The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication?
In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introd...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2018-09-01
|
| Series: | Mathematical Biosciences and Engineering |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2018057 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832590066563678208 |
|---|---|
| author | Alexander P. Krishchenko Konstantin E. Starkov |
| author_facet | Alexander P. Krishchenko Konstantin E. Starkov |
| author_sort | Alexander P. Krishchenko |
| collection | DOAJ |
| description | In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate upper bounds for all variables of this model and two lower bounds and, besides, study when dynamics of this model possesses a global attracting set. The nonexistence conditions of compact invariant sets are derived. We obtain bounds for treatment parameters $s_{1, 2}$ under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. Conditions imposed on under which the tumor population persists are presented as well. Finally, we compare tumor eradication/ persistence bounds and discuss our results. |
| format | Article |
| id | doaj-art-942b01720e06413394584d4b7451b63e |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2018-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-942b01720e06413394584d4b7451b63e2025-01-24T02:41:02ZengAIMS PressMathematical Biosciences and Engineering1551-00182018-09-011551243125410.3934/mbe.2018057The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication?Alexander P. Krishchenko0Konstantin E. Starkov1Bauman Moscow State Technical University, 2-aya Baumanskaya ul., 5, Moscow 105005, RussiaInstituto Politecnico Nacional, CITEDI, Avenida IPN N 1310, Nueva Tijuana, Tijuana 22510, B.C., MexicoIn this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate upper bounds for all variables of this model and two lower bounds and, besides, study when dynamics of this model possesses a global attracting set. The nonexistence conditions of compact invariant sets are derived. We obtain bounds for treatment parameters $s_{1, 2}$ under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. Conditions imposed on under which the tumor population persists are presented as well. Finally, we compare tumor eradication/ persistence bounds and discuss our results.https://www.aimspress.com/article/doi/10.3934/mbe.2018057kirschner-panetta tumor modelultimate dynamicsglobal stabilitycompact invariant settumor eradicationtumor persistence |
| spellingShingle | Alexander P. Krishchenko Konstantin E. Starkov The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication? Mathematical Biosciences and Engineering kirschner-panetta tumor model ultimate dynamics global stability compact invariant set tumor eradication tumor persistence |
| title | The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication? |
| title_full | The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication? |
| title_fullStr | The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication? |
| title_full_unstemmed | The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication? |
| title_short | The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication? |
| title_sort | four dimensional kirschner panetta type cancer model how to obtain tumor eradication |
| topic | kirschner-panetta tumor model ultimate dynamics global stability compact invariant set tumor eradication tumor persistence |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2018057 |
| work_keys_str_mv | AT alexanderpkrishchenko thefourdimensionalkirschnerpanettatypecancermodelhowtoobtaintumoreradication AT konstantinestarkov thefourdimensionalkirschnerpanettatypecancermodelhowtoobtaintumoreradication AT alexanderpkrishchenko fourdimensionalkirschnerpanettatypecancermodelhowtoobtaintumoreradication AT konstantinestarkov fourdimensionalkirschnerpanettatypecancermodelhowtoobtaintumoreradication |