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Chaos Induced by Delay in Model of the Immune Response

This Demonstration shows the effect of time delay in a basic model of the immune system.
The model consists of an ordinary differential equation coupled with a delay differential equation [1]:

with initial history functions and . Here represents the immune competence measured by the concentration of certain immune cells, for example cytotoxic lymphocytes or killer cells, represents the target population, bacteria or viruses, is time, and is a delay time necessary for the immune system to respond to the presence of targets; the parameters of the model are defined in [1] and are set here at . Increasing the delay time induces sustained oscillations and damped oscillations; larger delay times lead to chaotic oscillations similar to those seen in some data of the immune system [2].

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References
[1] E. de Souza, M. Lyra, and I. Gleria, "Critical Behavior of the Delay-Induced Chaos Transition in a Nonlinear Model of the Immune Response," Brazilian Journal of Physics, 39(2A), 2009 pp. 431–434. doi:10.1590/S0103-97332009000400015.
[2] H. Mayer, K. S. Zaenker, and U. an der Heiden, "A Basic Mathematical Model of the Immune Response," Chaos, 5(1), 1995 pp. 155–161. doi:10.1063/1.166098.
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