Mathematical Model of the Immune Response
This Demonstration shows a basic mathematical model of the immune response.
A basic mathematical model of the immune response  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
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.
 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.