Mathematical Model of the Immune Response

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.