93-07-048

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Abstract:

Immune Networks Modeled by Replicator Equations

## 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 on a single
clone. 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 equations. 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-Voterra 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.

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