Localization of compact invariant sets and global stability in analysis of one tumor growth model

In this paper, we study some features of global behavior of the four-dimensional superficial bladder cancer model with Bacillus Calmette-Guérin (BCG) immunotherapy described by Bunimovich-Mendrazitsky et al. in 2007 with the help of localization analysis of its compact invariant sets. Its dynamics is defined by the BCG treatment and by densities of three cells populations: effector cells, tumor infected cells by BCG, and tumor uninfected cells.We find upper bounds for ultimate dynamics of the whole state vector in the positive orthant and also under condition that there are no uninfected tumor cells. Further, we prove the existence of the bounded positively invariant domain in both of these two situations. Finally, by using these assertions, we derive our main result: sufficient conditions of global asymptotic stability of the tumor-free equilibrium point in the positive orthant.