Analysis of fractal-fractional Alzheimer's disease mathematical model in sense of Caputo derivative

Abstract

Alzheimer's disease stands as one of the most widespread neurodegenerative conditions associated with aging, giving rise to dementia and posing significant public health challenges. Mathematical models are considered as valuable tools to gain insights into the mechanisms underlying the onset, progression, and potential therapeutic approaches for AD. In this paper, we introduce a mathematical model for AD that employs the fractal fractional operator in the Caputo sense to characterize the temporal dynamics of key cell populations. This model encompasses essential elements, including amyloid-β ($\mathbb{ A_\beta }$), neurons, astroglia and microglia. Using the fractal fractional operator, we have established the existence and uniqueness of solutions for the model under consideration, employing Leray-Schaefer's theorem and the Banach fixed-point methods. Utilizing functional techniques, we have analyzed the proposed model stability under the Ulam-Hyers condition. The suggested model has been numerically simulated by using a fractional Adams-Bashforth approach, which involves a two-step Lagrange polynomial. For numerical simulations, different ranges of fractional order values and fractal dimensions are considered. This new fractal fractional operator in the form of the Caputo derivative was determined to yield better results than an ordinary integer order. Various outcomes are shown graphically by for different fractal dimensions and arbitrary orders.

Keywords: Alzheimer's disease; Ulam-Hyers stability; fractal-fractional derivatives; fractional Adams-Bashforth method.

Figure 1.. Flowchart of the model.

Figure 2.. Dynamical variation of proliferative reactive astrocytes of model (2.1) when F-D q and F-O p are equal (p, q ∈ [0, 1]).

Figure 3.. Dynamical variation of proliferative reactive astrocytes of model (2.1) for different F-D q and F-O p (p, q ∈ [0, 1]).

Figure 4.. Dynamical variation of quiescent (resting) astrocytes of model (2.1) when F-D q and F-O p are equal (p, q ∈ [0, 1]).

Figure 5.. Dynamical variation of quiescent (resting) astrocytes of model (2.1) for different F-D q and F-O p (p, q ∈ [0, 1]).

Figure 6.. Dynamical variation of aggregation-prone amyloid-β fibrils of model (2.1) when F-D q and F-O p are equal (p, q q ∈ [0, 1]).

Figure 7.. Dynamical variation of aggregation-prone amyloid-β fibrils of model (2.1) for different F-D q and F-O p (p, q ∈ [0, 1]).

Figure 8.. Dynamical variation of activated microglia in anti-inflammatory state of model (2.1) when F-D q and F-O p are equal (p, q ∈ [0, 1]).

Figure 9.. Dynamical variation of activated microglia in anti-inflammatory state of model (2.1) for different F-D q and F-O p (p, q ∈ [0, 1]).

Figure 10.. Dynamical variation of activated microglia in pro-inflammatory state of model (2.1) when F-D q and F-O p are equal (p, q ∈ [0, 1]).

Figure 11.. Dynamical variation of activated microglia in pro-inflammatory state of model (2.1) for different F-D q and F-O p (p, q ∈ [0, 1]).

Figure 12.. Dynamical variation of surviving neurons of model (2.1) when F-D q and F-O p are equal (p, q ∈ [0, 1]).

Figure 13.. Dynamical variation of surviving neurons of model (2.1) for different F-D q and F-O p (p, q ∈ [0, 1]).

Figure 14.. Dynamical variation of dead neurons of model (2.1) when F-D q and F-O p are equal (p, q ∈ [0, 1]).

Figure 15.. Dynamical variation of dead neurons of model (2.1) for different F-D q and F-O p (p, q ∈ [0, 1]).