# Periodic dynamics in a model of immune system

Applicationes Mathematicae (2000)

- Volume: 27, Issue: 1, page 113-126
- ISSN: 1233-7234

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topBodnar, Marek, and Foryś, Urszula. "Periodic dynamics in a model of immune system." Applicationes Mathematicae 27.1 (2000): 113-126. <http://eudml.org/doc/219255>.

@article{Bodnar2000,

abstract = {The aim of this paper is to study periodic solutions of Marchuk's model, i.e. the system of ordinary differential equations with time delay describing the immune reactions. The Hopf bifurcation theorem is used to show the existence of a periodic solution for some values of the delay. Periodic dynamics caused by periodic immune reactivity or periodic initial data functions are compared. Autocorrelation functions are used to check the periodicity or quasiperiodicity of behaviour.},

author = {Bodnar, Marek, Foryś, Urszula},

journal = {Applicationes Mathematicae},

keywords = {autocorrelation function; antibody; antigen; immune system organ-target; Hopf bifurcation; plasma cell; periodicity; periodic solutions; Marchuk's model},

language = {eng},

number = {1},

pages = {113-126},

title = {Periodic dynamics in a model of immune system},

url = {http://eudml.org/doc/219255},

volume = {27},

year = {2000},

}

TY - JOUR

AU - Bodnar, Marek

AU - Foryś, Urszula

TI - Periodic dynamics in a model of immune system

JO - Applicationes Mathematicae

PY - 2000

VL - 27

IS - 1

SP - 113

EP - 126

AB - The aim of this paper is to study periodic solutions of Marchuk's model, i.e. the system of ordinary differential equations with time delay describing the immune reactions. The Hopf bifurcation theorem is used to show the existence of a periodic solution for some values of the delay. Periodic dynamics caused by periodic immune reactivity or periodic initial data functions are compared. Autocorrelation functions are used to check the periodicity or quasiperiodicity of behaviour.

LA - eng

KW - autocorrelation function; antibody; antigen; immune system organ-target; Hopf bifurcation; plasma cell; periodicity; periodic solutions; Marchuk's model

UR - http://eudml.org/doc/219255

ER -

## References

top- [1] A. Asachenkov, G. I. Marchuk, R. Mohler, and S. Zuev, Disease Dynamics, Birkhäuser, Boston, 1994. Zbl0833.92007
- [2] L. N. Belykh, Analysis of Mathematical Models in Immunology, Nauka, Moscow, 1988 (in Russian). Zbl0663.92003
- [3] M. Bodnar and U. Foryś, The model of immune system with time-dependent immune reactivity, preprint Warsaw University, RW 99-04 (52), 1999. Zbl1166.34042
- [4] M. Bodnar and U. Foryś, The model of immune system with time-dependent immune reactivity, in: Proc. Fourth National Conf. on Application of Mathematics in Biology and Medicine, Warszawa, 1998. Zbl1166.34042
- [5] M. Bodnar and U. Foryś, Behaviour of solutions of Marchuk's model depending on time delay, Internat. J. Appl. Math. Comput. Sci. 10 (2000), to appear. Zbl0947.92015
- [6] F. Bofill, R. Quentalia and W. Szlenk, The Marchuk's model in the case of vaccination. Qualitative behaviour and some applications, preprint, Politecnico de Barcelona, 1996.
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- [9] U. Foryś, Global analysis of Marchuk's model in a case of weak immune system, Math. Comp. Model. 25 (1995), 97-106. Zbl0919.92022
- [10] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Acad. Publ., 1992. Zbl0752.34039
- [11] A. V. Kim and V. G. Pimenov, Numerical Methods for Delay Differential Equations. Application of i-smooth Calculus, notes of lectures at the Seoul National Univ., 1999.
- [12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, London, 1993.
- [13] G. I. Marchuk, Mathematical Models in Immunology, Nauka, Moscow, 1980 (in Russian). Zbl0505.92006
- [14] G. I. Marchuk, Mathematical Models in Immunology, Optimization Software, New York, 1983.
- [15] G. I. Marchuk, Mathematical Modelling of Immune Response in Infectious Diseases, Kluwer Acad. Publ., 1997. Zbl0876.92015
- [16] H. G. Schuster, Deterministic Chaos. An Introduction, VCH Verlagsgesellschaft, Weinheim, 1988. Zbl0707.58003
- [17] W. Szlenk and C. Vargas, Some remarks on Marchuk's mathematical model of immune system, preprint CINVESTAV Mexico, 1995.

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