Bifurcation analysis of a discrete-time tumor-immune system

The tumor-immune interaction plays a key role in tumor growth and its progression. Tumor is a group of diseases in which some of the human body cells start unchecked, uncontrolled and abnormal proliferation or division. Now a days, tumor-immune interaction is a great topic of interest from last few decades because immune cells interact in different ways, such as by releasing cytokines that activate other immune cells, directly killing the tumor cells, absorbing and presenting tumor antigen. The antitumor activity of helper T lymphocytes in providing help in generation and maintenance of CD8+ cytotoixc T cells and memory T cells are necessary for tumor control. So, in this work, we explore dynamics and bifurcation analysis of the discrete-time tumor-immune system in the interior of R_+^3. More precisely, it is proved that the discrete tumor-immune system has tumor-free equilibrium solution, tumor-dominant equilibrium solution and immune-control equilibrium solution under certain restrictions to the involved parameters. Then by linear stability theory, local dynamics with different topological classifications are investigated about tumor-free equilibrium solution, tumor-dominant equilibrium solution and immune-control equilibrium solution of the discrete tumor-immune system. Further, for the discrete tumor-immune system, existence of periodic points and convergence rate are also investigated. It is also investigated that the existence of possible bifurcations about tumor-free equilibrium solution and immune-control equilibrium solution, and proved that there exists no flip bifurcation about tumor-free equilibrium solution. Moreover, it is proved that about immune-control equilibrium solution there exist hopf and flip bifurcations, and we have studied these bifurcations by utilizing explicit criterion. Finally, numerically verified theoretical results.