Th1–Th2 regulation and allergy: Bifurcation analysis of the non-autonomous system

Abstract

A previously proposed mathematical model based on a simplified scheme of Th1– Th2 regulation mediated by the cytokine network which describes the population dynamics of allergen-specific naive T cells, Th1 and Th2 cells, autocrine and cross-suppressive cytokines, and allergen is more closely investigated. The model provides a theoretical explanation of the switch from a Th2-dominated response to a Th1-dominated response to allergen in allergic individuals as a result of a hyposensitization therapy. We focus here on the bifurcation analysis of the nonautonomous dynamical system driven by periodic allergen injections. The stability of the fixed points of a stroboscopic map is investigated. The set of unstable fixed points forms the dynamical separatrix between the regions of Th2-dominated response and Th1-dominated response which is crossed during a successful therapy. The maintenance phase of the therapy holds the system near the stable fixed point of the stroboscobic map.