9853

Mathematical Model of the Immune Response

This Demonstration shows a basic mathematical model of the immune response.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

A basic mathematical model of the immune response [1] is described by a system of two ordinary differential equations:
,
,
where represents the target, which may be any biological material subject to an immune response (bacteria in this case); is the elimination capacity of the immune system, here represented by cells; and are the rates of reproduction and destruction of the target; is the rate of cell death; and is time.
describes the formation of cells due to the presence of bacteria, and depicts an autocatalytic increase in cells. For the purposes of this Demonstration we use
and .
These equations correspond to a sigmoid shape for these functions and emphasize that the immune system may ignore very low bacterial concentrations, and that a critical number of immune cells may be necessary to obtain an autocatalytic effect. The constants and represent precursor cell pool sizes. These equations are solved using Mathematica's built-in functionNDSolve, and the results are presented in plots of and versus time and in the plane. At , no specialized cells are present and .
In response to an initial dose of bacteria, the active cells increase and converge toward the immune state where no bacteria, but only memory cells, are present. In a secondary infection (the dotted lines), the immune system responds faster; conversely, the immune system can be overwhelmed by bacteria that have a high reproduction or low destruction rate.
Reference
[1] H. Mayer, K. S. Zaenker, and U. an der Heiden, "A Basic Mathematical Model of the Immune Response," Chaos, Solitrons, and Fractals, 5(1), 1995 pp. 155–161. doi: 10.1063/1.166098.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+