Bifurcation analysis on immunotherapy of a tumor model without treatment

Immunotherapy is one of the most methodologies developed recently for reducing and alleviating the dangerous growth of tumors. In this paper, we explore bifurcation analysis on a tumor mathematical model including CD4$ ^+ $ T cells and cytokines. The case without treatment is studied in order to estimate the limitation of a successful therapy. Particular attention is focused on the existence of equilibrium solutions and their stability. Bifurcation diagrams are used to consider the inter-individual variability for a non-treatment situation, showing the existence of oscillating behaviors due to Hopf bifurcation. It is shown that patients with very low tumor antigenicity experience tumors expanding to its maximum size, regardless of whether the tumor destruction rate and cytokine production rate are high. Numerical simulations are present to illustrate the theoretical predictions.