Math. Model. Nat. Phenom.
Volume 2, Number 1, 2007Epidemiology
|Page(s)||44 - 61|
|Published online||15 June 2008|
Hopf Bifurcation Analysis of Pathogen-Immune Interaction Dynamics With Delay Kernel
Department of Economic Informatics, Mathematics and Statistics, Faculty of Economics
West University of Timişoara, str. Pestalozzi, nr. 16A, 300115, Timişoara, Romania
2 Department of Applied Mathematics, Faculty of Mathematics, West University of Timişoara Bd. V. Parvan, nr. 4, 300223, Timişoara, Romania
3 Department of Biophysics and Medical Informatics, University of Medicine and Pharmacy Piata Eftimie Murgu, nr. 3, 300041, Timişoara, Romania
Corresponding author: firstname.lastname@example.org
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.
Mathematics Subject Classification: 34C23 / 34C25 / 37G05 / 37G15 / 92D30
Key words: delay differential equation / stability / Hopf bifurcation / pathogen-immune interaction
© EDP Sciences, 2007
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