# American Institute of Mathematical Sciences

2015, 12(4): 789-801. doi: 10.3934/mbe.2015.12.789

## 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

Received  May 2014 Revised  January 2015 Published  April 2015

We consider the so-called prion equation with the general incidence term introduced in [14], and we investigate the stability of the steady states. The method is based on the reduction technique introduced in [11]. The argument combines a recent spectral gap result for the growth-fragmentation equation in weighted $L^1$ spaces and the analysis of a nonlinear system of three ordinary differential equations.
Citation: Pierre Gabriel. Global stability for the prion equation with general incidence. Mathematical Biosciences & Engineering, 2015, 12 (4) : 789-801. doi: 10.3934/mbe.2015.12.789
##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] J. S. Griffith, Nature of the scrapie agent: Self-replication and scrapie,, Nature, 215 (1967), 1043. doi: 10.1038/2151043a0. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, preprint,, , (). Google Scholar [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. Google Scholar [23] S. B. Prusiner, Novel proteinaceous infectious particles cause scrapie,, Science, 216 (1982), 136. doi: 10.1126/science.6801762. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [27] H. L. Smith, Monotone Dynamical Systems,, American Mathematical Society, (1995). Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] J. S. Griffith, Nature of the scrapie agent: Self-replication and scrapie,, Nature, 215 (1967), 1043. doi: 10.1038/2151043a0. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, preprint,, , (). Google Scholar [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. Google Scholar [23] S. B. Prusiner, Novel proteinaceous infectious particles cause scrapie,, Science, 216 (1982), 136. doi: 10.1126/science.6801762. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [27] H. L. Smith, Monotone Dynamical Systems,, American Mathematical Society, (1995). Google Scholar [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. Google Scholar
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