Mathematical analysis of neurological disorder under fractional order derivative

: Multiple sclerosis (MS) is a common neurological disorder that a ff ects the central nervous system (CNS) and can cause lesions that spread over space and time. Our study proposes a mathematical model that illustrates the progression of the disease and its likelihood of recurrence. We use Caputo fractional-order (FO) derivative operators to represent non-negative solutions and to establish a steady-state point and basic reproductive number. We also employ functional analysis to prove the existence of unique solutions and use the Ulam-Hyres (UH) notion to demonstrate the stability of the solution for the proposed model. Furthermore, we conduct numerical simulations using an Euler-type numerical technique to validate our theoretical results. Our findings are presented through graphs that depict various behaviors of the model for di ff erent parameter values.


Introduction and motivation
Multiple sclerosis (MS) is an autoimmune neurological disorder that affects the central nervous system (CNS) and primarily damages oligodendryocytes, which are myelin producing cells. This damage leads to dysfunction of neurons in the brain and spinal cord, causing various neurological symptoms. It is commonly accepted that MS is an unpredictable illness that can have permanent effects on the brain, spinal cord, and optic nerves. Symptoms may include difficulty with movement, cloudy vision, and further neurological issues. The severity and deductability of signs can vary greatly from patient to patient, ranging from subtle cases that require no medical intervention to severe cases that considers the specific uncertainties in space and time. We recall the following model proposed in [17] described below.
Let P(t) represent the ratio of normal (healthy) brain cells (NBC), I(t) represent the infected brain cells (IBC), and Q(t) represent the damaging (either immune or viral effectors) brain cells (DBC).
In addition, let γ denote the growth rate of NBC. Based on scientific observations, either viruses or immune cells contribute to the development of this disease. Therefore, we assume that infected cells I(t) release the virus Q(t) at a rate of δ and expire at a rate of α. The virus Q(t) is influenced by NBC P(t) at a rate of d and expires at a rate of κ. We assume that d is small and the control parameters β, γ, δ belong to that domain where P, I, Q, are non-negative. Therefore, the mathematical system can be expressed in the following form [17]: (1. 1) In the field of epidemiology, there has been a recent trend of using fractional-order (FO) derivatives and integral operators in mathematical models. These operators are more flexible than traditional deterministic models and have been a focus of current research in mathematics. Shah et al. investigated Pine Wilt disease and COVID-19 models under FO [18,19]. Beside this, FO operators have been used in the analysis and applications of delay differential equations. For instance, Xu et al. demonstrated a FO 4D neural network including two different delays [20] and three-triangle multi-delayed neural networks [21]. Additionally, we suggest two works on FO delay differential equations [22,23].
FO differential equations involve either fractional-order derivatives or integral operators that depend on past states as well as current states, making them a powerful tool for predicting future states. They are more effective than classical deterministic operators, and Caputo and Riemann-Liouville operators are commonly used in fractional-order differential equations. There are several applications of differential equations in science and technology. For instance, several authors have used FO to analyze wave behaviours in mathematical physics problems such as Schrodinger's equation [24], biophysics [25], nonlinear evolution equations [26], fractional wave equations [27], time-fractional Fornberg-Whitham and Klein-Gordon equations [28], and fluid dynamics [29]. FO can be applied to bifurcations and chaos analyses of dynamical systems. The advantage of FO over the integer order operator in chaos analysis is that it provides more advanced features in chaotic dynamical systems that are not attainable at integer order. Here, we give some applications of FO in bifurcation and chaos analysis, such as: 4D-hyperchaotic system [30], coupled memristive model [31], 4D-memristive system [32] and chaos in predator prey model [33].
In epidemiology, integer operators are basic tools for modelling the dynamics of a disease [34]. However, FO has been used by many researchers for an improved prediction and analysis of numerous disease models, such as a tumor-immune model [35], bacteria dependent disease [36], a COVID-19 model [37], Buruli ulcers [38], a COVID-19 model for a case study in Saudi Arabia [39], HBV and HCV co-infection models [40], a Vector-host disease model [41], a compartmental disease model [42], and many more [43,44].
In the next section, we propose a simple mathematical model consisting of three simultaneous fractional differential equations that can replicate the clinical presentation of MS. In developing our model, we draw upon previous research on Ulam-Hyers stability and the instability of steady state solutions and pattern formation. We derive conditions for the existence, uniqueness, and stability of steady states both at equilibrium and away from equilibrium. We then use numerical simulation techniques to verify our theoretical results. Finally, in the concluding section, we summarize our findings and conclude our study.
The structure of the paper is as follows: In Section 2, we introduce the fundamental concepts of fractional calculus. In Section 3, we present the primary findings of our research, comprised of a theoretical and numerical analysis of the proposed model. In Section 4, we present the numerical simulations. Finally, in Section 5, we provide the concluding remarks.

Preliminaries
Here, we provide a definition of some fractional operators.

Main work
Here, the classical derivative in the model (1.1) is replaced by the Caputo fractional-order RRMS model, described as: along with initial conditions: P(0) = P 0 , I(0) = I 0 , Q(0) = Q 0 . Since model (3.1) is nonlinear and lacks a time-dependent explicit solution, it is studied over a prolonged time period. By setting the derivatives of model (3.1) equal to zero, we obtain the following equilibria: , also, the basic reproduction number is given as:

Non-negativity of solutions
This portion focuses on the existence of a non-negative solution for the fractional order RRMS model. It is evident from biological considerations that each state variable in the model (3.1) represents a non-negative real-valued function. Stated differently, (P (t) , First, we will demonstrate that all solutions of the model (3.1) are non-negative.
Using fractional integral, we will get the solution of the above systems. The solution will be nonnegative since there is no negative term in the system. □

Existence and uniqueness theorems
In this part, we will find the existence of a solution for the considered FO RRMS model (3.1) via the concepts of the fixed-point theory. Define a norm as: , where (c [0, T]) denotes the Banach space of real value continues mapping on T in the related supremum norm. Let us express the model (3.1) as: By applying the definition of fractional integral of model (3.2) and using the properties of fractional calculus we get where t ∈ [0, T] and 0 < µ ≤ 1 with the following theorem, we will assume that ∥ P (t) ∥≤ c 1 , ∥ I (t) ∥≤ c 2 , and ∥ Q (t) ∥≤ c 3 where c i , i = 1, 2, 3 are positive constants Theorem 3.2. Assume that 0 ≤ max {m 1 , m 2 , m 3 } < 1, then we show that kernels Q 1 , Q 2 , Q 3 agrees to fulfill the Lipschitz constraint and are contractions under some mappings.
Proof. Consider P 1 and P 2 be any functions, then we have Therefore, the Lipschitz condition is fulfilled. Similarly, the kernels Q 2 , Q 3 , can be fined using {I 1 , In addition, since 0 ≤ max {m 1 , m 2 , m 3 } < 1, the kernel are contractions. Recursively, system (3.3) can be given as Using (3.5), we obtain the following formation: The difference between the successive terms of model (3.2) in the recursive form is defined as: with the initial condition P 0 (t) = P (0) , I 0 (t) = I (0) , Q 0 (t) = Q (0) . By using the norm of the equation, we have According to the Lipschitz condition (3.4), we get Thus, we have Similarly, the Lipschitz property for the remaining equation of the model can be obtained with The properties of the kernel, as shown in Eq (3.4), have been established and holds. By analysing the remainder of Eqs (3.7) and (3.8), and applying the recursive access, we get the output as: Therefore, the existing and given sequence follows ∥Φ 1n (t)∥ → 0, ∥Φ 2n (t)∥ → 0 and ∥Φ 3n (t)∥ → 0 as n → ∞. Therefore, at least one solution exists for the considered model.
We propose the following results, which guarantees the unique solution of model (3.2). To prove the uniqueness of the solution, considering Eq (3.9) and taking triangular inequality for any i = 1, 2, 3, we have where 1 Γ(1−µ) m 1 p 1−µ < 1 by assertion and m i = 1 Γ(1−µ) m 1 p 1−µ n , i = 1, 2, 3. Therefore, the sequences P n , I n , Q n behave as Cauchy sequences in M (k), and as such, they converge uniformly. By applying the limit theory to Eq (3.10) as n → ∞, the limit of these sequences gives a unique solution to model (3.1). Thus, the existence of a unique solution for model (3.1) has been proven. □

Ulam-Hyers stability
This section will cover the various forms of UH stability (UHS) and gives definitions for some UHS criteria for the proposed system. Let K ∈ A be any solution of the considered FO RRMS model (3.1), and ε > 0 and L Z : [0, T] × R 3 → R + shows a continuous function. We are presented with a set of inequalities: where t ∈ [0, T] and ε = max(ε i ) T for i = 1, 2, 3.
Definition 3.4. The given FO RRMS model (3.1) will be UHS if ∀ ε > 0, and ∀ solution M ∈ A of system (3.3) ∃ J Z > 0 such that Definition 3.6. The given RRMS model (3.1) will be a UHRS if ∃ U H Z > 0 such that Definition 3.7. The considered RRMS model will be a generalized UHRS if ∃ U H Z > 0 such that where t ∈ [0, T].
• c D µ t M(t) = Z(t, M(t)) + Q(t). Remark 3.9. M ∈ A be a solution of (3.12) if and only if there exist H ∈ A, the following properties hold: To concentrate our discussion on the Ulam's stabilities of the proposed model, we will first establish some necessary results. Additionally, we introduce an assumption that could prove useful in our subsequent analysis. We will assume that: Lemma 3.10. For M ∈ A, given inequality below holds: Proof. Since M ∈ A satisfies (3.11), according to the 2nd condition of Remark 3.8, one can conclude that (3.14) By utilizing the fractional integral, we can obtain the solution of system (3.14), which is expressed as follows: Based on the initial assumption, we can use the first condition of Remark 3.8 to obtain the following result: Hence, this ends the proof. □ Lemma 3.11. If M ∈ A satisfies (3.12), then Proof. Let M ∈ A be the solution of (3.12). Using the second condition of Remark 3.9, we can express it as follows: In the context of first part of Remark 3.9, we can express the above equation as: This ends the proof. □ Now, we are ready to verify the UH and Ulam-Hyers-Rassias (UHR) stability of the proposed RRMS model (3.1).
Proof. Let M ∈ A be any solution of Eq (3.11) and K ∈ A represent a unique solution of the proposed model (3.5), via Eq (3.6) and Lemma 3.10, one has After simplification, we obtain |M(t) − K(t)| ≤ εJ Z , where . Therefore, the result of the UHS is achieved. Hence, the RRMS system (3.5) is a UHRS. Next, set H Z (ε) = εJ Z so that H Z (0) = 0. Hence, the considered FO RRMS system is a generalized UHS. □ The given system exhibits a UHRS and a generalized UHRS according to the following theorem. Proof. Let M ∈ A be any solution of (3.13), and K ∈ A denotes a unique solution of (3.5) via Eq (3.6) along with Lemma 3.11, one achieves After simplification, one achieves . By supposing: , we get the required result: Consequently, the proposed system is a UHRS. Next, by setting ε = 1 in Eq (3.15) along with H Z (0) = 0, the suggested RRMS system is a generalized UHRS. □

Numerical results of RRMS model via Euler method
Finding exact or explicit solutions for FO differential equations remains a challenging task in computational and applied mathematics. In this section, we propose an approach to obtain approximate solutions for the FO model using fractional Euler's methods. These techniques focus on either the primary Taylor's function or a power series expansion. The strategy for the fractional Euler's numerical solution will be discussed.
By taking an IVP as (3.16) where c = t 0 , t 1 , ..., t n = p such that t i = c + i ▽ . Let j = 0, 1, 2, ..., n, and ▽ = p−c n . Suppose that C D µ t S(t) and C D 2µ t S(t) are continuous functions on c, p . Utilizing Taylor , s expansion for i = 0, 1, 2, ..., n, one reaches: For a small enough time interval, ▽, the term ▽ 2µ (3.17) can be omitted. Therefore, from the Eq (3.17), we acquire the numerical method in the form of: Agarwal et al. utilized Euler's methods to develop a similar numerical scheme that provides the solution for Eq (3.16) in the following form: The discrete Eqs (3.18) and (3.19) differ only in the denominator of their right-hand sides. In this paper, we utilize the numerical method presented in Eq (3.19) and can now employ the previous scheme for a simplified computation. We employ the numerical approximation scheme Eq (3.19), and each of the equations in the model (3.1). First, we consider the equation for 0 < µ < 1, t > 0, with the initial values S (0) = S 0 = 0.4, I (0) = I 0 = 0.3, Q (0) = Q 0 = 0.1, the total population N = 0.8. Suppose g (t, S(t)) = γP(1 − P) − βPQ, then we define C D µ t P(t) = g (t, S(t)) with S 0 = 0.4, 0 < µ < 1, t > 0. Next, utilizing the numerical scheme in Eq (3.19), one obtains where g (t i , S(t i )) is described as , and define C D µ t I(t i ) = g (t, I(t i )) where I 0 = 0.3, 0 < µ < 1, t > 0. Utilizing the numerical solution scheme in Eq (3.10), we obtain where g (t i , I(t i )) is described as for i = 0, 1, 2, ..., n − 1. Lastly, Utilizing the numerical scheme in Eq (3.10), we obtain where g (t i , Q(t i )) is described as for i = 0, 1, 2, ..., n − 1.

Numerical simulations
This section presents a simulation of the proposed mathematical model to analyze the behavior of three classes of a system for different cases. The initial conditions for these cases are chosen as   The simulation results reveal that the system exhibits oscillatory dynamics of the considered disease, and this behavior is presented in Figure 3, where sub-figures (a) − (c) depict the evolution of each class of the proposed model for a few FOs.
Furthermore, we analyze the chaotic dynamics of the proposed model in Figures 4 and 5. We choose different values of FO to illustrate the memory and heredity features of the model. From these simulations, we observe that the evolution of the disease varies as the value of the FO changes. We note that the model achieves stability more rapidly at lower FOs compared to higher orders. Moreover, the graphs of each class get closer to the dynamics of the integer order model as the FO increases. Therefore, we conclude that the proposed model is more advanced and generalized than the classical model.
From a biological perspective, mathematical models are an important tool to study the dynamics of diseases and can help to understand how they spread and how they can be controlled. The proposed model is used to simulate the behavior of a disease affecting brain cells. The simulations provide insight into how the disease progresses under different conditions, such as at the DFE point, the EE point, and with oscillatory dynamics. We also analyze the chaotic dynamics of the model and how the disease evolution changes as the value of the FO changes. These findings could provide valuable information for developing strategies to control and treat the disease.

Conclusions and future recommendation
In this research, we have explored a mathematical model to assess the level of disease progression and recurrence. To provide non-negative solutions, we have expanded the model by including Caputo fractional-order derivative operators. Additionally, we have also established steady-state points and basic reproductive number. Moreover, we have identified the necessary conditions for the existence of a unique fractional system. Different forms of UHS have been demonstrated to show that the solution of the considered FO system is a UHS. To obtain an approximate solution for the FO system, we have used an effective Euler-type numerical technique.
The study presents the results of three different cases, where the behavior of the model is analyzed for a few fractional values. We have concluded that the proposed system is more generalized than the classical model, as it provides the behavior of the model for fractional values. Overall, this research contributes to the understanding of disease progression and recurrence by providing a mathematical model that can be utilized to simulate and analyze the behavior of disease models. The inclusion of Caputo fractional-order derivative operators allows for a more comprehensive analysis of the model's stability and recurrence, making it a valuable tool for researchers and health care professionals.
In the near future, we will study the proposed model using different operators and concepts, such as fractal fractional operators, fuzzy operators, and stochastic concepts.

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