Immune network behavior—I. From stationary states to limit cycle oscillations

Abstract

We develop a model for the idiotypic interaction between two B cell clones. This model takes into account B cell proliferation. B cell maturation, antibody production, the formation and subsequent elimination of antibody-antibody complexes and recirculation of antibodies between the spleen and the blood. Here we investigate, by means of stability and bifurcation analysis. how each of the processes influences the model's behavior. After appropriate nondimensionalization. the model consists of eight ordinary differential equations and a number of parameters. We estimate the parameters from experimental sources. Using a coordinate system that exploits the pairwise symmetry of the interactions between two clones, we analyse two simplified forms of the model and obtain bifurcation diagrams showing how their five equilibrium states are related. We show that the so-called immune states lose stability if B cell and antibody concentrations change on different time scales. Additionally, we derive the structure of stable and unstable manifolds of saddle-type equilibria, pinpoint their (global) bifurcations and show that these bifurcations play a crucial role in determining the parameter regimes in which the model exhibits oscillatory behavior.

Global dynamics of B cells and anti-idiotipic B cells and its application to autoimmunity

“…We can then show that P2 and P3 are locally asymptotically stable similarly to [1]. For the parameter values near that used in [1] and small u, we can show that Po is locally asymptotically stable and Pl and P4 are unstable.…”

“…The qualitative features of (3) are investigated also in [1]. For estimated parameter values, system (3) has five equilibria Qo(go, qo), QI (qi, qi), Q2(g3, q2), Q3(g2• o•^), and Q4(g4, q4), where qo < qI < q4 and q2 < q3.…”

“…2 (c) shows injection does not immediately affect the orbit because the change of B1 , keeping B2 unchanged, affects the orbit little, but after that the remaining antigen affects the orbit like in the case (b). Note that (B 1 …”

“…De Boer, Perelson, and Kevrekidis [1] consider the behavior of immune network interaction with various kinds of models. One of them is the B model, which describes the interaction between B cell clones.…”

Growth and Recruitment in the Immune Network

“…2b. In this respect the behavior resembles the oscillatory/chaotic behavior of the continuous model , 1992c). …”