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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Analysis of a model for the dynamics of prions

Pages: 225 - 235, Volume 6, Issue 1, January 2006      doi:10.3934/dcdsb.2006.6.225

 
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Jan Prüss - Fachbereich Mathematik und Informatik, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle, Germany (email)
Laurent Pujo-Menjouet - Department of Mathematics, Vanderbilt University, Nashville, Tennessee TN 37240, United States (email)
G.F. Webb - Department of Mathematics, Vanderbilt University, Nashville, TN 37340, United States (email)
Rico Zacher - Fachbereich Mathematik und Informatik, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle, Germany (email)

Abstract: A mathematical model for the dynamics of prion proliferation is analyzed. The model involves a system of three ordinary differential equations for the normal prion forms, the abnormal prion forms, and polymers comprised of the abnormal forms. The model is a special case of a more general model, which is also applicable to other models of infectious diseases. A theorem of threshold type is derived for this general model. It is proved that below and at the threshold, there is a unique steady state, the disease-free equilibrium, which is globally asymptotically stable. Above the threshold, the disease-free equilibrium is unstable, and there is another steady state, the disease equilibrium, which is globally asymptotically stable.

Keywords:  prions, proliferation, epidemics, viral-host interaction.
Mathematics Subject Classification:  92D25, 92C60.

Received: May 2005;      Revised: October 2005;      Published: October 2005.