On the solvability of a mathematical model for prion proliferation

Abstract

We show that a model describing the interaction between normal and infectious prion proteins admits global solutions. More precisely, supposing the involved degradation rates to be bounded, we prove global existence and uniqueness of classical solutions. Based on this existence theory, we provide sufficient conditions for the existence of global weak solutions in the case of unbounded splitting rates. Moreover, we prove global stability of the disease-free steady state.

Well-posedness for a model of prion proliferation dynamics

“…Let us mention at this point that our uniqueness results do not require the (weak) differentiability of u 0 (i.e. u 0 ∈ W 1 1 (Y )) in contrast to [9,11] but make use of the differentiability of β and µ, an assumption which is not needed in [9,11]. This fact is actually a consequence of our approach.…”

“…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]. In these papers it is proven that bounded degradation rates ensure well-posedness of (1)-(4) in a classical sense, while the problem admits weak solutions for unbounded degradation rates with certain growth rates.…”

“…The present paper aims at deriving uniqueness results for monomer-preserving weak solutions and extending the existence results of [9,11] to a wider class of degradation rates. Recall that the total number of monomers can increase during time evolution merely due to the constant background source λ and decrease only by metabolic degradation of monomers and polymers.…”

Mathematical Aspects of Coagulation-Fragmentation Equations

“…As an example that has recently received some attention, we can point to models of growth and proliferation of prions (i.e., of proteins with transmissible pathological conformations) responsible for the Bovine Spongiform Encephalopathy ("Mad Cow Disease") [102,187] and several math-ematical models have already been object of a rigorous analysis [136,198,216]. According to the contemporary biological understanding, there are two basic prion forms, a normal, non-infectious, monomeric one (denoted by PrP C in the literature) and an infectious polymeric form (PrP Sc ) formed by the polymerization of the monomeric form.…”

“…Continuous mass versions of these equations were also considered in the literature [102,136,198,216]. …”