Ultimate tumor dynamics and eradication using oncolytic virotherapy

•Convergence dynamics of a cancer model with virotherapy is studied.•Ultimate sizes of interacting cell populations are computed.•Tumor eradication conditions are derived.•Upper bounds of virus replication rate that destroys the tumor are got. In this paper we study ultimate dynamics of one three-dimensional model for tumor growth under oncolytic virotherapy which describes interactions between cytotoxic T-cells, uninfected tumor cells and infected tumor cells. Using the localization theorem of compact invariant sets we derive ultimate upper bounds for all cell populations and establish the property of the existence of the attracting set. Next, we find several conditions under which our system demonstrates convergence dynamics to equilibrium points located in invariant planes corresponding cases of the absence of uninfected or infected tumor cells. These assertions mean global eradication of uninfected or infected tumor cell populations and are presented as algebraic inequalities respecting virus replication rate θ. In particular, we find in Theorems 4 and 5the following curious phenomenon. Namely, when we vary θ from the instability range of the infected tumor free equilibrium point to the stability range we obtain in the latter range convergence dynamics to one of tumor free equilibrium points; this means that the local eradication of infected tumor cells implies their global eradication. Besides, we give conditions under which the infected tumor cell population persists. Our theoretical studies are supplied by results of numerical simulation.