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Global stability for the prion equation with general incidence
1. | Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, 45 Avenue de États-Unis, 78035 Versailles cedex |
References:
[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.
doi: 10.3934/krm.2013.6.219. |
[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.
|
[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.
doi: 10.1016/j.matpur.2011.01.003. |
[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.
doi: 10.1080/17513750902935208. |
[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.
doi: 10.1016/j.mbs.2008.10.007. |
[6] |
M. Doumic, T. Goudon and T. Lepoutre, Scaling limit of a discrete prion dynamics model,, Comm. Math. Sci., 7 (2009), 839.
doi: 10.4310/CMS.2009.v7.n4.a3. |
[7] |
M. Doumic Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model,, Math. Models Methods Appl. Sci., 20 (2010), 757.
doi: 10.1142/S021820251000443X. |
[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.
doi: 10.1016/j.jmaa.2005.11.021. |
[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.
doi: 10.1016/j.anihpc.2004.06.001. |
[10] |
P. Gabriel, The shape of the polymerization rate in the prion equation,, Math. Comput. Modelling, 53 (2011), 1451.
doi: 10.1016/j.mcm.2010.03.032. |
[11] |
P. Gabriel, Long-time asymptotics for nonlinear growth-fragmentation equations,, Commun. Math. Sci., 10 (2012), 787.
doi: 10.4310/CMS.2012.v10.n3.a4. |
[12] |
P. Gabriel and F. Salvarani, Exponential relaxation to self-similarity for the superquadratic fragmentation equation,, Appl. Math. Lett., 27 (2014), 74.
doi: 10.1016/j.aml.2013.08.001. |
[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.
doi: 10.1016/j.jtbi.2006.04.010. |
[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.
doi: 10.1137/06066076X. |
[15] |
J. S. Griffith, Nature of the scrapie agent: Self-replication and scrapie,, Nature, 215 (1967), 1043.
doi: 10.1038/2151043a0. |
[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.
doi: 10.1016/0092-8674(93)90635-4. |
[17] |
P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation,, Commun. Math. Sci., 7 (2009), 503.
doi: 10.4310/CMS.2009.v7.n2.a12. |
[18] |
P. Laurençot and C. Walker, Well-posedness for a model of prion proliferation dynamics,, J. Evol. Equ., 7 (2007), 241.
doi: 10.1007/s00028-006-0279-2. |
[19] |
J. Masel, V. Jansen and M. Nowak, Quantifying the kinetic parameters of prion replication,, Biophysical Chemistry, 77 (1999), 139.
doi: 10.1016/S0301-4622(99)00016-2. |
[20] |
P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models,, J. Math. Pures Appl., 84 (2005), 1235.
doi: 10.1016/j.matpur.2005.04.001. |
[21] |
S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, preprint,, , (). |
[22] |
B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation,, J. Differential Equations, 210 (2005), 155.
doi: 10.1016/j.jde.2004.10.018. |
[23] |
S. B. Prusiner, Novel proteinaceous infectious particles cause scrapie,, Science, 216 (1982), 136.
doi: 10.1126/science.6801762. |
[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.
|
[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.
doi: 10.1038/nature03989. |
[26] |
G. Simonett and C. Walker, On the solvability of a mathematical model for prion proliferation,, J. Math. Anal. Appl., 324 (2006), 580.
doi: 10.1016/j.jmaa.2005.12.036. |
[27] |
H. L. Smith, Monotone Dynamical Systems,, American Mathematical Society, (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.
|
show all references
References:
[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.
doi: 10.3934/krm.2013.6.219. |
[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.
|
[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.
doi: 10.1016/j.matpur.2011.01.003. |
[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.
doi: 10.1080/17513750902935208. |
[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.
doi: 10.1016/j.mbs.2008.10.007. |
[6] |
M. Doumic, T. Goudon and T. Lepoutre, Scaling limit of a discrete prion dynamics model,, Comm. Math. Sci., 7 (2009), 839.
doi: 10.4310/CMS.2009.v7.n4.a3. |
[7] |
M. Doumic Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model,, Math. Models Methods Appl. Sci., 20 (2010), 757.
doi: 10.1142/S021820251000443X. |
[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.
doi: 10.1016/j.jmaa.2005.11.021. |
[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.
doi: 10.1016/j.anihpc.2004.06.001. |
[10] |
P. Gabriel, The shape of the polymerization rate in the prion equation,, Math. Comput. Modelling, 53 (2011), 1451.
doi: 10.1016/j.mcm.2010.03.032. |
[11] |
P. Gabriel, Long-time asymptotics for nonlinear growth-fragmentation equations,, Commun. Math. Sci., 10 (2012), 787.
doi: 10.4310/CMS.2012.v10.n3.a4. |
[12] |
P. Gabriel and F. Salvarani, Exponential relaxation to self-similarity for the superquadratic fragmentation equation,, Appl. Math. Lett., 27 (2014), 74.
doi: 10.1016/j.aml.2013.08.001. |
[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.
doi: 10.1016/j.jtbi.2006.04.010. |
[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.
doi: 10.1137/06066076X. |
[15] |
J. S. Griffith, Nature of the scrapie agent: Self-replication and scrapie,, Nature, 215 (1967), 1043.
doi: 10.1038/2151043a0. |
[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.
doi: 10.1016/0092-8674(93)90635-4. |
[17] |
P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation,, Commun. Math. Sci., 7 (2009), 503.
doi: 10.4310/CMS.2009.v7.n2.a12. |
[18] |
P. Laurençot and C. Walker, Well-posedness for a model of prion proliferation dynamics,, J. Evol. Equ., 7 (2007), 241.
doi: 10.1007/s00028-006-0279-2. |
[19] |
J. Masel, V. Jansen and M. Nowak, Quantifying the kinetic parameters of prion replication,, Biophysical Chemistry, 77 (1999), 139.
doi: 10.1016/S0301-4622(99)00016-2. |
[20] |
P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models,, J. Math. Pures Appl., 84 (2005), 1235.
doi: 10.1016/j.matpur.2005.04.001. |
[21] |
S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, preprint,, , (). |
[22] |
B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation,, J. Differential Equations, 210 (2005), 155.
doi: 10.1016/j.jde.2004.10.018. |
[23] |
S. B. Prusiner, Novel proteinaceous infectious particles cause scrapie,, Science, 216 (1982), 136.
doi: 10.1126/science.6801762. |
[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.
|
[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.
doi: 10.1038/nature03989. |
[26] |
G. Simonett and C. Walker, On the solvability of a mathematical model for prion proliferation,, J. Math. Anal. Appl., 324 (2006), 580.
doi: 10.1016/j.jmaa.2005.12.036. |
[27] |
H. L. Smith, Monotone Dynamical Systems,, American Mathematical Society, (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.
|
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