Analysis Of The Dynamic Model For Amyloid Protein In Alzheimer’s Disease

Alzheimer’s disease（AD）is a worldwide disease of dementia that not only seriously affects the quality of life of the elderly,but also brings huge economic burden to the patients and the society.Although the reason why AD occurs is still unclear,there exist a large number of experimental results indicate that the beta-amyloid（Aβ）can trigger the progression of AD.Hence,in this thesis,several types of mathematical model are built to study the evolution of Aβ.Increasing evidences show that there is a positive loop between the level of calcium and the level of Aβ.In addition,the both of the evolutions for calcium and Aβcan be affected by the stochastic noise.Based on the above consider-ations,in the second chapter,stochastic noises are incorporated into a minimal model of Alzheimer’s disease which focuses upon the evolution of calcium and Aβ.Mathematical analysis indicates that solutions of the model without stochastic noises converge either to a unique equilibrium or to bistable equilibria.Analytical conditions for the stochastic P-bifurcation are derived by means of technique of slow-fast dynamical systems.A formula is presented to approximate the mean switching time from a normal state to a pathological state.A disease index is also proposed to predict the risk to transit from a normal state to a disease state.Further numerical simulations reveal how the parameters influence the evolution-ary outcomes of beta-amyloid and calcium.Especially,the noise from calcium declines the progression of Alzheimer’disease.However,the noise from Aβcan accelerate the progression of AD.Becauseγ-secretor is a major proteolytic enzyme in the progression to the production of Aβ,γ-secretase becomes one of candidate therapeutic targets for preventing the pathological progress of AD and variousγ-secretase inhibitors were designed to limit the production of Aβ.However,there exists an apparent-ly paradoxical behavior called‘Aβrise’effect（or biphasic behavior）where the lower concentration of inhibitor evolves the production of Aβ.This paradoxical behavior blocks the development of related drugs.Hence,in the third and fourth chapters,the mechanism of‘Aβrise’effect is studied by mathematical models under different biological assumptions.In the third chapter,a general mathematical model considering different en-zyme is built to find the mechanism for‘Aβrise’effect.Assume thatγ-secretase operates for substrate C₈₃in the first order kinetic region and the conversion rate for C₉₉to C₈₃is small.Then the analysis results show that the inhibit effect of C₉₉to theα-secretase cleavage of APP（amyloid precursor protein）and the low-erγ-secretase independent degradation of C₉₉are two factors for the occurence of‘Aβrise’effect.In the fourth chapter,assume that both theγ-secretase andα-secretase operate in the first order kinetic region,we built a mathematical model.Math-ematical analysis show that the system undergoes saddle-node bifurcation and has the bistable behavior.Furthermore,in this case,we obtain the condition for the occurence of‘Aβrise’effect.An interesting phenomenon where the number of steady state for Aβchanges as the increase of concentration ofγ-inhibitor can be found.In addition,with the increase of concentration ofγ-inhibitor,the level of the higher steady state for Aβfirstly rises and then decreases.Based on the analysis results,we find that enhancing another cleavage pathway of APP or C₉₉γ-secretase independent degradation can prevent the occurence of‘Aβrise’effect.