Dynamics of the tumor-immune-virus interactions: Convergence conditions to tumor-only or tumor-free equilibrium points

Konstantin E. Starkov; Giovana Andres Garfias; Konstantin E. Starkov; Giovana Andres Garfias

doi:10.3934/mbe.2019020

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 1: 421-437. doi: 10.3934/mbe.2019020

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Research article

Dynamics of the tumor-immune-virus interactions: Convergence conditions to tumor-only or tumor-free equilibrium points

Instituto Politecnico Nacional-CITEDI,Avenida IPN N 1310, Nueva Tijuana,Tijuana 22510, B.C., Mexico.

Received: 17 April 2018 Accepted: 05 September 2018 Published: 17 December 2018

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 modification of Eftimie et al. model in which the immune cells injection is included. This modified system possesses the global tumor cells eradication property if the influx 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.
- tumor model,
- convergence dynamics,
- localization,
- compact invariant set,
- tumor-free equilibrium point,
- tumor-only equilibrium point
Citation: Konstantin E. Starkov, Giovana Andres Garfias. Dynamics of the tumor-immune-virus interactions: Convergence conditions to tumor-only or tumor-free equilibrium points[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 421-437. doi: 10.3934/mbe.2019020

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Abstract

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 modification of Eftimie et al. model in which the immune cells injection is included. This modified system possesses the global tumor cells eradication property if the influx 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.

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