On the Dynamics of Immune-Tumor Conjugates in a Four-Dimensional Tumor Model

We examine the ultimate dynamics of the four-dimensional model describing interactions between host cells, immune cells, tumor cells, and immune-tumor conjugate cells proposed by Abernethy and Gooding in 2018. In our paper, the ultimate upper bounds for all variables of this model are obtained. Formulas for positively invariant sets are deduced. Using these results, we establish conditions for the existence of the global attractor, derive formulas for its location, and present conditions under which immune and immune-tumor conjugate cells asymptotically die out. Next, we study equilibrium points, including the stability property for most of the equilibrium points. We discuss the existence of very low cancer-burden equilibrium points. Next, parametric conditions are derived under which the derivative of the density of the immune-tumor conjugate cell population eventually tends to zero; this mathematically rigorously confirms the correctness of the application of model reduction for this model in studies of its ultimate dynamics. In the final section, we summarize the results of this work and outline how to continue this study.