# Hopf Bifurcation Analysis of Pathogen-Immune Interaction Dynamics With Delay Kernel

• Volume: 2, Issue: 1, page 44-61
• ISSN: 0973-5348

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## Abstract

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The aim of this paper is to study the steady states of the mathematical models with delay kernels which describe pathogen-immune dynamics of infectious diseases. In the study of mathematical models of infectious diseases it is important to predict whether the infection disappears or the pathogens persist. The delay kernel is described by the memory function that reflects the influence of the past density of pathogen in the blood and it is given by a nonnegative bounded and normated function k defined on [ 0, ∞ ). By using the coefficient of the kernel k, as a bifurcationparameter, the models are found to undergo a sequence of Hopf bifurcation. The direction and the stability criteria of bifurcation periodic solutions are obtained by applying the normal form theory and the center manifold theorems. Some numerical simulation examples for justifying the theoretical results are also given.

## How to cite

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Neamţu, M., et al. "Hopf Bifurcation Analysis of Pathogen-Immune Interaction Dynamics With Delay Kernel." Mathematical Modelling of Natural Phenomena 2.1 (2010): 44-61. <http://eudml.org/doc/222214>.

@article{Neamţu2010,
abstract = { The aim of this paper is to study the steady states of the mathematical models with delay kernels which describe pathogen-immune dynamics of infectious diseases. In the study of mathematical models of infectious diseases it is important to predict whether the infection disappears or the pathogens persist. The delay kernel is described by the memory function that reflects the influence of the past density of pathogen in the blood and it is given by a nonnegative bounded and normated function k defined on [ 0, ∞ ). By using the coefficient of the kernel k, as a bifurcationparameter, the models are found to undergo a sequence of Hopf bifurcation. The direction and the stability criteria of bifurcation periodic solutions are obtained by applying the normal form theory and the center manifold theorems. Some numerical simulation examples for justifying the theoretical results are also given. },
author = {Neamţu, M., Buliga, L., Horhat, F. R., Opriş, D.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {delay differential equation; stability; Hopf bifurcation; pathogen-immune interaction},
language = {eng},
month = {3},
number = {1},
pages = {44-61},
publisher = {EDP Sciences},
title = {Hopf Bifurcation Analysis of Pathogen-Immune Interaction Dynamics With Delay Kernel},
url = {http://eudml.org/doc/222214},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Neamţu, M.
AU - Buliga, L.
AU - Horhat, F. R.
AU - Opriş, D.
TI - Hopf Bifurcation Analysis of Pathogen-Immune Interaction Dynamics With Delay Kernel
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/3//
PB - EDP Sciences
VL - 2
IS - 1
SP - 44
EP - 61
AB - The aim of this paper is to study the steady states of the mathematical models with delay kernels which describe pathogen-immune dynamics of infectious diseases. In the study of mathematical models of infectious diseases it is important to predict whether the infection disappears or the pathogens persist. The delay kernel is described by the memory function that reflects the influence of the past density of pathogen in the blood and it is given by a nonnegative bounded and normated function k defined on [ 0, ∞ ). By using the coefficient of the kernel k, as a bifurcationparameter, the models are found to undergo a sequence of Hopf bifurcation. The direction and the stability criteria of bifurcation periodic solutions are obtained by applying the normal form theory and the center manifold theorems. Some numerical simulation examples for justifying the theoretical results are also given.
LA - eng
KW - delay differential equation; stability; Hopf bifurcation; pathogen-immune interaction
UR - http://eudml.org/doc/222214
ER -

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