Stability and bifurcation analysis of delay induced tumor immune interaction model

A modified mathematical model of Kuznetsov et al. (1994) representing tumor-immune interaction with discrete time delay is proposed in this paper. By choosing the discrete time delay as the bifurcation parameter, we establish the sufficient condition for local stability of interior steady state, Hopf bifurcation occurs when the time delay passes a critical value. The estimation of the length of delay to preserve stability has been derived. Moreover, based on the normal form theory and center manifold theorem, we establish explicit expressions to determine the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are obtained to validate our analytical findings by varying the system parameter. A two parameters bifurcation diagrams are also shown by varying the system parameters, which bring out the rich biological dynamics of the system.