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 function

NDSolve, 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.

[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.