Using perturbation theory to explain the existence of infected equilibrium point of immune-cervical cancer model

Abstract

Immunotherapy give a new hope for cervical cancer treatment. The Kirschner-Panetta model describe the interaction between effector cells, cancer cells, and interleukin-2(IL-2) with two immunotherapy, i.e. Adoptive Cellular Immunotherapy(ACI) and Cytokine therapy. The infected equilibrium point can give an idea of the cure level, but no one has discussed analytically. The function of cancer in steady state is a quintic polynomial that cannot be solved analytically. This study discusses the existence and bifurcation of the infected equilibrium point. Both, can explain the level of cure through analysis of the amount of cancer cells. We use the Singular Perturbation Method because the presents of the small parameter in the leading coefficient. The combination of Naive expansion and dominant balance technique are used. A consistent polynomial rescale is used to find the lost root due to Naive expansion solutions. The four infected equilibrium points get from the Naive and one from the dominant balance technique. ACI and cytokine therapy alone can reduce the cancer cells but with an imbalance of effector cells and IL-2. With both therapies, the cancer cells are close to zero which indicates a good level of cure. It is necessary to study further regarding the bifurcation causes other important parameters besides antigenicity.