Immune networks modeled by replicator equations.
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.
Stadler
P F
PF
Institut für Theoretische Chemie, Universität Wien, Austria.
Schuster
P
P
Perelson
A S
AS
D000818:Animals; D001402:B-Lymphocytes / Q000276:immunology*; D007130:Immunoglobulin Idiotypes / Q000276:immunology; D007156:Immunologic Memory; D008213:Lymphocyte Activation; D018448:Models, Immunological; D008962:Models, Theoretical
D007130:Immunoglobulin Idiotypes