Dynamics aspects and bifurcations of a tumor-immune system interaction under stationary immunotherapy

Abstract

We consider a three-dimensional mathematical model that describes the interaction between the effector cells, tumor cells, and the cytokine (IL-2) of a patient. This is called the Kirschner–Panetta model. Our objective is to explain the tumor oscillations in tumor sizes as well as long-term tumor relapse. We then explore the effects of adoptive cellular immunotherapy on the model and describe under what circumstances the tumor can be eliminated or can remain over time but in a controlled manner. Nonlinear dynamics of immunogenic tumors are given, for example: we prove that the trajectories of the associated system are bounded and defined for all positive time; there are some invariant subsets; there are open subsets of parameters, such that the system in the first octant has at most five equilibrium solutions, one of them is tumor-free and the others are of co-existence. We are able to prove the existence of transcritical and pitchfork bifurcations from the tumor-free equilibrium point. Fixing an equilibrium and introducing a small perturbation, we are able to show the existence of a Hopf periodic orbit, showing a cyclic behavior among the population, with a strong dominance of the parental anomalous growth cell population. The previous information reveals the effects of the parameters. In our study, we observe that our mathematical model exhibits a very rich dynamic behavior and the parameter $\tilde{\mu }$ (death rate of the effector cells) and ${\tilde{p}}_1$ (production rate of the effector cell stimulated by the cytokine IL-2) plays an important role. More precisely, in our approach the inequality ${\tilde{\mu }}_2>{\tilde{p}}_1$ is very important, that is, the death rate of the effector cells is greater than the production rate of the effector cell stimulated by the cytokine IL-2. Finally, medical implications and a set of numerical simulations supporting the mathematical results are also presented.