93-07-048
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.
Return to 1993 working papers list.