Analysis of a model for the dynamics of prions II

Abstract

A new mathematical model for the dynamics of prion proliferation involving an ordinary differential equation coupled with a partial integro-differential equation is analyzed, continuing the work in [J. Prüss, L. Pujo-Menjouet, G.F. Webb, R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. 6 (2006) 225-235]. We show the well-posedness of this problem in its natural phase space Z + := R + × L + 1 ((x 0 , ∞); x dx), i.e., there is a unique global semiflow on Z + associated to the problem. A theorem of threshold type is derived for this model which is typical for mathematical epidemics. If a certain combination of kinetic parameters is below or at the threshold, there is a unique steady state, the disease-free equilibrium, which is globally asymptotically stable in Z + ; above the threshold it is unstable, and there is another unique steady state, the disease equilibrium, which inherits that property.

Well-posedness for a model of prion proliferation dynamics

“…On the one hand, part (b) of Theorem 1.2 guarantees uniqueness of monomer-preserving weak solutions in the natural state space L 1 (Y, ydy) and includes as a particular case the uniqueness result obtained in [3] for constant rates τ , β, µ and k 0 ≡ 1. Our proof is however completely different and does not rely on the special structure of (1)- (2) in that case.…”

“…Some of the results have subsequently been improved in [8]. The pde problem corresponding to (10), (11) has been studied in [3], where well-posedness of certain mild solutions and global asymptotic stability of the steady states are shown. Results for the original equations (1)-(4) not imposing data of the form (10), (11) can be found in [9,11].…”

Mathematical Modeling of Prion Disease

“…Prüss and colleagues [45] demonstrated that the prion phenotypes were globally asymptotically stable and not merely locally stable, through deriving a Lyapunov function. Engler et al [46] analyzed the well-posedness of the generalization of the NPM where aggregate sizes were continuous, instead of discrete. As such, rather than an infinite system of ordinary differential equations, the system consisted of a single ODE for protein in the normal configuration and a PDE specifying the distribution of aggregate sizes.…”

“…While this formulation departs from the physically discrete nature of aggregates, in the limit of large aggregate sizes these formalisms are provably equivalent [47] and the use of PDEs permits a wider array of mathematical techniques. Most notably, the continuous relaxation on aggregate sizes has permitted determination of the explicit asymptotic density [44,46] ] prion system [53][54][55], but linking experimental outcomes uniquely to specific kinetic parameters remains challenging.…”

Preface

“…In these papers, positivity and the spectral theory of C 0 -semigroups, but also Volterra equations, played an important role. Mathematical biology reappeared several times in Jan's work [33,35,87] and in particular in his important papers [67,71] with G. Webb and others about the longterm behavior of prion models. Moreover, he commonly treated biological systems in his teaching in ODE and dynamical systems, as witnessed in the introductory book Mathematical Models in Biology [B3] (in German).…”