`a`
Mathematical Biosciences and Engineering (MBE)
 

Global stability for the prion equation with general incidence
Pages: 789 - 801, Issue 4, August 2015

doi:10.3934/mbe.2015.12.789      Abstract        References        Full text (377.6K)                  Related Articles

Pierre Gabriel - Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, 45 Avenue de États-Unis, 78035 Versailles cedex, France (email)

1 D. Balagué, J. A. Cañizo and P. Gabriel, Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates, Kinetic Related Models, 6 (2013), 219-243.       
2 M. J. Cáceres, J. A. Cañizo and S. Mischler, Rate of convergence to self-similarity for the fragmentation equation in $L^1$ spaces, Comm. Appl. Ind. Math., 1 (2010), 299-308.       
3 M. J. Cáceres, J. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl., 96 (2011), 334-362.       
4 V. Calvez, N. Lenuzza, M. Doumic, J.-P. Deslys, F. Mouthon and B. Perthame, Prion dynamic with size dependency - strain phenomena, J. Biol. Dyn., 4 (2010), 28-42.       
5 V. Calvez, N. Lenuzza, D. Oelz, J.-P. Deslys, P. Laurent, F. Mouthon and B. Perthame, Size distribution dependence of prion aggregates infectivity, Math. Biosci., 217 (2009), 88-99.       
6 M. Doumic, T. Goudon and T. Lepoutre, Scaling limit of a discrete prion dynamics model, Comm. Math. Sci., 7 (2009), 839-865.       
7 M. Doumic Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783.       
8 H. Engler, J. Prüss and G. Webb, Analysis of a model for the dynamics of prions ii, J. Math. Anal. Appl., 324 (2006), 98-117.       
9 M. Escobedo, S. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125.       
10 P. Gabriel, The shape of the polymerization rate in the prion equation, Math. Comput. Modelling, 53 (2011), 1451-1456.       
11 P. Gabriel, Long-time asymptotics for nonlinear growth-fragmentation equations, Commun. Math. Sci., 10 (2012), 787-820.       
12 P. Gabriel and F. Salvarani, Exponential relaxation to self-similarity for the superquadratic fragmentation equation, Appl. Math. Lett., 27 (2014), 74-78.       
13 M. L. Greer, L. Pujo-Menjouet and G. F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theoret. Biol., 242 (2006), 598-606.       
14 M. L. Greer, P. van den Driessche, L. Wang and G. F. Webb, Effects of general incidence and polymer joining on nucleated polymerization in a model of prion proliferation, SIAM J. Appl. Math., 68 (2007), 154-170.       
15 J. S. Griffith, Nature of the scrapie agent: Self-replication and scrapie, Nature, 215 (1967), 1043-1044.
16 J. T. Jarrett and P. T. Lansbury, Seeding "one-dimensional crystallization'' of amyloid: A pathogenic mechanism in alzheimer's disease and scrapie?, Cell, 73 (1993), 1055-1058.
17 P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7 (2009), 503-510.       
18 P. Laurençot and C. Walker, Well-posedness for a model of prion proliferation dynamics, J. Evol. Equ., 7 (2007), 241-264.       
19 J. Masel, V. Jansen and M. Nowak, Quantifying the kinetic parameters of prion replication, Biophysical Chemistry, 77 (1999), 139-152.
20 P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.       
21 S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, preprint, arXiv:1310.7773.
22 B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210 (2005), 155-177.       
23 S. B. Prusiner, Novel proteinaceous infectious particles cause scrapie, Science, 216 (1982), 136-144.
24 J. Prüss, L. Pujo-Menjouet, G. Webb and R. Zacher, Analysis of a model for the dynamics of prion, Dis. Cont. Dyn. Sys. Ser. B, 6 (2006), 225-235.       
25 J. Silveira, G. Raymond, A. Hughson, R. Race, V. Sim, S. Hayes and B. Caughey, The most infectious prion protein particles, Nature, 437 (2005), 257-261.
26 G. Simonett and C. Walker, On the solvability of a mathematical model for prion proliferation, J. Math. Anal. Appl., 324 (2006), 580-603.       
27 H. L. Smith, Monotone Dynamical Systems, American Mathematical Society, Providence, RI, 1995.       
28 C. Walker, Prion proliferation with unbounded polymerization rates, in Proceedings of the Sixth Mississippi State-UBA Conference on Differential Equations and Computational Simulations, 15 (2007), 387-397.       

Go to top