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Analysis of a model for the dynamics of prions. II. (English) Zbl 1103.92024

Summary: 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 part I by J. Prüss, L. Pujo-Menjouet, G. F. Webb and R. Zacher [Discrete Contin. Dyn. Syst., Ser. B 6, No. 1, 225–235 (2006; Zbl 1088.92043)]. We show the well-posedness of this problem in its natural phase space \(Z_+: =\mathbb{R}_+\times L_1^+((x_0,\infty);xdx)\), 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.

MSC:

92C50 Medical applications (general)
45K05 Integro-partial differential equations
47N60 Applications of operator theory in chemistry and life sciences
47D03 Groups and semigroups of linear operators
92C60 Medical epidemiology

Citations:

Zbl 1088.92043
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References:

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[7] Prüss, J.; Pujo-Menjouet, L.; Webb, G. F.; Zacher, R., Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst., 6, 225-235 (2006) · Zbl 1088.92043
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