Abstract
We introduce a mathematical model of the in vivo progression of Alzheimer’s disease with focus on the role of prions in memory impairment. Our model consists of differential equations that describe the dynamic formation of \(\upbeta \)-amyloid plaques based on the concentrations of A\(\upbeta \) oligomers, PrPC proteins, and the A\(\upbeta \)-\(\times \)-PrPCcomplex, which are hypothesized to be responsible for synaptic toxicity. We prove the well-posedness of the model and provided stability results for its unique equilibrium, when the polymerization rate of \(\upbeta \)-amyloid is constant and also when it is described by a power law.
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Acknowledgments
The authors thanks the Reviewers for their usefull comments and suggestions. E.H. thanks A. Rambaud for helpful discussions, which improved the paper.
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This work was supported by ANR grant MADCOW no. 08-JCJC-0135-CSD5. E. H. was partially supported by FONDECYT Postoctoral Grant no. 3130318 (Chile).
Appendices
Appendix A: Characteristic polynomials of the linearized ODE system
Here we give the coefficient \(a_i\), \(i=1,\ldots ,4\) for the characteristic polynomial of the linearized system in Theorem 1:
Appendix B: Lyapunov functional
Here we detail a Lyapunov function \(\varPhi \) which is the key ingredient to prove global stability of system (8–11) in Proposition 2. This function appears to be a bit tricky, but determining it rest upon the backward method described for instance Chapter 4, p. 120, in the book by Khalil (1996). It consists in investigate an expression of the derivative \(\varPhi '\) and then going back to chose the parameters \(\varPhi \) such as \(\varPhi '\) is negative definite. After tedious calculus, a Liapunov function \(\varPhi \) for system (8–11) is given by
where \(\theta _1=A-A_\infty \), \(\theta _2=u-u_\infty \), \(\theta _3=p-p_\infty \), \(\theta _4=b-b_\infty \), with \(s_1=\max (T_1,T_2)\) such that
and \(T_2 = \varGamma \left( \frac{\delta }{2\gamma _p}\right) ^2 T_2'\) with
and
We remark that \(T_1>0\) so that \(s_1 >0\), and then we deduce that the Lyapunov function \(\varPhi \) is positive when condition \(\left( 1+2\frac{\delta +\gamma _u}{\sigma }\right) >\frac{\delta }{2\gamma _p}>\frac{\gamma _p}{\sigma }\) holds true. In such case, its derivative along the solutions of system (8–11) is given by
and remains nonpositive. Furthermore, \(\varPhi '=0\) if and only if \(\theta _1=\theta _2=\theta _3=\theta _4=0\). The conclusion holds by the LaSalle Invariance Principle LaSalle (1976).
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Helal, M., Hingant, E., Pujo-Menjouet, L. et al. Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions. J. Math. Biol. 69, 1207–1235 (2014). https://doi.org/10.1007/s00285-013-0732-0
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DOI: https://doi.org/10.1007/s00285-013-0732-0