# A Mathematical Investigation of Sex Differences in Alzheimer’s Disease

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model

#### 2.2. Numerical Scheme

**lsqnonneg**which is a numerical implementation of the Kuhn–Tucker conditions [71]. The code was validated using the case $q=1$: the solution generated by

**lsqnonneg**coincides with the solution to the system of first order differential Equations (1)–(7) obtained using Matlab’s built-in function

**ode45**which was constrained to satisfy the NonNegative option selected with the function

**odeset**. Since an analytic solution of the initial value problem (13) and (9) is not known, the convergence of the numerical scheme (15) was observed numerically: the same results were obtained for $h\in \{0.1,\phantom{\rule{0.166667em}{0ex}}0.05,\phantom{\rule{0.166667em}{0ex}}0.01,\phantom{\rule{0.166667em}{0ex}}0.005,\phantom{\rule{0.166667em}{0ex}}0.001\}$. Thus, the step size for the numerical simulations presented in the next section was chosen to be $h=0.005$. Lastly, the system of Equations (1)–(7) for constant fractional order q was solved numerically using Matlab’s function

**fde_pi1_ex**written by Garrappa [72]. Details on the convergence and accuracy of the numerical scheme (15) and the one implemented in

**fde_pi1_ex**are given in [70] and [72] respectively.

## 3. Results

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AD | Alzheimer’s disease |

${A}_{p}$ | Proliferative reactive astrocytes (in activated state) |

${A}_{q}$ | Quiescent astrocytes (in resting state) |

$A\beta $ | Amyloid-$\beta $ |

${M}_{1}$ | Activated microglia in pro-inflammatory state |

${M}_{2}$ | Activated microglia in anti-inflammatory state |

${N}_{s}$ | Surviving neurons |

${N}_{d}$ | Dead neurons |

## References

- Pospich, S.; Raunser, S. The molecular basis of Alzheimer’s plaques. Science
**2017**, 358, 45–46. [Google Scholar] [CrossRef] [PubMed] - Nebel, R.A.; Aggarwal, N.T.; Barnes, L.L.; Gallagher, A.; Goldstein, J.M.; Kantarci, K.; Mallampalli, M.P.; Mormino, E.C.; Scott, L.; Yu, W.H.; et al. Understanding the impact of sex and gender in Alzheimer’s disease: A call to action. Alzheimers Dement.
**2018**, 14, 1171–1183. [Google Scholar] [CrossRef] [PubMed] - Alzheimer’s Disease Facts and Figures. Available online: https://www.alz.org/alzheimers-dementia/facts-figures (accessed on 28 June 2022).
- Dementia. Available online: https://www.who.int/news-room/fact-sheets/detail/dementia (accessed on 28 June 2022).
- Wang, Y.; Mishra, A.; Brinton, R.D. Transitions in metabolic and immune systems from pre-menopause to post-menopause: Implications for age-associated neurodegenerative diseases. F1000Research
**2020**, 9, 68. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zarate, S.; Stevnsner, T.; Gredilla, R. Role of estrogen and other sex hormones in brain aging. Neuroprotection and DNA repair. Front Aging Neurosci.
**2017**, 9, 430. [Google Scholar] [CrossRef] [Green Version] - Mishra, A.; Brinton, R.D. Inflammation: Bridging age, menopause and APOEϵ4 genotype to Alzheimer’s disease. Front. Aging Neurosci.
**2018**, 10, 312. [Google Scholar] [CrossRef] [Green Version] - Liu, C.C.; Liu, C.C.; Kanekiyo, T.; Xu, H.; Bu, G. Apolipoprotein E and Alzheimer disease: Risk, mechanisms and therapy. Nat. Rev. Neurol.
**2013**, 9, 106–118. [Google Scholar] [CrossRef] [Green Version] - Dubal, D.B. Sex difference in Alzheimer’s disease: An updated, balanced and emerging perspective on differing vulnerabilities. Handb. Clin. Neurol.
**2020**, 175, 261–273. [Google Scholar] [CrossRef] - Pontifex, M.G.; Martinsen, A.; Saleh, R.N.M.; Harden, G.; Tejera, N.; Muller, M.; Fox, C.; Vauzour, D.; Minihane, A.-M. APOE4 genotype exacerbates the impact of menopause on cognition and synaptic plasticity in APOE-TR mice. FASEB J.
**2021**, 35, e21583. [Google Scholar] [CrossRef] - Corder, E.H.; Ghebremedhin, E.; Taylor, M.G.; Thal, D.R.; Ohm, T.G.; Braak, H. The biphasic relationship between regional brain senile plaque and neurofibrillary tangle distributions: Modification by age, sex, and APOE polymorphism. Ann. N. Y. Acad. Sci.
**2004**, 1019, 24–28. [Google Scholar] [CrossRef] - Delage, C.I.; Simoncicova, E.; Tremblay, M.E. Microglial heterogeneity in aging and Alzheimer’s disease: Is sex relevant? J. Pharmacol. Sci.
**2021**, 146, 169–181. [Google Scholar] [CrossRef] - Navakkode, S.; Gaunt, J.R.; Pavon, M.V.; Bansal, V.A.; Abraham, R.P.; Chong, Y.S.; Ch’ng, T.H.; Sajikumar, S. Sex-specific accelerated decay in time/activity-dependent plasticity and associative memory in an animal model of Alzheimer’s disease. Aging Cell.
**2021**, 20, e13502. [Google Scholar] [CrossRef] [PubMed] - Kim, Y.; Park, J.; Choi, Y.K. The role of astrocytes in the central nervous system focused on BK channel and heme oxygenase metabolites: A review. Antioxidants
**2019**, 8, 121. [Google Scholar] [CrossRef] [Green Version] - Wake, H.; Moorhouse, A.J.; Jinno, S.; Kohsaka, S.; Nabekura, J. Resting microglia directly monitor the functional state of synapses in vivo and determine the fate of ischemic terminals. J. Neurosci.
**2009**, 29, 3974–3980. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jung, Y.-J.; Chung, W.-S. Phagocytic roles of glial cells in healthy and diseased brains. Biomol. Ther.
**2018**, 26, 350–357. [Google Scholar] [CrossRef] [PubMed] - Dubal, D.B.; Wise, P.M. Estrogen and neuroprotection: From clinical observations to molecular mechanisms. Dialogues Clin. Neurosci.
**2002**, 4, 149–161. [Google Scholar] [CrossRef] - Brann, D.W.; Dhandapani, K.; Wakade, C.; Mahesh, V.B.; Khan, M.M. Neurotrophic and neuroprotective actions of estrogen: Basic mechanisms and clinical implications. Steroids
**2007**, 72, 381–405. [Google Scholar] [CrossRef] [Green Version] - What Are the Symptoms of High Estrogen? Available online: https://www.medicalnewstoday.com/articles/323280#treatment (accessed on 30 June 2022).
- Lynch, A.M.; Murphy, K.J.; Deighan, B.F.; O’Reilly, J.A.; Gun’ko, Y.K.; Cowley, T.R.; Gonzalez-Reyes, R.E.; Lynch, M.A. The impact of glial activation in the aging brain. Aging Dis.
**2010**, 1, 262–278. [Google Scholar] - Chun, H.; Marriott, I.; Lee, C.J.; Cho, H. Elucidating the interactive roles of glia in Alzheimer’s disease using established and newly developed experimental models. Front. Neurol.
**2018**, 9, 797. [Google Scholar] [CrossRef] [Green Version] - Jiwaji, Z.; Tiwari, S.S.; Aviles-Reyes, R.X.; Hooley, M.; Hampton, D.; Torvell, M.; Johnson, D.A.; McQueen, J.; Baxter, P.; Sabari-Sankar, K.; et al. Reactive astrocytes acquire neuroprotective as well as deleterious signatures in response to Tau and Aβ pathology. Nat Commun.
**2022**, 13, 135. [Google Scholar] [CrossRef] - Varnum, M.M.; Ikezu, T. The classification of microglial activation phenotypes on neurodegeneration and regeneration in Alzheimer’s disease brain. Arch. Immunol. Ther. Exp.
**2012**, 60, 251–266. [Google Scholar] [CrossRef] - Klohs, J. An integrated view on vascular dysfunction in Alzheimer’s disease. Neurodegener. Dis.
**2019**, 19, 109–127. [Google Scholar] [CrossRef] [PubMed] - Korte, N.; Nortley, R.; Attwell, D. Cerebral blood flow decrease as an early pathological mechanism in Alzheimer’s disease. Acta Neuropathol.
**2020**, 140, 793–810. [Google Scholar] [CrossRef] [PubMed] - Taylor, J.L.; Pritchard, H.A.T.; Walsh, K.R.; Strangward, P.; White, C.; Hill-Eubanks, D.; Alakrawi, M.; Hennig, G.W.; Allan, S.M.; Nelson, M.T.; et al. Functionally linked potassium channel activity in cerebral endothelial and smooth muscle cells is compromised in Alzheimer’s disease. Pharmacology
**2022**, 119, e2204581119. [Google Scholar] [CrossRef] - Wang, D.; Chen, F.; Han, Z.; Yin, Z.; Ge, X.; Lei, P. Relationship between amyloid-β deposition and blood-brain barrier dysfunction in Alzheimer’s disease. Front Cell Neurosci.
**2021**, 15, 695479. [Google Scholar] [CrossRef] - Prins, N.; Scheltens, P. White matter hyperintensities, cognitive impairment and dementia: An update. Nat. Rev. Neurol.
**2015**, 11, 157–165. [Google Scholar] [CrossRef] [PubMed] - Lohner, V.; Pehlivan, G.; Sanroma, G.; Miloschewski, A.; Schirmer, M.D.; Stocker, T.; Reuter, M.; Breteler, M.M.B. The relation between sex, menopause, and white matter hyperintensities: The Rhineland study. Neurology
**2022**. [Google Scholar] [CrossRef] - Alzheimer, A. Über einen eigenartigen schweren Erkrankungsprozeß der Hirnrincle. Neurol. Central.
**1906**, 25, 1134. [Google Scholar] - Pallitto, M.M.; Murphy, R.M. A mathematical model of the kinetics of beta-amyloid fibril growth from the denatured state. Biophys J.
**2001**, 81, 1805–1822. [Google Scholar] [CrossRef] [Green Version] - Helal, M.; Hingant, E.; Pujo-Menjouet, L.; Webb, G.F. Alzheimer’s disease: Analysis of a mathematical model incorporating the role of prions. J. Math. Biol.
**2014**, 69, 1207–1235. [Google Scholar] [CrossRef] [Green Version] - Craft, D.L.; Wein, L.M.; Selkoe, D.J. A mathematical model of the impact of novel treatments on the A beta burden in the Alzheimer’s brain, CSF and plasma. Bull. Math. Biol.
**2002**, 64, 1011–1031. [Google Scholar] [CrossRef] [PubMed] - Puri, I.K.; Li, L. Mathematical modeling for the pathogenesis of Alzheimer’s disease. PLoS ONE
**2010**, 5, e15176. [Google Scholar] [CrossRef] [PubMed] - Hao, W.; Friedman, A. Mathematical model on Alzheimer’s disease. BMC Syst. Biol.
**2016**, 10, 108. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bertsch, M.; Franchi, B.; Marcello, N.; Tesi, M.C.; Tosin, A. Alzheimer’s disease: A mathematical model for onset and progression. Math. Med. Biol.
**2017**, 34, 193–214. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pal, S.; Melnik, R. Nonlocal models in the analysis of brain neurodegenerative protein dynamics with application to Alzheimer’s disease. Sci. Rep.
**2022**, 12, 7328. [Google Scholar] [CrossRef] - Vosoughi, A.; Sadigh-Eteghad, S.; Ghorbani, M.; Shahmorad, S.; Farhoudi, M.; Rafi, M.A.; Omidi, Y. Mathematical models to shed light on amyloid-beta and tau protein dependent pathologies in Alzheimer’s disease. Neuroscience
**2020**, 424, 45–57. [Google Scholar] [CrossRef] [PubMed] - Jack, C.R., Jr.; Holtzman, D.M. Biomarker modeling of Alzheimer’s disease. Neuron
**2013**, 80, 1347–1358. [Google Scholar] [CrossRef] [Green Version] - Young, A.L.; Oxtoby, N.P.; Daga, P.; Cash, D.M.; Fox, N.C.; Ourselin, S.; Schott, J.M.; Alexander, D.C. A data-driven model of biomarker changes in sporadic Alzheimer’s disease. Brain J. Neurol.
**2014**, 137, 2564–2577. [Google Scholar] [CrossRef] [Green Version] - Macdonald, A.; Pritchard, D. A mathematical model of Alzheimer’s disease and the Apoe gene. ASTIN Bull.
**2000**, 30, 69–110. [Google Scholar] [CrossRef] [Green Version] - Hane, F.; Augusta, C.; Bai, O. A hierarchical Bayesian model to predict APOE4 genotype and the age of Alzheimer’s disease onset. PLoS ONE
**2018**, 13, e0200263. [Google Scholar] [CrossRef] - Perez, C.; Ziburkus, J.; Ullah, G. Analyzing and modeling the dysfunction of inhibitory neurons in Alzheimer’s disease. PLoS ONE
**2016**, 11, e0168800. [Google Scholar] [CrossRef] - Proctor, C.J.; Boche, D.; Gray, D.A.; Nicoll, J.A.R. Investigating interventions in Alzheimer’s disease with computer simulation models. PLoS ONE
**2013**, 8, e73631. [Google Scholar] [CrossRef] [Green Version] - Hadjichrysanthou, C.; Ower, A.K.; de Wolf, F.; Anderson, R.M. Alzheimer’s disease neuroimaging initiative. The development of a stochastic mathematical model of Alzheimer’s disease to help improve the design of clinical trials of potential treatments. PLoS ONE
**2018**, 13, e0190615. [Google Scholar] [CrossRef] [Green Version] - Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives; Gordon and Breach: Yverdon, Switzerland, 1993. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Mathematics in Science and Engineering 198; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific: River Edge, NJ, USA, 2000. [Google Scholar]
- Oldham, K.B.; Spanier, J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order; Dover: Mineola, NY, USA, 2006. [Google Scholar]
- Tarasov, V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media; Springer: Heidelberg, Germany, 2010. [Google Scholar]
- Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models; Imperial College Press: London, UK, 2010. [Google Scholar]
- Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J.J. Fractional Calculus: Models and Numerical Methods; Series on Complexity, Nonlinearity and Chaos 3; World Scientific: Hackensack, NJ, USA, 2012. [Google Scholar]
- West, B.J. Fractional Calculus View of Complexity Tomorrow’S Science; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- West, B.J. Nature’s Patterns and the Fractional Calculus; Series Fractional Calculus in Applied Sciences and Engineering 2; De Gruyter: Berlin, Germany, 2017. [Google Scholar]
- Evangelista, L.R.; Lenzi, E.K. Fractional Diffusion Equations and Anomalous Diffusion; Cambridge University Press: Cambridge, UK, 2018. [Google Scholar]
- Yang, X.-Y.; Yang, G.J. General Fractional Derivatives with Applications in Viscoelasticity; Academic Press: Cambridge, MA, USA, 2020. [Google Scholar]
- Patnaik, S.; Hollkamp, J.P.; Semperlotti, F. Applications of variable-order fractional operators: A review. Proc. R. Soc. A
**2020**, 476, 20190498. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wangersky, P.J. Lotka-Volterra population models. Ann. Rev. Ecol. Syst.
**1978**, 9, 189–218. [Google Scholar] [CrossRef] - Cushing, J.M. Volterra integrodifferential equations in population dynamics. In Mathematics of Biology; Iannelli, M., Ed.; CIME Summer Schools 80; Springer: Berlin/Heidelberg, Germany, 2010; pp. 81–148. [Google Scholar]
- Ruan, S. Delay differential equations in single species dynamics. In Delay Differential Equations and Applications; Arino, O., Hbid, M.L., Ait Dads, E., Eds.; NATO science series. Series II, Mathematics, physics, and chemistry 205; Springer: Dordrecht, The Netherlands, 2006; pp. 477–517. [Google Scholar]
- Roberts, R.O.; Aakre, J.A.; Kremers, W.K.; Vassilaki, M.; Knopman, D.S.; Mielke, M.M.; Alhurani, R.; Geda, Y.E.; Machulda, M.M.; Coloma, P.; et al. Prevalence and outcomes of amyloid positivity among persons without dementia in a longitudinal, population-based detting. JAMA Neurol.
**2018**, 75, 970–979. [Google Scholar] [CrossRef] - Sturchioa, A.; Dwivedic, A.K.; Youngd, C.B.; Malme, T.; Marsilia, L.; Sharmaa, J.S.; Mahajana, A.; Hilla, E.J.; Andaloussif, S.E.L.; Postond, K.L.; et al. High cerebrospinal amyloid-β 42 is associated with normal cognition in individuals with brain amyloidosis. E. Clin. Med.
**2021**, 38, 100988. [Google Scholar] [CrossRef] - Song, Y.; Li, S.; Li, X.; Chen, X.; Wei, Z.; Liu, Q.; Cheng, Y. The effect of estrogen replacement therapy on Alzheimer’s disease and Parkinson’s disease in post-menopausal women: A meta-analysis. Front. Neurosci.
**2020**, 14, 157. [Google Scholar] [CrossRef] - Vinogradova, Y.; Dening, T.; Hippisley-Cox, J.; Taylor, L.; Moore, M.; Coupland, C. Use of menopausal hormone therapy and risk of dementia: Nested case-control studies using QResearch and CPRD databases. BMJ
**2021**, 374, n2182. [Google Scholar] [CrossRef] - Lorenzo, C.F.; Hartley, T.T. Variable order and distributed order fractional operators. Nonlinear Dyn.
**2002**, 29, 57–98. [Google Scholar] [CrossRef] - Sun, H.; Chen, W.; Wei, H.; Chen, Y. A comparative study of constant-order and variable- order fractional models in characterizing memory property of systems. Eur. Phys. J. Spec. Top.
**2011**, 193, 185–192. [Google Scholar] [CrossRef] - Ramirez, L.; Coimbra, C. On the selection and meaning of variable order operators for dynamic modeling. Int. J. Diff. Equ.
**2010**, 2010, 846107. [Google Scholar] [CrossRef] [Green Version] - De Barros, L.C.; Lopes, M.M.; Pedro, F.S.; Esmi, E.; dos Santos, J.P.C.; Sanchez, D.E. The memory effect on fractional calculus: An application in the spread of COVID-19. Comput. Appl. Math.
**2021**, 40, 72. [Google Scholar] [CrossRef] - When Does Menopause Start? Understanding the Symptoms by Age. Available online: https://www.healthpartners.com/blog/menopause-symptoms-by-age/ (accessed on 26 June 2022).
- Moghaddam, B.P.; Machado, J.A.T. Extended algorithms for approximating variable order fractional derivatives with applications. J. Sci. Comput.
**2017**, 71, 1351–1374. [Google Scholar] [CrossRef] - Lawson, C.L.; Hanson, R.J. Solving Least Squares Problems; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1995; p. 161. [Google Scholar]
- Garrappa, R. Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics
**2018**, 6, 16. [Google Scholar] [CrossRef] [Green Version] - Almeida, R.; Malinowska, A.B.; Odzijewicz, T. On systems of fractional differential equations with the ψ-Caputo derivative and their applications. Math. Meth. Appl. Sci.
**2021**, 44, 8026–8041. [Google Scholar] [CrossRef] - Almeida, R.; Torres, D.F.M. An expansion formula with higher-order derivatives for fractional operators of variable order. Sci. World J.
**2013**, 2013, 915437. [Google Scholar] [CrossRef] [Green Version] - Hale, J. Functional Differential Equations; Applied Mathematical Sciences 3; Springer: New York, NY, USA, 1971. [Google Scholar]
- Berezansky, L.; Braverman, E. On the existence of positive solutions for systems of differential equations with a distributed delay. Comput. Math. Appl.
**2012**, 63, 1256–1265. [Google Scholar] [CrossRef] [Green Version] - Berezansky, L.; Braverman, E. On nonoscillation and stability for systems of differential equations with a distributed delay. Automatica
**2012**, 48, 612–618. [Google Scholar] [CrossRef] - Teschl, G. Ordinary Differential Equations and Dynamical Systems; Graduate Studies in Mathematics 140; American Mathematical Society: Providence, RI, USA, 2012. [Google Scholar]
- Pooseh, S.; Almeida, R.; Torres, D.F.M. A numerical scheme to solve fractional optimal control problems. Conf. Pap. Sci.
**2013**, 2013, 165298. [Google Scholar] [CrossRef] - Rivers-Auty, J.; Mather, A.E.; Peters, R.; Lawrence, C.B.; Brough, D. Anti-inflammatories in Alzheimer’s disease-potential therapy or spurious correlate? Brain Commun.
**2020**, 2, fcaa109. [Google Scholar] [CrossRef] - Arevalo-Rodriguez, I.; Smailagic, N.; Roqué-Figuls, M.; Ciapponi, A.; Sanchez-Perez, E.; Giannakou, A.; Pedraza, O.L.; Bonfill Cosp, X.; Cullum, S. Mini-Mental State Examination (MMSE) for the early detection of dementia in people with mild cognitive impairment (MCI). Cochrane Database Syst Rev.
**2021**, 7, CD010783. [Google Scholar] [CrossRef] [PubMed] - Jack, C.R., Jr.; Knopman, D.S.; Jagust, W.J.; Shaw, L.M.; Aisen, P.S.; Weiner, M.W.; Petersen, R.C.; Trojanowski, J.Q. Hypothetical model of dynamic biomarkers of the Alzheimer’s pathological cascade. Lancet Neurol.
**2010**, 9, 119–128. [Google Scholar] [CrossRef] [Green Version] - Makin, S. The amyloid hypothesis on trial. Nature
**2018**, 559, S4–S7. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Furman, J.L.; Sama, D.M.; Gant, J.C.; Beckett, T.L.; Murphy, M.P.; Bachstetter, A.D.; Van Eldik, L.J.; Norris, C.M. Targeting astrocytes ameliorates neurologic changes in amouse model of Alzheimer’s disease. J Neurosci.
**2012**, 32, 16129–16140. [Google Scholar] [CrossRef] [Green Version] - Chou, R.C.; Kane, M.; Ghimire, S.; Gautam, S.; Gui, J. Treatment for rheumatoid arthritis and risk of Alzheimer’s disease: A nested case-control analysis. CNS Drugs
**2016**, 30, 1111–1120. [Google Scholar] [CrossRef] [Green Version] - Furman, D.; Campisi, J.; Verdin, E.; Carrera-Bastos, P.; Targ, S.; Franceschi, C.; Ferrucci, L.; Gilroy, D.W.; Fasano, A.; Miller, G.W.; et al. Chronic inflammation in the etiology of disease across the life span. Nat. Med.
**2019**, 25, 1822–1832. [Google Scholar] [CrossRef] - Yaribeygi, H.; Atkin, S.L.; Pirro, M.; Sahebkar, A. A review of the anti-inflammatory properties of antidiabetic agents providing protective effects against vascular complications in diabetes. J. Cell Physiol.
**2019**, 234, 8286–8294. [Google Scholar] [CrossRef] - Munoz-Jimenez, M.; Zaarkti, A.; Garcia-Arnes, J.A.; Garcia-Casares, N. Antidiabetic drugs in Alzheimer’s disease and mild cognitive impairment: A systematic review. Dement. Geriatr. Cogn. Disord.
**2020**, 49, 423–434. [Google Scholar] [CrossRef] - Chowen, J.A.; Garcia-Segura, L.M. Role of glial cells in the generation of sex differences in neurodegenerative diseases and brain aging. Mech. Ageing Dev.
**2021**, 196, 111473. [Google Scholar] [CrossRef]

**Figure 1.**Plots of the variable fractional orders $q\left(t\right)$ modeling neuroprotection loss in post-menopausal females (${q}_{f}\left(t\right)$ given by expression (10)) and males (${q}_{m}\left(t\right)$ given by expression (11)) due to AD progression with age, and therapy (${q}_{treat}\left(t\right)$ given by formula (12)) for females starting 4 years after the AD onset.

**Figure 2.**Temporal variations of populations of $A\beta $ fibrils for (

**a**) variable fractional orders $q\left(t\right)$ and (

**b**) constant fractional orders $0<q\le 1$.

**Figure 3.**Temporal variations of populations of (

**a**) surviving neurons ${N}_{s}$, and (

**b**) dead neurons ${N}_{d}$, for variable fractional orders $q\left(t\right)$ modeling females’ and, respectively, males’ neuroprotection loss due to the AD progression with age.

**Figure 4.**Temporal variations of populations of glial cells: (

**a**) pro-inflammatory microglia ${M}_{1}$, (

**b**) anti-inflammatory microglia ${M}_{2}$, (

**c**) proliferative reactive astrocytes ${A}_{p}$, and (

**d**) quiescent astrocytes ${A}_{q}$ for variable fractional orders $q\left(t\right)$ modeling female’s and, respectively, male’s neuroprotection loss due to the AD progression with age.

**Figure 5.**Temporal variations of populations of (

**a**) surviving neurons ${N}_{s}$, and (

**b**) dead neurons ${N}_{d}$, for variable fractional orders $q\left(t\right)$ modeling neuroprotection loss in post-menopausal females due to AD progression with age and therapy given to females starting 4 years after the AD onset.

**Figure 6.**Temporal variations of populations of glial cells: (

**a**) pro-inflammatory microglia ${M}_{1}$, (

**b**) anti-inflammatory microglia ${M}_{2}$, (

**c**) proliferative reactive astrocytes ${A}_{p}$, and (

**d**) quiescent astrocytes ${A}_{q}$ for variable fractional orders $q\left(t\right)$ modeling neuroprotection loss in post-menopausal females due to AD progression with age and therapy given to females starting 4 years after the AD onset.

**Figure 7.**Temporal variations of populations of (

**a**) surviving neurons ${N}_{s}$, and (

**b**) dead neurons ${N}_{d}$, for variable fractional orders $q\left(t\right)$ modeling neuroprotection loss in post-menopausal females due to AD progression with age and therapies given to females starting 2 years, 4 years, or 10 years after the AD onset.

**Figure 8.**Temporal variations of populations of (

**a**) surviving neurons ${N}_{s}$, and (

**b**) dead neurons ${N}_{d}$, for constant fractional orders $0<q\le 1$.

**Figure 9.**Temporal variations of populations of glial cells: (

**a**) pro-inflammatory microglia ${M}_{1}$, (

**b**) anti-inflammatory microglia ${M}_{2}$, (

**c**) proliferative reactive astrocytes ${A}_{p}$, and (

**d**) quiescent astrocytes ${A}_{q}$ for constant fractional orders $0<q\le 1$.

Rate (Pathway) | Value [ ${\mathit{y}\mathit{e}\mathit{a}\mathit{r}}^{-\mathit{q}\left(\mathit{t}\right)}$ ] | Rate (Pathway) | Value [ ${\mathit{y}\mathit{e}\mathit{a}\mathit{r}}^{-\mathit{q}\left(\mathit{t}\right)}$ ] |
---|---|---|---|

${\alpha}_{1}$ (${A}_{q}\to {N}_{s}$) | ${10}^{-5}$ | ${\alpha}_{10}$ (${N}_{d}\to {M}_{1}$) | ${10}^{-2}$ |

${\alpha}_{2}$ (${A}_{p}\to {N}_{d}$) | ${10}^{-3}$ | ${\alpha}_{11}$ (${N}_{s}\u22a3{M}_{1}$) | ${10}^{-2}$ |

${\alpha}_{3}$ (${M}_{1}\to {N}_{d}$) | ${10}^{-2}$ | ${\alpha}_{12}$ (${A}_{q}\u22a3{M}_{1}$) | ${10}^{-4}$ |

${\alpha}_{4}$ (${M}_{2}\to {A}_{q}$) | ${10}^{-4}$ | ${\alpha}_{13}$ ($A\beta \to {M}_{1}$) | ${10}^{-2}$ |

${\alpha}_{5}$ (${M}_{1}\to {A}_{p}$) | ${10}^{-2}$ | ${\alpha}_{14}$ (${M}_{2}\u22a3{M}_{1}$) | ${10}^{-4}$ |

${\alpha}_{6}$ (${N}_{s}\to {M}_{2}$) | ${10}^{-2}$ | ${\alpha}_{15}$ (${N}_{s}\to A\beta $) | 1 |

${\alpha}_{7}$ (${A}_{q}\to {M}_{2}$) | ${10}^{-4}$ | ${\alpha}_{16}$ (${M}_{2}\u22a3A\beta $) | ${10}^{-2}$ |

${\alpha}_{8}$ ($A\beta \u22a3{M}_{2}$) | ${10}^{-2}$ | ${\alpha}_{r}$ (${M}_{2}\u22a3A\beta $) | 1 |

${\alpha}_{9}$ (${M}_{1}\u22a3{M}_{2}$) | ${10}^{-2}$ |

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Drapaca, C.S.
A Mathematical Investigation of Sex Differences in Alzheimer’s Disease. *Fractal Fract.* **2022**, *6*, 457.
https://doi.org/10.3390/fractalfract6080457

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Drapaca CS.
A Mathematical Investigation of Sex Differences in Alzheimer’s Disease. *Fractal and Fractional*. 2022; 6(8):457.
https://doi.org/10.3390/fractalfract6080457

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2022. "A Mathematical Investigation of Sex Differences in Alzheimer’s Disease" *Fractal and Fractional* 6, no. 8: 457.
https://doi.org/10.3390/fractalfract6080457