Stationary distribution and extinction of a stochastic Alzheimer’s disease model

: In this paper, a stochastic Alzheimer’s disease model with the e ﬀ ect of calcium on amyloid beta is proposed. The Lyapunov function is constructed, followed by the feasibility and positivity and the existence of a stationary distribution for the positive solutions of the proposed model. The su ﬃ cient conditions for the extinction of the stochastic Alzheimer’s disease model are derived through the Lyapunov function. This indicates that beta-amyloid plaque and the complex of beta-amyloid oligomers with prion protein may go extinct and there is a possibility of a cure for the disease. Furthermore, our numerical simulations show that as the intensity of the random disturbance increases, the time it takes for the disease to go extinct decreases.


Introduction
Alzheimer's disease (AD), a major form of dementia, is accompanied by cognitive decline, memory impairment, impaired ability to learn new information and language dysfunction. As one of the top 10 causes of death worldwide today, it will severely affect the daily life of the patient [1]. According to the global burden of disease study in 2019 (GBD 2019), the number of people with Alzheimer'slike dementia has 50 million in 2018 and it will reach 152 million by 2050 [2]. In the USA, total payments for medicare, long-term care, and hospice services for dementia are estimated to be up to $335 billion in 2021 [3]. With no reliable and effective treatment, dementia will affect the patient's ability to perform daily live by impairing cognitive function and pose an increasing challenge to health care systems worldwide [4][5][6].
In the earliest phase of Alzheimer's disease (cellular phase), amoid beta (Aβ) accumulate in the brain, along with the spread of tau pathology [7]. The peptide Aβ, obtained by amyloid precursor protein (APP), can form Aβ oligomers (two main Aβ forms, Aβ 40 and Aβ 42 ), which will reduce the number of synapses and decrease glucose metabolism in the brain. This process will finally lead to brain atrophy [8]. To discuss how Aβ peptide aggregates into Aβ oligomers, Masoud Hoore et al. [9] developed a model of Aβ fibrillation on a minimal scale. The results showed that Aβ monomers rapidly increased once Aβ oligomers produced. Furthermore, by considered Aβ 40 and Aβ 42 as two forms of Aβ oligomers, Li and Zhao [10] proved that the targeted therapeutic drug Aducanumab of Aβ cannot completely cure AD. However, many studies have found that the prion protein (PrP C ) inhibits the activity of the protease that cleaves APP and slows the proliferation of Aβ [11,12]. To understand the dynamics of PrP C , Helal et al. [13] devised an in vitro model to study the role of protein and analyzed the kinetics of Aβ plaques, Aβ oligomers, PrP C and Aβ−x−PrP C complex. Considering the process of diffusion of these substances, Hu et al. [14] focused on the dynamic behaviors of the system in a finite time interval and under what conditions the state value may exceed a certain value.
Various factors are involved in the transmission of neural signals. For example, in the cerebrospinal fluid, the level of Aβ oligomers is affected by Ca 2+ , microglia activity, reactive oxygen species and Na + concentration etc. [15][16][17][18]. For example, Caluwé and Dupont [19] designed a positive loop between Aβ and Ca 2+ to explore the role of Ca 2+ on Aβ oligomers during the progression of a healthy pathological state to a severe pathology. All the factors always fluctuate in a small range over long periods which will affect the level of Aβ oligomers and the pathological status of AD. Therefore, stochastic perturbations cannot be ignored and parameters are often assumed in biomathematics to be perturbed by linear functions of white noise, a phenomenon described by stochastic differential equations (SDE) [20][21][22][23]. Hu et al. [24] formulated a stochastic model of the in vivo progression of AD incorporating the role of prions derived from Helal et al. [13] and discussed the existence of the ergodic stationary distribution of the model.
Studies have been done on minimizing the concentrations of Aβ plaques and Aβ−x−PrP C complex in Alzheimer's disease models, but the conditions are complex and not well measured in many practical situations [14]. And the random factors in the interstitial fluid (ISF) cannot be neglected, therefore it is necessary to study stochastic models of Alzheimer's disease to explore the convergence to extinction in a probabilistic sense. For this purpose, we introduce calcium ions into the system based on Helal et al. [12] and consider the effect of random noise on Brownian motion in the environment. The main contributions of this paper are as follows: (i) A stochastic Alzheimer's disease model is formulated by taking the influence of calcium ions and environmental noise on Aβ oligomers into account. (ii) The sufficient conditions for extinction of the model are established.
The remaining paper is organized as follows. In the next section, the mathematical model of Alzheimer's disease with Ca 2+ is established. Section 3 shows the existence, uniqueness and boundedness of the solution of the model. The conditions for the existence of a steady state distribution are derived in Section 4. Section 5 focuses on the threshold conditions for the extinction of plaques and complex and shows how random noise affects the development of Alzheimer's disease. In Section 6, a numerical simulation is performed to prove the validity of the theoretical derivation. In the ending section, we present our conclusion.

Mathematical model
In this section, we introduce the model and then give the necessary definitions and lemmas.

Model formation
To explore the role of prions in memory impairment, Helal et al. [13] introduced a mathematical model of in vivo Alzheimer's disease progression that explains the relationship between Aβ plaque, Aβ oligomers, PrP C and Aβ−x−PrP C complex. The model is as follows Where A(t), u(t), p(t) and b(t) represent the concentration of Aβ plaque, Aβ oligomers, PrP C and Aβ−x−PrP C complex. Where α is the rate of formation of oligomers, η is the rate of degradation of a plaque, τ is the rate of binding of Aβ oligomers to PrP C , σ is the rate of unbinding of Aβ−x−PrP C , ρ is the conversion rate of oligomers to plaque, k i (i = 2, 3, 4) is the degradation of Aβ oligomers, PrP C and Aβ−x−PrP C complex, λ i (i = 2, 3) is the source of PrP C and Aβ oligomers.
In this paper, by considering that the presence of PrP C can optimize and control Ca 2+ input [11] and this process is affected by the level of Ca 2+ , it can be assumed to be a bilinear model [25,26]. Furthermore, there is positive feedback between the level of Ca 2+ and the level of Aβ [19], so Ca 2+ is introduced into the model. Moreover, due to the randomness of real life, especially in the neurobiological environment, there exist various random factors involved in signaling. In many stochastic models of infectious diseases, factors such as noise, Brownian motion, pollution, etc. have been considered [27][28][29]. Then, we assume that the white noise in the environment is proportional to the variables C(t), u(t), p(t), b(t), and A(t). The stochastic differential model can be written as Where λ 1 is the source of Ca 2+ , v 2 is the acceleration of Ca 2+ due to Aβ, v 3 is the limitation of Ca 2+ due to PrP C , k 1 is the degradation of Ca 2+ , v 1 is the maximal rate of the positive feedback of Ca 2+ on Aβ and k is half-saturation constant, B i (t) denote independent and standard Brownian motions and ξ 2 i are the intensities of the white noise, i = 1, 2, 3, 4, 5. The other parameters in model (2.2) have identical significance as in model (2.1). Our main purpose is to explore the threshold related to epidemic transmission and try to establish the threshold dynamics of model (2.2).

Preliminaries
Throughout this paper, we let (Ω, F , {F } l 0 , P) be a complete probability space with filtration {F } l 0 satisfying the usual conditions (that is to say, it is increasing and right continuous while F 0 contains all P -null sets). Let B i (t)(i = 1, 2, 3...) denote the independent standard Brownian motions defined on this probability space. We also denote R d + = x ∈ R d : x i > 0 for all 1 i d and a ∧ b = min{a, b}. Generally speaking, consider the d-dimensional stochastic differential equation (SDE) where f (t, x(t)) is a function in R d defined in [t 0 , ∞] × R d and g(x(t), t) is a d × m matrix, f , g are locally Lipschitz functions in x. B t denotes an m-dimensional standard Brownian motion (B t = (B 1 (t), B 2 (t), ..., B m (t)) T , B i (t)(i = 1, 2, ..., m) is standard normal distribution and B i (t) ∼ N(0, t)) defined on the complete probability space (Ω, F , {F } t≥0 , P). Denote by , ∞] such that they are continuously twice differentiable in x and once in t. We define the differential operator L of Eq (2.3) by [30] Here are some definitions and lemmas what we will use in the following text.
, where p(t, x) means the probability density function of x(t) at t.

Definition 2.
[30] For a set Ω k composed of elementary random events ω, the indicator function of Ω k , denoted by 1 Ω k , is the random variable, where 3) is said to be stochastically ultimately bounded if for any ε ∈ (0, 1), there is a positive constant χ = χ(ε) such that for any initial data If for any 0 < ε < 1, there exists a pair of positive constants θ = θ(ε) and χ = χ(ε) such that then the species i is said to be stochastically permanent.

Lemma 1.
[23] (Chebychev inequality) Let X = {X t } t 0 be a nonnegative random variable, its mean value is noted as E(X), for a given r > 0. Then, Lemma 2. [21,28] The Markov process X(t) has a unique ergodic stationary distribution µ(·) if there exists a bound D ⊂ R d with regular boundary Γ and the following conditions: (1) In the domain D and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix A(x) is bounded away from zero.
(2) There exists a nonnegative C 2 -function V such that LV is negative for any R d \D. Then, for all x ∈ R d , where f is a function integrable with respect to the measure µ.

Existence and uniqueness of the solution
Theorem 1. For any initial value there exists a positive salutation X(t) = (C(t), u(t), p(t), b(t), A(t)) of the stochastic model (2.2) for t 0 and the solution will hold in R 5 + with probability one. Proof. We can easily know that the coefficients of model (2.2) are locally Lipschitz continuous. Then, for any given initial value (C(0), u(0), p(0), b(0), A(0)) ∈ R 5 + , there exists a unique local solution (C(0), u(0), p(0), b(0), A(0)) on t ∈ [0, τ e ), where τ e is the explosion time (see [20]). To prove that the solution is global, all you have to do is to prove τ e = ∞ almost surely. Let k 0 0 be sufficiently large so that (C(0), u(0), p(0), b(0), A(0)) all lie within the interval 1 k 0 , k 0 . For each integer k k 0 , define the following stopping time: Where throughout this paper, we set inf ∅ = ∞ (as usual ∅ denotes the empty set). According to the definition of the stopping time, τ k is increasing as Namely, we need to show that τ ∞ = ∞ almost surely. If τ ∞ ∞, we assumed that there exists a pair of constants T > 0 and ∈ (0, 1) such that As a result, there is an integer k 1 k 0 such that where m i (i = 1, 2, 3, 4) are positive constants to be determined below. Then, by using the Itô's formula, we have Choosing And there exists a constant K such that LV K, where K is define as follows Integration of the above inequality from 0 to τ k ∧ T and taking the expectation on both sides, we get the following inequality Now, we set Ω k = {τ k T }, k k 1 . It follows from the inequality (3.1) that P (Ω k ) ε. Note that for each ω ∈ Ω k , C (τ k , ω), u (τ k , ω), p (τ k , ω), b (τ k , ω), A (τ k , ω) equals either k or 1 k . Consequently, where 1 Ω k is the indicator function of Ω k . Letting k → ∞ leads to This is a contradiction. As a consequence, τ ∞ = ∞ a.s. The proof is completed.
Proof. For facilitate calculation, define N = nA + mC + u + p + 2b, choosing Λ = By using the Itô's formula, we have Then, by a similar proof of Theorem 4.3 in literature [32] we can get the X(t) of model (2.2) is Vgeometrically ergodic. And through a simple calculation we have It follows that Note that, That means, Thus, According to Definition 3 and Definition 6, model (2.2) is stochastically ultimately bounded and permanent. The proof is completed.

The stationary distribution of Alzheimer's disease model
In this section, we will consider whether there is a unique stationary distribution of the model (2.2) that allows the disease to persist rather than die off. Theorem 3. If there exist constants c i (i = 1, 2, 3) such that inequality (4.1) holds then for any initial value X(0) = (C(0), u(0), p(0), b(0), A(0)) ∈ R 5 + , the model (2.2) admits a unique stationary distribution µ(·) and it has the ergodic feature.
Proof. According to Lemma 5, the diffusion matrix of model (2.2) is given by for any (C, u, p, b, A) ∈ D, θ = (θ 1 , θ 2 , θ 3 , θ 4 , θ 5 ) ∈ R 5 + . Then the condition (1) in Lemma 2 is satisfied. To prove condition (2) of Lemma 2 is fulfilled, we need to develop a non-negative C 5 −function V: To do this, we first define By using the Itô's formula in the proposed model (2.2), we obtain Therefore, we have In addition, we can obtain For the sake of simplicity, we define Also, Applying the Itô's formula and using the proposed model, we get The next step is to define the set where 0 < < 1 is a constant that is sufficiently small and satisfies the following Eq (4.2) We divide the domain R 5 + \D into the ten regions is follows In what follows, we will prove that LW(C, u, p, b, A) < 0, for any (C, u, p, b, A) ∈ R 5 + .
We divide the proof into ten cases.
Including the analysis from Cases 1 to 10, we can derive that LW(C, u, p, b, A) < 0, for any (C, u, p, b, A) ∈ R 5 + . Consequently, condition (2) in Lemma 2 is satisfied. This finishes the proof.

Stochastic extinction dynamics
In this section we are going to discuss under what conditions the disease will be extinct, for convenient, we define X(t) = 1 t t 0 x(r)dr, and define another threshold parameter as follows: , Theorem 4. If R 1 < 1 and R 2 < 1 hold, the b(t) and A(t) will die out with probability one, moreover Proof. By using the Itô's formula to the equation of model (2.2), we can get Integration both sides of the equation above from 0 to t, we get By simple calculation, we can obtain where the value of φ 1 (t) is defined via the subsequent equation With the large number theorem as stated in Lemma 3 and local martingales, lim t→∞ φ 1 (t) = 0. Similarly, we also can get where φ 2 (t) is defined by Similarly, lim t→∞ φ 2 (t) = 0.
Likewise, we integrate both sides of the last two equations of the proposed model (2.2), yielding these equations and With a simple calculation, we can get Clearly, lim t→∞ φ 3 (t) = 0.
In the same way, by applying the Itô's formula to the last equation of model (2.2), we can obtain, Obviously, Therefore when R 2 < 1, we obtain It implies that, lim t→∞ nA(t) = 0, a.s.
This completes the proof.
Remark 1. Theorem 4 reveals that the extinction or not of the disease depends on the sign of R 1 and R 2 . With R i < 1(i = 1, 2), both the Aβ oligomers and Aβ-x-PrP C complex incline to go extinct. That is, stochastic perturbations of the environment are beneficial to the extinction of both materials. This means that in real life, it is useful to pay attention to the physical condition of the patient and improve the internal environment of the body [33]. A more interesting result is that such random perturbations may lead to disease extinction. This provides a theoretical basis for disease cure.

Numerical simulations
To illustrate the theoretical results obtained, we give some examples in this section. Using the Milstein's higher order method developed in [34], we present our results. Let us consider the corresponding discretizing equations, Where j,i j = 1, 2, 3, 4, 5 are the realization of five independent Gaussian random variables with distribution N(0, 1) and time step ∆t = 0.01. Using MATLAB, numerical simulations were performed on the proposed stochastic Alzheimer's disease model (2.2) and an approximate solution of the model is obtained. In addition, it is shown that noise intensity has a significant influence. By assuming numerical values of the parameters related to their biological feasibility, we verified the extinction of the disease and the existence of a stationary distribution.
Next, based on the previous assumptions, we change λ 1 , v 1 , ξ 1 , k 2 , ξ 2 , λ 3 , ξ 3 , η, ξ 4 , ξ 5 to be λ 1 = 0.02, v 1 = 0.08, ξ 1 = 2.8, k 2 = 3, ξ 2 = 4, λ 3 = 0.85, ξ 3 = 5, η = 0.12, ξ 4 = 0.6 and ξ 5 = 1.6. We can easily calculate the basic reproduction number R 1 = 0.8556 < 1 and R 2 = 0.2357 < 1. And according to Theorem 4 the solution of model (2.2) must obey This means that the disease will die out in this case and the numerical simulation of Figure 2 confirms our theoretical results. Figure 2 shows that the stochastic equation (2.2) and the deterministic equation have differences in their behavior. By this, we can point out that the disease tends towards the extinction with environmental noise. The numerical simulation shows that the surrounding noise have a very large effect on the mentioned disease. That is, the environmental interference will cause the Aβ plaque and Aβ−x−PrP C complex to disappear.
Finally, to simulate the effect of different intensities of environmental interference, we fix the parameters above except ξ 4 and ξ 5 . We change the values of ξ 4 and ξ 5 in Figure 3. As the intensity of white noise increases, Aβ plaques and Aβ-x-PrP C complex will accelerate extinction.

Conclusions
During neural signaling, the concentration of Aβ is influenced by a number of stochastic factors. For example, calcium ions can regulate of Aβ levels in the interstitial fluid (ISF) by affecting the permeability of the cell membrane. We established a random Alzheimer's disease model containing Ca 2+ and investigated the transmission dynamics with changing biological environment. Using the stochastic Lyapunov functions theory, the existence and positivity were proved. The extinction and the stationary distribution were also discussed, the related conditions implied that the random parameters such as the random of Ca 2+ concentration will lead to disease's extinction. In contrast to the optimal control conditions proposed by Hu et al. [14], this paper directly derives more explicit and simple conditions for the extinction of Aβ plaques and Aβ−x−PrP C complex, which will form the basis in formulating novel therapeutic solutions for control strategies regarding AD pathology. In the future, the model can be further extended by adding drugs. One can also talk about the drug-target kinetics of the model by adding drugs and the influence of toxicological effects of drugs on therapeutic efficacy.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.