Open AccessDynamics of the tumor-immune-virus interactions: Convergence conditions to tumor-only or tumor-free equilibrium points
Author(s) -
Konstantin E. Starkov,
Giovana Andres Garfias
Publication year - 2019
Publication title -
mathematical biosciences and engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.451
H-Index - 45
eISSN - 1551-0018
pISSN - 1547-1063
DOI - 10.3934/mbe.2019020
Subject(s) - invariant (physics) , mathematics , convergence (economics) , equilibrium point , invariance principle , mathematical analysis , mathematical physics , differential equation , linguistics , philosophy , economic growth , economics
In the present paper convergence dynamics of one tumor-immune-virus model is examined with help of the localization method of compact invariant sets and the LaSalle theorem. This model was elaborated by Eftimie et al. in 2016. It is shown that this model possesses the Lagrange stability property of positive half-trajectories and ultimate upper bounds for compact invariant sets are obtained. Conditions of convergence dynamics are found. It is explored the case when any trajectory is attracted to one of tumor-only equilibrium points or tumor-free equilibrium points. Further, it is studied ultimate dynamics of one modication of Eftimie et al. model in which the immune cells injection is included. This modied system possesses the global tumor cells eradication property if the inux rate of immune cells exceeds some value which is estimated. Main results are expressed in terms simple algebraic inequalities imposed on model and treatment parameters.