Global stability for the prion equation with general incidence.

Pierre Gabriel

doi:10.3934/mbe.2015.12.789

Abstract

We consider the so-called prion equation with the general incidence term introduced in [14], and we investigate the stability of the steady states. The method is based on the reduction technique introduced in [11]. The argument combines a recent spectral gap result for the growth-fragmentation equation in weighted L1 spaces and the analysis of a nonlinear system of three ordinary differential equations.

Highlights

Prion diseases, referred to as transmissible spongiform encephalopathies, are infectious and fatal neurodegenerative diseases
To understand more qualitatively this mechanism, a mathematical model consisting in a infinite number of coupled ordinary differential equations (ODEs) was introduced in [19]
The existence of an endemic equilibria (EE) as well as the stability of the disease free equilibrium (DFE) depend on the basic reproduction rate R0 of Equation (1), which indicates the average number of new infections caused by a single infective introduced to an entirely susceptible population

Summary

Introduction

Referred to as transmissible spongiform encephalopathies, are infectious and fatal neurodegenerative diseases. The existence of an EE as well as the stability of the DFE depend on the basic reproduction rate R0 of Equation (1), which indicates the average number of new infections caused by a single infective introduced to an entirely susceptible population. To find this parameter R0, we linearize the equation on u about the DFE (V , 0) and we test the resulting equation against x to obtain d dt xu(t, x) dx ≃ Vτ xu(t, x) dx − μ xu(t, x) dx. The paper is organized as follows: In Section 2 we explain the method which allows to reduce Equation (1) to a system of ODEs, and in Section 3 we take advantage of this reduction to prove Theorem 2

Reduction to a system of ODEs

Proof of Theorem 2

Conclusion