Global stability for the prion equation with general incidence

Pierre Gabriel; Pierre Gabriel

doi:10.3934/mbe.2015.12.789

Mathematical Biosciences and Engineering

2015, Volume 12, Issue 4: 789-801. doi: 10.3934/mbe.2015.12.789

Previous Article Next Article

Global stability for the prion equation with general incidence

Pierre Gabriel ¹

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: 01 May 2014 Accepted: 29 June 2018 Published: 01 April 2015
MSC : Primary: 92D25; Secondary: 35B35, 35B40, 35Q92, 45K05.

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.

Keywords:

Citation: Pierre Gabriel. Global stability for the prion equation with general incidence[J]. Mathematical Biosciences and Engineering, 2015, 12(4): 789-801. doi: 10.3934/mbe.2015.12.789

Related Papers:

[1]	H. T. Banks, Cammey E. Cole, Paul M. Schlosser, Hien T. Tran . Modeling and optimal regulation of erythropoiesis subject to benzene intoxication. Mathematical Biosciences and Engineering, 2004, 1(1): 15-48. doi: 10.3934/mbe.2004.1.15
[2]	Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito . Finite difference approximations for measure-valued solutions of a hierarchicallysize-structured population model. Mathematical Biosciences and Engineering, 2015, 12(2): 233-258. doi: 10.3934/mbe.2015.12.233
[3]	Azmy S. Ackleh, H.T. Banks, Keng Deng, Shuhua Hu . Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations. Mathematical Biosciences and Engineering, 2005, 2(2): 289-315. doi: 10.3934/mbe.2005.2.289
[4]	Yuganthi R. Liyanage, Nora Heitzman-Breen, Necibe Tuncer, Stanca M. Ciupe . Identifiability investigation of within-host models of acute virus infection. Mathematical Biosciences and Engineering, 2024, 21(10): 7394-7420. doi: 10.3934/mbe.2024325
[5]	Lernik Asserian, Susan E. Luczak, I. G. Rosen . Computation of nonparametric, mixed effects, maximum likelihood, biosensor data based-estimators for the distributions of random parameters in an abstract parabolic model for the transdermal transport of alcohol. Mathematical Biosciences and Engineering, 2023, 20(11): 20345-20377. doi: 10.3934/mbe.2023900
[6]	Scott R. Pope, Laura M. Ellwein, Cheryl L. Zapata, Vera Novak, C. T. Kelley, Mette S. Olufsen . Estimation and identification of parameters in a lumped cerebrovascular model. Mathematical Biosciences and Engineering, 2009, 6(1): 93-115. doi: 10.3934/mbe.2009.6.93
[7]	Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd . Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences and Engineering, 2012, 9(3): 553-576. doi: 10.3934/mbe.2012.9.553
[8]	Kimberlyn Roosa, Ruiyan Luo, Gerardo Chowell . Comparative assessment of parameter estimation methods in the presence of overdispersion: a simulation study. Mathematical Biosciences and Engineering, 2019, 16(5): 4299-4313. doi: 10.3934/mbe.2019214
[9]	Maria Pia Saccomani, Karl Thomaseth . Calculating all multiple parameter solutions of ODE models to avoid biological misinterpretations. Mathematical Biosciences and Engineering, 2019, 16(6): 6438-6453. doi: 10.3934/mbe.2019322
[10]	Keenan Hawekotte, Susan E. Luczak, I. G. Rosen . Deconvolving breath alcohol concentration from biosensor measured transdermal alcohol level under uncertainty: a Bayesian approach. Mathematical Biosciences and Engineering, 2021, 18(5): 6739-6770. doi: 10.3934/mbe.2021335

Abstract

References

[1]	Kinetic Related Models, 6 (2013), 219-243.
[2]	Comm. Appl. Ind. Math., 1 (2010), 299-308.
[3]	J. Math. Pures Appl., 96 (2011), 334-362.
[4]	J. Biol. Dyn., 4 (2010), 28-42.
[5]	Math. Biosci., 217 (2009), 88-99.
[6]	Comm. Math. Sci., 7 (2009), 839-865.
[7]	Math. Models Methods Appl. Sci., 20 (2010), 757-783.
[8]	J. Math. Anal. Appl., 324 (2006), 98-117.
[9]	Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125.
[10]	Math. Comput. Modelling, 53 (2011), 1451-1456.
[11]	Commun. Math. Sci., 10 (2012), 787-820.
[12]	Appl. Math. Lett., 27 (2014), 74-78.
[13]	J. Theoret. Biol., 242 (2006), 598-606.
[14]	SIAM J. Appl. Math., 68 (2007), 154-170.
[15]	Nature, 215 (1967), 1043-1044.
[16]	Cell, 73 (1993), 1055-1058.
[17]	Commun. Math. Sci., 7 (2009), 503-510.
[18]	J. Evol. Equ., 7 (2007), 241-264.
[19]	Biophysical Chemistry, 77 (1999), 139-152.
[20]	J. Math. Pures Appl., 84 (2005), 1235-1260.
[21]	arXiv:1310.7773.
[22]	J. Differential Equations, 210 (2005), 155-177.
[23]	Science, 216 (1982), 136-144.
[24]	Dis. Cont. Dyn. Sys. Ser. B, 6 (2006), 225-235.
[25]	Nature, 437 (2005), 257-261.
[26]	J. Math. Anal. Appl., 324 (2006), 580-603.
[27]	American Mathematical Society, Providence, RI, 1995.
[28]	in Proceedings of the Sixth Mississippi State-UBA Conference on Differential Equations and Computational Simulations, 15 (2007), 387-397.

This article has been cited by:

1.	Shawn P. Reese, Benjamin J. Ellis, Jeffrey A. Weiss, 2013, Chapter 157, 978-3-642-36481-5, 103, 10.1007/8415_2012_157
2.	Jason M. Graham, Bruce P. Ayati, Sarah A. Holstein, James A. Martin, Luc Malaval, The Role of Osteocytes in Targeted Bone Remodeling: A Mathematical Model, 2013, 8, 1932-6203, e63884, 10.1371/journal.pone.0063884
3.	Marc D. Ryser, Kevin A. Murgas, Bone remodeling as a spatial evolutionary game, 2017, 418, 00225193, 16, 10.1016/j.jtbi.2017.01.021
4.	Bing Ji, Jie Chen, Changqing Zhen, Qing Yang, Nengwang Yu, Mathematical modelling of the role of Endo180 network in the development of metastatic bone disease in prostate cancer, 2020, 117, 00104825, 103619, 10.1016/j.compbiomed.2020.103619
5.	Ariel Camacho, Silvia Jerez, Bone metastasis treatment modeling via optimal control, 2019, 78, 0303-6812, 497, 10.1007/s00285-018-1281-3
6.	Yang Zhang, Enrico Dall’Ara, Marco Viceconti, Visakan Kadirkamanathan, Jose Manuel Garcia Aznar, A new method to monitor bone geometry changes at different spatial scales in the longitudinal in vivo μCT studies of mice bones, 2019, 14, 1932-6203, e0219404, 10.1371/journal.pone.0219404
7.	S. Jerez, B. Chen, Stability analysis of a Komarova type model for the interactions of osteoblast and osteoclast cells during bone remodeling, 2015, 264, 00255564, 29, 10.1016/j.mbs.2015.03.003
8.	Corey J. Miller, Silvia Trichilo, Edmund Pickering, Saulo Martelli, Peter Delisser, Lee B. Meakin, Peter Pivonka, Cortical Thickness Adaptive Response to Mechanical Loading Depends on Periosteal Position and Varies Linearly With Loading Magnitude, 2021, 9, 2296-4185, 10.3389/fbioe.2021.671606
9.	Carley V. Cook, Ariel M. Lighty, Brenda J. Smith, Ashlee N. Ford Versypt, A review of mathematical modeling of bone remodeling from a systems biology perspective, 2024, 4, 2674-0702, 10.3389/fsysb.2024.1368555
10.	Peter Pivonka, José Luis Calvo-Gallego, Stephan Schmidt, Javier Martínez-Reina, Advances in mechanobiological pharmacokinetic-pharmacodynamic models of osteoporosis treatment – Pathways to optimise and exploit existing therapies, 2024, 186, 87563282, 117140, 10.1016/j.bone.2024.117140

Reader Comments

Your name:*

Email:*
© 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)