<p style='text-indent:20px;'>The present work is based on the coupling of prion proliferation system together with chaperone which consists of two ODEs and a partial integro-differential equation. The existence and uniqueness of a positive global classical solution of the system is proved for the bounded degradation rates by the idea of evolution system theory in the state space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R} \times \mathbb{R} \times L_{1}(Z,zdz). $\end{document}</tex-math></inline-formula> Moreover, the global weak solutions for unbounded degradation rates are discussed by weak compactness technique.</p>
A mathematical model for the dynamics of prion proliferation in the presence of chaperone involving a coupled system consisting of an ordinary differential equation and a partial integro-differential equation is analyzed. For bounded reaction rates, we prove the existence and uniqueness of positive classical solutions with the help of the theory of evolution system. In the case of unbounded reaction rates, the model is set up into a semilinear evolution equation form in the product Banach space R × L 1 ((z 0 , ∞); (q + z)dz) and the existence of a unique positive local mild solution is established by using C 0-semigroups theory of operators. KEYWORDS classical and mild solutions, C 0-semigroups, prions proliferation, semilinear evolution equations MSC CLASSIFICATION 45K05; 47H07 • S(t) = Population of PrP C monomers at time t. • u(t, z) = Population of PrP Sc polymers of length z at time t, where z > z 0. • = Constant rate of production of normal PrP C in the system.
In the present work, a mathematical model which consists of a nonlinear partial integro-differential equation coupled with two ordinary differential equations (ODEs) is analyzed. This model describes the relation between infectious, noninfectious prion proteins, and chaperone. The well-posedness of the system is proved for bounded kernels by using evolution operator theory in the state space R×R×L 1 (Z, zdz). The existence of a global weak solution for unbounded kernels is also discussed by a weak compactness argument. In addition, we investigated the stability analysis results theoretically and effect of chaperone on prion proliferation numerically.
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