An ODE Model of Th1/Th2 Immune Switching and Microglial Dynamics Reveals Rate-Dependent Thresholds Governing Neuronal Stability and Relapse

Abstract

In this study, we develop a mathematical model based on ordinary differential equations (ODEs) to investigate how immune responses influence neuronal stability and decline. The model focuses on interactions between Th1 and Th2 lymphocytes and their effects on microglial polarization, in which microglia can adopt either a pro-inflammatory (M1) or an anti-inflammatory (M2) phenotype. Through these interactions, immune activity can create conditions that either damage or protect neuronal populations. Our results indicate that when the immune response is dominated by Th1 activity, microglia tend to shift toward the M1 state, producing a pro-inflammatory environment that can progressively damage neurons. In contrast, stronger Th2 responses promote the M2 state, which supports anti-inflammatory and protective processes that help maintain neuronal stability. Sensitivity analysis further reveals that key parameters, including cytokine signaling rates and neuronal loss rates, play an important role in shaping these outcomes. Importantly, the model reveals a threshold-driven transition between neuronal stability and neurodegeneration, governed by the balance between pro-inflammatory and anti-inflammatory responses. This transition is associated with a change in the stability of the coexistence equilibrium, providing a mechanistic explanation for the shift between healthy and relapse states. By highlighting how immune balance regulates neuronal survival, the proposed model provides a quantitative framework for understanding how immune dysregulation may contribute to neuronal decline in neurodegenerative conditions such as Parkinson’s disease. The analysis identifies parameter conditions that govern the transition from a stable neuronal state to neurodegeneration and highlights key factors that may influence neuronal relapse.