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 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.
B-cells - immune system - mutation - replicator equations
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