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Predator-Prey Equations Simulating an Immune Response

The immune response to a replicating antigen may be viewed as a problem of interacting populations of antigens and antibodies. This Demonstration analyzes a simple mathematical model of the immune response: antigen plays the role of prey and antibody the role of predator.

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The model is governed by two differential equations [1]:
,
,
where and denote the concentrations of antigen and antibody, is time, is the antigen replication rate, is the antibody decay rate, is an equilibrium constant, is the rate of antigen elimination, is the rate of antibody production, and is the maximum antibody concentration the host is able to produce.
In this Demonstration, equations 1a, 1b, and 4 of [1] are solved with , , , , and ; the solution is presented in plots of antigen and antibody concentrations versus time and in a antigen-antibody phase plot.
If , periodic oscillations of the antigen and antibody are found; if , both populations execute damped oscillations and approach a steady state, and if , both populations execute oscillations of increasing amplitude. During such oscillations in a real biological system, presumably either the antigen would be eliminated during an antigen minimum or the host would be eliminated during an antigen maximum.
Reference
[1] G. I. Bell, "Predator-Prey Equations Simulating an Immune Response," Mathematical Biosciences, 16(3–4),1973 pp. 291–314. doi:10.1016/0025-5564(73)90036-9.
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