Global stability for the prion equation with general incidence.

Area of Science:

• Mathematical Biology
• Dynamical Systems
• Partial Differential Equations

Background:

• The prion equation models the dynamics of infectious proteins.
• Understanding the stability of steady states is crucial for predicting disease progression.
• Previous studies have utilized various mathematical techniques to analyze such models.

Purpose of the Study:

• To investigate the stability of steady states for the prion equation with a general incidence term.
• To apply a reduction technique to simplify the analysis of the prion equation.
• To leverage recent advancements in spectral gap analysis for fragmentation equations.

Main Methods:

• Reduction technique applied to the prion equation.
• Spectral gap analysis in weighted L1 spaces for growth-fragmentation equations.
• Analysis of a nonlinear system of three ordinary differential equations.

Main Results:

• The study establishes conditions for the stability of steady states in the prion equation.
• The reduction technique effectively simplifies the complex prion equation.
• The spectral gap result provides crucial insights into the equation's behavior.

Conclusions:

• The stability of prion equation steady states can be rigorously determined using the presented methods.
• This work contributes to a deeper mathematical understanding of prion diseases.
• The combined approach offers a powerful framework for analyzing similar biological models.