Theoretical analysis of an immune response model

In this paper, we develop a two-component model for antigen–immune interactions and present new theoretical insights into immunological diseases, particularly the dynamical mechanism of cytokine storms. First, we prove the existence of multiple positive equilibria, implying different immunological states. Second, we analyze the stability of these equilibria and summarize the conditions for bistability, which forms a reversible switch between stable states. This reversible switch enables abrupt activation or inactivation of the immune system during normal immune response. Third, we show that bistable but irreversible equilibria also exist, where the system cannot return to the healthy state even after antigen removal, and this case captures cytokine storm development induced by immune system overactivation. Finally, monostable equilibrium emerges under stronger negative feedback, promoting immune escape during tumor progression. These theoretical results elucidate the mechanisms underlying distinct response behaviors observed in our previous work.