94-12-099

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Abstract:

Immune Networks Modeled by Replicator Equations

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Peter F. Stadler, Peter Schuster, and Alan S. Perelson

In order to evaluate the role of idiotypic networks in the operation
of the immune system a number of mathematical models have been
formulated. Here we examine a class of B-cell models in which cell
proliferation is governed by a non-negative, unimodal, symmetric
response function *f(h)*, where the field *h* summarizes the
effect of the network on a single clone. We show that by transforming
into relative concentrations, the B-cell network equations can be
brought into a form that closely resembles the replicator equation.
We then show that when the total number of clones in a network is
conserved, the dynamics of the network can be represented by the
dynamics of a replicator equation. The number of equilibria and their
stability are then characterized using methods developed for the study
of second-order replicator equations. Analogies with standard
Lotka-Volterra equations are also indicated. A particularly
interesting result of our analysis is the fact that even though the
immune network equations are not second-order, the number and
stability of their equilibria can be obtained by a superposition of
second-order replicator systems. As a consequence, the problem of
finding all of the equilibrium points of the nonlinear network
equations can be reduced to solving linear equations.
**keywords:**
B-cells - immune system - mutation - replicator equations

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