# Modelling tumour-immunity interactions with different stimulation functions

Petar Zhivkov; Jacek Waniewski

International Journal of Applied Mathematics and Computer Science (2003)

- Volume: 13, Issue: 3, page 307-315
- ISSN: 1641-876X

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topZhivkov, Petar, and Waniewski, Jacek. "Modelling tumour-immunity interactions with different stimulation functions." International Journal of Applied Mathematics and Computer Science 13.3 (2003): 307-315. <http://eudml.org/doc/207645>.

@article{Zhivkov2003,

abstract = {Tumour immunotherapy is aimed at the stimulation of the otherwise inactive immune system to remove, or at least to restrict, the growth of the original tumour and its metastases. The tumour-immune system interactions involve the stimulation of the immune response by tumour antigens, but also the tumour induced death of lymphocytes. A system of two non-linear ordinary differential equations was used to describe the dynamic process of interaction between the immune system and the tumour. Three different types of stimulation functions were considered: (a) Lotka-Volterra interactions, (b) switching functions dependent on the tumour size in the Michaelis-Menten form, and (c) Michaelis-Menten switching functions dependent on the ratio of the tumour size to the immune capacity. The linear analysis of equilibrium points yielded several different types of asymptotic behaviour of the system: unrestricted tumour growth, elimination of tumour or stabilization of the tumour size if the initial tumour size is relatively small, otherwise unrestricted tumour growth, global stabilization of the tumour size, and global elimination of the tumour. Models with switching functions dependent on the tumour size and the tumour to the immune capacity ratio exhibited qualitatively similar asymptotic behaviour.},

author = {Zhivkov, Petar, Waniewski, Jacek},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {stability analysis; immunotherapy; critical points; ordinary differential equations},

language = {eng},

number = {3},

pages = {307-315},

title = {Modelling tumour-immunity interactions with different stimulation functions},

url = {http://eudml.org/doc/207645},

volume = {13},

year = {2003},

}

TY - JOUR

AU - Zhivkov, Petar

AU - Waniewski, Jacek

TI - Modelling tumour-immunity interactions with different stimulation functions

JO - International Journal of Applied Mathematics and Computer Science

PY - 2003

VL - 13

IS - 3

SP - 307

EP - 315

AB - Tumour immunotherapy is aimed at the stimulation of the otherwise inactive immune system to remove, or at least to restrict, the growth of the original tumour and its metastases. The tumour-immune system interactions involve the stimulation of the immune response by tumour antigens, but also the tumour induced death of lymphocytes. A system of two non-linear ordinary differential equations was used to describe the dynamic process of interaction between the immune system and the tumour. Three different types of stimulation functions were considered: (a) Lotka-Volterra interactions, (b) switching functions dependent on the tumour size in the Michaelis-Menten form, and (c) Michaelis-Menten switching functions dependent on the ratio of the tumour size to the immune capacity. The linear analysis of equilibrium points yielded several different types of asymptotic behaviour of the system: unrestricted tumour growth, elimination of tumour or stabilization of the tumour size if the initial tumour size is relatively small, otherwise unrestricted tumour growth, global stabilization of the tumour size, and global elimination of the tumour. Models with switching functions dependent on the tumour size and the tumour to the immune capacity ratio exhibited qualitatively similar asymptotic behaviour.

LA - eng

KW - stability analysis; immunotherapy; critical points; ordinary differential equations

UR - http://eudml.org/doc/207645

ER -

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