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

1.   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's-like 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 fanxiexian_myfh335 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 ($A\beta$) accumulate in the brain, along with the spread of tau pathology ^[7]. The peptide $A\beta$, obtained by amyloid precursor protein (APP), can form $A\beta$ oligomers (two main $A\beta$ forms, $A\beta_{40}$ and $A\beta_{42}$), 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 $A\beta$ peptide aggregates into $A\beta$ oligomers, Masoud Hoore et al. ^[9] developed a model of $A\beta$ fibrillation on a minimal scale. The results showed that $A\beta$ monomers rapidly increased once $A\beta$ oligomers produced. Furthermore, by considered $A\beta_{40}$ and $A\beta_{42}$ as two forms of $A\beta$ oligomers, Li and Zhao ^[10] proved that the targeted therapeutic drug Aducanumab of $A\beta$ 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 $A\beta$ ^[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 $A\beta$ plaques, $A\beta$ oligomers, $PrP^{C}$ and $A\beta-$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 $A\beta$ 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 $A\beta$ and $Ca^{2+}$ to explore the role of $Ca^{2+}$ on $A\beta$ 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 $A\beta$ 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 $A\beta$ plaques and $A\beta-$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 $A\beta$ 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.

2.   Mathematical model

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

2.1.   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 $A\beta$ plaque, $A\beta$ oligomers, $PrP^{C}$ and $A\beta-$x$-PrP^{C}$ complex. The model is as follows

Where $A(t), \:u(t), \:p(t)$ and $b(t)$ represent the concentration of $A\beta$ plaque, $A\beta$ oligomers, $PrP^{C}$ and $A\beta-$x$-PrP^{C}$ complex. Where $\alpha$ is the rate of formation of oligomers, $\eta$ is the rate of degradation of a plaque, $\tau$ is the rate of binding of $A\beta$ oligomers to $PrP^{C}$, $\sigma$ is the rate of unbinding of $A\beta-$x$-PrP^{C}$, $\rho$ is the conversion rate of oligomers to plaque, $k_{i}(i = 2, 3, 4)$ is the degradation of $A\beta$ oligomers, $PrP^{C}$ and $A\beta-$x$-PrP^{C}$ complex, $\lambda_{i}(i = 2, 3)$ is the source of $PrP^{C}$ and $A\beta$ 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 $A\beta$ ^[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 $\lambda_{1}$ is the source of $Ca^{2+}$, $v_{2}$ is the acceleration of $Ca^{2+}$ due to $A\beta$, $v_{3}$ 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 $A\beta$ and $k$ is half-saturation constant, $B_{i}(t)$ denote independent and standard Brownian motions and $\xi_{i}^{2}$ 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).

2.2.   Preliminaries

Throughout this paper, we let $\left(\Omega, \mathcal{F}, \{\mathcal{F}\}_{l \geqslant 0}, \mathbb{P}\right)$ be a complete probability space with filtration $\{\mathcal{F}\}_{l \geqslant 0}$ satisfying the usual conditions (that is to say, it is increasing and right continuous while $\mathcal{F}_{0}$ contains all $\mathbb{P}$ -null sets). Let $B_{i}(t)(i = 1, 2, 3...)$ denote the independent standard Brownian motions defined on this probability space. We also denote $\mathbb{R}_{+}^{d} = \left\{x \in \mathbb{R}^{d}: x_{i} > 0 \text{ for all } 1 \leqslant i \leqslant d \right\}$ and $a \wedge b = \min\{a, b\}$.

Generally speaking, consider the d-dimensional stochastic differential equation (SDE)

where $f(t, x(t))$ is a function in $\mathbb{R}^{d}$ defined in $\left[t_{0}, \infty\right]\times \mathbb{R}^{d}$ and $g(x(t), t)$ is a $d \times m$ 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 $B_{i} (t)\sim N (0, t))$ defined on the complete probability space $\left(\Omega, \mathcal{F}, \{\mathcal{F}\}_{t \geq 0}, \mathbb{P}\right)$. Denote by $C^{2, 1}\left(\mathbb{R}^{d} \times\right. \left.\left[t_{0}, \infty\right]; \mathbb{R}_{+}\right)$ the family of all nonnegative functions $V(x(t), t)$ defined on $\mathbb{R}^{d} \times\left[t_{0}, \infty\right]$ such that they are continuously twice differentiable in $x$ and once in $t$.

We define the differential operator $\mathrm{\; L}$ of Eq (2.3) by ^[30]

If $\mathrm{\; L}$ acts on a function $V \in C^{2, 1}\left(\mathbb{R}^{d} \times\left[t_{0}, \infty\right], \mathbb{R}_{+}\right)$ then

where $V_{t}(x, t) = \frac{\partial V}{\partial t}$, $V_{x}(x, t) = \left(\frac{\partial V}{\partial x_{i}}, \ldots, \frac{\partial V}{\partial x_{d}}\right), V_{x x}(x, t) = \left(\frac{\partial^{2} V}{\partial x_{i} \partial x_{j}}\right) _{d \times d}$. From Itô's formula, if $x(t) \in \mathbb{R}^{d}$, then

Here are some definitions and lemmas what we will use in the following text.

Definition 1. ^[21] (Fokker-Plank equation) The respective Fokker-Plank equation for an unknown probability density function (PDF) in variables $x(t)$ can be assigned to Eq (2.3):

where $p(t, x)$ means the probability density function of $x(t)$ at $t$.

Definition 2. ^[30] For a set $\Omega_{k}$ composed of elementary random events $\omega$, the indicator function of $\Omega_{k}$, denoted by $1_{\Omega_{k}}$, is the random variable, where

Definition 3. ^[31] The SDE (2.3) is said to be stochastically ultimately bounded if for any $\varepsilon \in(0, 1)$, there is a positive constant $\chi = \chi(\varepsilon)$ such that for any initial data $\{x(t):-\tau \leqslant t \leqslant 0\} \in C\left([-\tau, 0]; \mathbb{R}_{+}^{d}\right)$, the solution $x(t)$ of Eq (2.3) has the property that

Definition 4. ^[30] For the Markov process $\{X(t), t\geqslant0\}$, the state space is $S = \{1, 2, ..., T\}$, if there exists a positive integer $m$ such that

then $X(t)$ has the ergodic feature.

Definition 5. ^[23,24,27] The diffusion matrix of system (2.3) is defined as follows:

Definition 6. ^[23] Let $N(t) = \left(N_{i}(t)\right)^{T}\: (i = 1, 2, ..., d)$ be the solution of model (2.2) with initial value $N(0) \in \mathbb{R}_{+}^{d}$. If for any $0 < \varepsilon < 1$, there exists a pair of positive constants $\theta = \theta(\varepsilon)$ and $\chi = \chi(\varepsilon)$ such that

then the species i is said to be stochastically permanent.

Definition 7. ^[20,22,28] For model (2.3), the infected individuals $x_{i}(t)$ are said to be extinctive if $\lim \limits_{t \rightarrow \infty} x_{i}(t) = 0$, almost surely (a.s.).

Lemma 1. ^[23] (Chebychev inequality) Let $X = {\{X_{t}\}}_ {t\geqslant0}$ be a nonnegative random variable, its mean value is noted as $\mathbb{E}(X)$, for a given $r > 0$. Then,

Lemma 2. ^[21,28] The Markov process $X(t)$ has a unique ergodic stationary distribution $\mu(\cdot)$ if there exists a bound $D \subset \mathbb{R}^{d}$ with regular boundary $\Gamma$ 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 $\mathrm{\; L} V$ is negative for any $\mathbb{R}_{d} \backslash D$. Then,

for all $x \in \mathbb{R}^{d}$, where $f$ is a function integrable with respect to the measure $\mu$.

Lemma 3. ^[20] (Strong Law of Large Number) Let $M = \{M\}_{t \leqslant 0}$ be continuous and real-valued local martingale, which vanishes as $t \rightarrow 0$, then

3.   Existence and uniqueness of the solution

Theorem 1. For any initial value

there exists a positive salutation $X(t) = \left(C(t), u(t), p(t), b(t), A(t)\right)$ of the stochastic model (2.2) for $t \geqslant 0$ and the solution will hold in $\mathbb{R}_{+}^{5}$ 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 $(C(0), u(0), p(0), b(0), A(0)) \in \mathbb{R}_{+}^{5}$, there exists a unique local solution $(C(0), u(0), p(0), b(0), A(0))$ on $t \in\left[0, \tau_{e}\right)$, where $\tau_{e}$ is the explosion time (see ^[20]). To prove that the solution is global, all you have to do is to prove $\tau_{e} = \infty$ almost surely. Let $k_{0} \geqslant 0$ be sufficiently large so that $(C(0), u(0), p(0), b(0), A(0))$ all lie within the interval $\left[\frac{1}{k_{0}}, k_{0}\right]$. For each integer $k \geqslant k_{0}$, define the following stopping time:

Where throughout this paper, we $\operatorname{set} \inf \emptyset = \infty$ (as usual $\emptyset$ denotes the empty set). According to the definition of the stopping time, $\tau_{k}$ is increasing as $k \rightarrow \infty$. Set $\tau_{\infty} = \lim \limits_{t \rightarrow \infty} \tau_{k}$ with $\tau_{\infty} \leqslant \tau_{e}$ almost surely. Namely, we need to show that $\tau_{\infty} = \infty$ almost surely. If $\tau_{\infty} \neq\infty$, we assumed that there exists a pair of constants $T > 0$ and $\epsilon \in(0, 1)$ such that

As a result, there is an integer $k_{1} \geqslant k_{0}$ such that

Now, we define a $C^{5}$-function $V(C, u, p, b, A) \in \mathbb{R}_{+}^{5}$ as follows

where $m_{i}(i = 1, 2, 3, 4)$ are positive constants to be determined below. Then, by using the Itô's formula, we have

where

Choosing

such that

And there exists a constant $K$ such that $L V \leqslant K$, where $K$ is define as follows

Integration of the above inequality from 0 to $\tau_{k} \wedge T$ and taking the expectation on both sides, we get the following inequality

Now, we set $\Omega_{k} = \left\{\tau_{k} \leqslant T\right\}$, $k \geqslant k_{1}$. It follows from the inequality (3.1) that $P\left(\Omega_{k}\right) \geqslant \varepsilon$. Note that for each $\omega \in \Omega_{k}$, $C \left(\tau_{k}, \omega\right)$, $u \left(\tau_{k}, \omega\right)$, $p \left(\tau_{k}, \omega\right)$, $b \left(\tau_{k}, \omega\right)$, $A \left(\tau_{k}, \omega\right)$ equals either $k$ or $\dfrac{1}{k}$. Consequently,

From (3.2) and (3.3) we get

where $1_{\Omega_{k}}$ is the indicator function of $\Omega_{k}$. Letting $k \rightarrow \infty$ leads to

This is a contradiction. As a consequence, $\tau_{\infty} = \infty$ a.s. The proof is completed. □

Theorem 2. For any initial value $X(0) = \left(C(0), u(0), p(0), b(0), A(0)\right) \in \mathbb{R}_{+}^{5}$, the solutions of the model (2.2) are stochastically ultimately bounded and permanent.

Proof. For facilitate calculation, define $N = n A+m C+u+p+2 b$, choosing $\Lambda = \min\left\{ \eta, k_{1}, k_{2}-m v_{2}, k_{3}+m v_{3}, k_{4} \right\}$, $0 < m < \min \left\{ \dfrac{k_{2}}{v_{2}}, \dfrac{k_{3}}{v_{3}} \right\}$ and define

By using the Itô's formula, we have

where

Then, by a similar proof of Theorem 4.3 in literature ^[32] we can get the $X(t)$ of model (2.2) is V-geometrically ergodic. And through a simple calculation we have

It follows that

Thus, $\lim \limits_{t \rightarrow \infty} \sup \mathbb{E}[V(t)] \leqslant H$, we chose a constant $\chi$ which is sufficiently large, such that $\dfrac{H}{\chi} < 1$. By using Chebychev inequality in Lemma {1},

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. □

4.   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

the model (2.2) admits a unique stationary distribution $\mu(\cdot)$ and it has the ergodic feature.

Proof. According to Lemma 5, the diffusion matrix of model (2.2) is given by

Choose

we can get that

for any $(C, u, p, b, A) \in D, \:\theta = (\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}, \theta_{5})\in \mathrm{\; R}_{+}^{5}$. 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 $V$: $\mathrm{\; R}_{+}^{5} \rightarrow \mathrm{\; R}$. To do this, we first define

By using the Itô's formula in the proposed model (2.2), we obtain

Therefore, we have

Let

In addition, we can obtain

For the sake of simplicity, we define

Also,

Now we define a $C^{5}$-function $V(C, u, p, b, A) \in \mathbb{R}_{+}^{5}$ as follows

Applying the Itô's formula and using the proposed model, we get

The next step is to define the set

where $0 < \epsilon < 1$ is a constant that is sufficiently small and satisfies the following Eq (4.2)

We divide the domain $\mathbb{R}_{+}^{5} \backslash D$ into the ten regions is follows

Obviously, $\Theta_{\varepsilon}^{c} = \mathbb{R}_{+}^{5} / \Theta^{c} = D_{i = 1}^{10} \Theta_{i}$. In what follows, we will prove that

We divide the proof into ten cases.

Case 1: If $(C, u, p, b, A) \in D_{1}$, we can derive that

According to (4.2),

Case 2: If $(C, u, p, b, A) \in D_{2}$, we can derive that

According to (4.2),

Case 3: If $(C, u, p, b, A) \in D_{3}$, we can derive that

According to (4.2),

Case 4: If $(C, u, p, b, A) \in D_{4}$, we can derive that

According to (4.2),

Case 5: If $(C, u, p, b, A) \in D_{5}$, we can derive that

According to (4.2),

Case 6: If $(C, u, p, b, A) \in D_{6}$, we can derive that

According to (4.2),

Case 7: If $(C, u, p, b, A) \in D_{7}$, we can derive that

According to (4.2),

Case 8: If $(C, u, p, b, A) \in D_{8}$, we can derive that

According to (4.2),

Case 9: If $(C, u, p, b, A) \in D_{9}$, we can derive that

According to (4.2),

Case 10: If $(C, u, p, b, A) \in D_{10}$, we can derive that

According to (4.2),

Including the analysis from Cases 1 to 10, we can derive that

Consequently, condition (2) in Lemma 2 is satisfied. This finishes the proof. □

5.   Stochastic extinction dynamics

In this section we are going to discuss under what conditions the disease will be extinct, for convenient, we define $\langle X(t)\rangle = \dfrac{1}{t} \int_{0}^{t} x(r) d r$, 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 $\phi_{1}(t)$ is defined via the subsequent equation

With the large number theorem as stated in Lemma 3 and local martingales, $\lim \limits_{t \rightarrow \infty} \phi_{1}(t) = 0$. Similarly, we also can get

where $\phi_{2}(t)$ is defined by

Similarly, $\lim \limits_{t \rightarrow \infty} \phi_{2}(t) = 0$.

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

where

Clearly, $\lim \limits_{t \rightarrow \infty} \phi_{3}(t) = 0$.

By using the Itô's formula on the fourth equation of model (2.2), we have

Obviously,

Therefore when $R_{1} < 1$, we obtain

That implies that,

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,

That is to say

With $\langle p\rangle = \dfrac{\lambda_{3}}{k_{3}}-\dfrac{k_{4}}{k_{3}}\langle b\rangle+\phi_{3}(t)$ above, we can get that

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 $A\beta$ oligomers and $A\beta$-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.

6.   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 $\varpi_{j, i} \: j = 1, 2, 3, 4, 5$ are the realization of five independent Gaussian random variables with distribution $N(0, 1)$ and time step $\Delta t = 0.01$. 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.

First, we choose $\lambda_1 = 0.2$, $v_1 = 1$, $v_2 = 0.6$, $v_3 = 0.4$, $k_1 = 7$, $\xi_{1} = 0.1$, $k = 0.3$, $k_2 = 0.35$, $\rho = 0.5$, $\xi_{2} = 0.25$, $\lambda_3 = 0.5$, $k_3 = 0.2$, $\xi_{3} = 0.2$, $\tau = 0.85$, $\sigma = 0.6$, $\eta = 0.8$, $\alpha = 0.3$, $n = 3$, $\xi_{5} = 0.5$, $\xi_{4} = 0.1$. Furthermore, we consider the initial size of population density as $X(0) = \left(C(0), u(0), p(0), b(0), A(0)\right) = (0.2, 0.5, 0.5, 1.2, 1)$. These assumptions satisfy the Theorem 3, which implies that model (2.2) has a unique stationary distribution as shown in Figure 1 and means the disease will be persistent.

Next, based on the previous assumptions, we change $\lambda_1$, $v_1$, $\xi_{1}$, $k_2$, $\xi_{2}$, $\lambda_3$, $\xi_{3}$, $\eta$, $\xi_{4}$, $\xi_{5}$ to be $\lambda_1 = 0.02$, $v_1 = 0.08$, $\xi_{1} = 2.8$, $k_2 = 3$, $\xi_{2} = 4$, $\lambda_3 = 0.85$, $\xi_{3} = 5$, $\eta = 0.12$, $\xi_{4} = 0.6$ and $\xi_{5} = 1.6$. 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

and

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 $A\beta$ plaque and $A\beta-$x$-PrP^{C}$ complex to disappear.

Finally, to simulate the effect of different intensities of environmental interference, we fix the parameters above except $\xi_{4}$ and $\xi_{5}$. We change the values of $\xi_{4}$ and $\xi_{5}$ in Figure 3. As the intensity of white noise increases, $A\beta$ plaques and $A\beta$-x-$PrP^{C}$ complex will accelerate extinction.

7.   Conclusions

During neural signaling, the concentration of $A\beta$ is influenced by a number of stochastic factors. For example, calcium ions can regulate of $A\beta$ 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 $A\beta$ plaques and $A\beta-$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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 12271068, 11961024), the Group Building Scientific Innovation Project for Universities in Chongqing (No. CXQT21021), the Joint Training Base Construction Project for Graduate Students in Chongqing (No. JDLH PYJD2021016), the Innovation Project for Graduate Research in Chongqing Jiaotong University (No. 2023ST011).

Conflict of interest

The authors declare that they have no conflicts of interest related to this article.