilibria are
unstable, while the behavior of the positive equilibriums is closely
related to the ratio of the immune killing rate to tumor volume growth
rate. The positive equilibrium is more likely to be locally
asymptotically stable when the ratio is smaller than a critical value,
and unstable otherwise. We also derive conditions to guarantee the
existence of Hopf bifurcation at the positive equilibrium. Applying the
center manifold reduction and normal form method, we obtain explicit
formulas to determine the properties of Hopf bifurcations. The global
continuation of local Hopf bifurcation is studied based on the
coincidence degree theory. The results reveal that long immune delay can
lead to oscillation dynamics. We also carry out detailed numerical
analysis for parameters to illustrate our qualitative analysis.
Numerically, we find that a shorter immune response time leads to a
longer patient survival time and the period and amplitude of a stable
periodic solution increase with the immune response time. When CTLs
recruitment rate and death rate vary, we observe how the ratio metioned
above and the first bifurcation value change numerically, which yields
further insights to the tumor-immune dynamics.