Periodic dynamics in a model of immune system

Marek Bodnar; Urszula Foryś

Applicationes Mathematicae (2000)

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

Abstract

top
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.

How to cite

top

Bodnar, 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. [1] A. Asachenkov, G. I. Marchuk, R. Mohler, and S. Zuev, Disease Dynamics, Birkhäuser, Boston, 1994. Zbl0833.92007
  2. [2] L. N. Belykh, Analysis of Mathematical Models in Immunology, Nauka, Moscow, 1988 (in Russian). Zbl0663.92003
  3. [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. [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. [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. [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. 
  7. [7] J. Hale, Theory of Functional Differential Equations, Springer, New York, 1977. Zbl0352.34001
  8. [8] U. Foryś, Global analysis of Marchuk's model in case of strong immune system, J. Biol. Sys., to appear. 
  9. [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. [10] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Acad. Publ., 1992. Zbl0752.34039
  11. [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. [12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, London, 1993. 
  13. [13] G. I. Marchuk, Mathematical Models in Immunology, Nauka, Moscow, 1980 (in Russian). Zbl0505.92006
  14. [14] G. I. Marchuk, Mathematical Models in Immunology, Optimization Software, New York, 1983. 
  15. [15] G. I. Marchuk, Mathematical Modelling of Immune Response in Infectious Diseases, Kluwer Acad. Publ., 1997. Zbl0876.92015
  16. [16] H. G. Schuster, Deterministic Chaos. An Introduction, VCH Verlagsgesellschaft, Weinheim, 1988. Zbl0707.58003
  17. [17] W. Szlenk and C. Vargas, Some remarks on Marchuk's mathematical model of immune system, preprint CINVESTAV Mexico, 1995. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.