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Article Dans Une Revue Journal of Mathematical Biology Année : 2019

Stability analysis of a steady state of a model describing Alzheimer’s disease and interactions with prion proteins

Résumé

Alzheimer's disease (AD) is a neuro-degenerative disease affecting more than 46 million people worldwide in 2015. AD is in part caused by the accumulation of Aβ peptides inside the brain. These can aggregate to form insoluble oligomers or fibrils. Oligomers have the capacity to interact with neurons via membrane receptors such as prion proteins (PrP C). This interaction leads PrP C to be misfolded in oligomeric prion proteins (PrP ol), transmitting a death signal to neurons. In this work, we propose a new mathematical model bringing together different mechanisms: Aβ polymerization, including formation of oligomers and fibrils, and interaction between Aβ oligomers and prion proteins. The model is based on Becker-Döring equations for the polymer-ization process, with delayed differential equations accounting for Aβ/PrP C interactions. We analyse the well-posedness of the model and show existence, uniqueness and non-negativity of 2 Helal et al. solutions. Moreover, we demonstrate that this model admits a non-trivial steady state, which is found to be globally stable thanks to a Lyapunov function. We finally present numerical simulations and discuss the impact of model parameters on the whole dynamics, which could constitute the main targets for pharmaceutical industry.
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Dates et versions

hal-01946610 , version 1 (07-12-2018)

Identifiants

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Mohammed Helal, Angélique Igel-Egalon, Abdelkader Lakmeche, Pauline Mazzocco, Angélique Perrillat-Mercerot, et al.. Stability analysis of a steady state of a model describing Alzheimer’s disease and interactions with prion proteins. Journal of Mathematical Biology, 2019, 78 (1-2), pp.57-81. ⟨10.1007/s00285-018-1267-1⟩. ⟨hal-01946610⟩
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