Classification of Alzheimer's disease using robust TabNet neural networks on genetic data

Yu Jin; Zhe Ren; Wenjie Wang; Yulei Zhang; Liang Zhou; Xufeng Yao; Tao Wu; Yu Jin; Zhe Ren; Wenjie Wang; Yulei Zhang; Liang Zhou; Xufeng Yao; Tao Wu

doi:10.3934/mbe.2023366

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 8358-8374. doi: 10.3934/mbe.2023366

Previous Article Next Article

Research article

Classification of Alzheimer's disease using robust TabNet neural networks on genetic data

1.
College of Medical Imaging, Jiading District Central Hospital affiliated Shanghai University of Medicine and Health Sciences, Shanghai 201318, China
2.
School of Health Science and Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

Academic Editor: Yi Jiang
† These authors equally contributed to this study.

Received: 28 December 2022 Revised: 13 February 2023 Accepted: 16 February 2023 Published: 02 March 2023

Alzheimer's disease (AD) is one of the most common neurodegenerative diseases and its onset is significantly associated with genetic factors. Being the capabilities of high specificity and accuracy, genetic testing has been considered as an important technique for AD diagnosis. In this paper, we presented an improved deep learning (DL) algorithm, namely differential genes screening TabNet (DGS-TabNet) for AD binary and multi-class classifications. For performance evaluation, our proposed approach was compared with three novel DLs of multi-layer perceptron (MLP), neural oblivious decision ensembles (NODE), TabNet as well as five classical machine learnings (MLs) including decision tree (DT), random forests (RF), gradient boosting decision tree (GBDT), light gradient boosting machine (LGBM) and support vector machine (SVM) on the public data set of gene expression omnibus (GEO). Moreover, the biological interpretability of global important genetic features implemented for AD classification was revealed by the Kyoto encyclopedia of genes and genomes (KEGG) and gene ontology (GO). The results demonstrated that our proposed DGS-TabNet achieved the best performance with an accuracy of 93.80% for binary classification, and with an accuracy of 88.27% for multi-class classification. Meanwhile, the gene pathway analyses demonstrated that there existed two most important global genetic features of AVIL and NDUFS4 and those obtained 22 feature genes were partially correlated with AD pathogenesis. It was concluded that the proposed DGS-TabNet could be used to detect AD-susceptible genes and the biological interpretability of susceptible genes also revealed the potential possibility of being AD biomarkers.

Keywords:

Citation: Yu Jin, Zhe Ren, Wenjie Wang, Yulei Zhang, Liang Zhou, Xufeng Yao, Tao Wu. Classification of Alzheimer's disease using robust TabNet neural networks on genetic data[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8358-8374. doi: 10.3934/mbe.2023366

Related Papers:

[1]	Abeer Alshareef . Quantitative analysis of a fractional order of the $SEI_{c}\, I_{\eta} VR$ epidemic model with vaccination strategy. AIMS Mathematics, 2024, 9(3): 6878-6903. doi: 10.3934/math.2024335
[2]	Turki D. Alharbi, Md Rifat Hasan . Global stability and sensitivity analysis of vector-host dengue mathematical model. AIMS Mathematics, 2024, 9(11): 32797-32818. doi: 10.3934/math.20241569
[3]	A. E. Matouk, Ismail Gad Ameen, Yasmeen Ahmed Gaber . Analyzing the dynamics of fractional spatio-temporal $\mathrm{SEIR}$ epidemic model. AIMS Mathematics, 2024, 9(11): 30838-30863. doi: 10.3934/math.20241489
[4]	Xinjie Zhu, Hua Liu, Xiaofen Lin, Qibin Zhang, Yumei Wei . Global stability and optimal vaccination control of SVIR models. AIMS Mathematics, 2024, 9(2): 3453-3482. doi: 10.3934/math.2024170
[5]	Rahman Ullah, Muhammad Farooq, Faiz Faizullah, Maryam A Alghafli, Nabil Mlaiki . Fractional stochastic functional differential equations with non-Lipschitz condition. AIMS Mathematics, 2025, 10(3): 7127-7143. doi: 10.3934/math.2025325
[6]	Muhammad Altaf Khan, Sajjad Ullah, Saif Ullah, Muhammad Farhan . Fractional order SEIR model with generalized incidence rate. AIMS Mathematics, 2020, 5(4): 2843-2857. doi: 10.3934/math.2020182
[7]	Yasir Nawaz, Muhammad Shoaib Arif, Wasfi Shatanawi, Muhammad Usman Ashraf . A new unconditionally stable implicit numerical scheme for fractional diffusive epidemic model. AIMS Mathematics, 2022, 7(8): 14299-14322. doi: 10.3934/math.2022788
[8]	Jutarat Kongson, Chatthai Thaiprayoon, Weerawat Sudsutad . Analysis of a mathematical model for the spreading of the monkeypox virus with constant proportional-Caputo derivative operator. AIMS Mathematics, 2025, 10(2): 4000-4039. doi: 10.3934/math.2025187
[9]	Minghung Lin, Yiyou Hou, Maryam A. Al-Towailb, Hassan Saberi-Nik . The global attractive sets and synchronization of a fractional-order complex dynamical system. AIMS Mathematics, 2023, 8(2): 3523-3541. doi: 10.3934/math.2023179
[10]	Mdi Begum Jeelani, Abeer S Alnahdi, Rahim Ud Din, Hussam Alrabaiah, Azeem Sultana . Mathematical model to investigate transmission dynamics of COVID-19 with vaccinated class. AIMS Mathematics, 2023, 8(12): 29932-29955. doi: 10.3934/math.20231531

Abstract

1. Introduction

Mathematics was initially utilized in biology in the $13^{th}$ century by Fibonacci, who developed the famed Fibonacci series to explain an increasing population. Daniel Bernoulli utilized mathematics to describe the impact on tiny shapes. Johannes Reinke coined the phrase "bio maths" in 1901. Biomathematics involves the theoretical examination of mathematical models to understand the principles governing the formation and functioning of biological systems.

Mathematical models are utilized to investigate specific questions related to the studied disease. For example, epidemiological models are important for predicting how infectious diseases spread, and help to control them by identifying important factors in the community. In this case, we want to analyze a particular model to understand how the COVID-19 virus behaves. This virus appeared in late 2019, and continues to be a global challenge. To gain a deeper insight into the underlying physical processes, we delve into the realm of fractional calculus. Previous literature has introduced various operators through the framework of fractional calculus ^[1,2]. In the field of C $^\eta$ -Calculus, Golmankhaneh et al. ^[3] provided an explanation of the Sumudu transform and Laplace transform. Additionally, in 2019, Goyal ^[4] proposed a fractional model that demonstrated the potential to manage the Lassa hemorrhagic fever disease.

Advancements in technology have led to significant progress in the field of epidemiology, enabling the examination of various infectious diseases for treatment, control, and cure ^[5]. It is essential to underscore that mathematical biology plays a pivotal role in investigating numerous diseases. Significant strides have been taken in the mathematical modeling of infectious diseases in recent decades, as evidenced by various studies ^[6,7]. Over the last thirty years, mathematical modeling has gained prominence in research, making substantial contributions to the development of effective public health strategies for disease control ^[8,9]. Mathematical models serve as invaluable tools for analyzing spatiotemporal patterns and the dynamic behavior of infections. Acknowledging their significance, researchers have approached the study of COVID-19 from diverse perspectives in the last three years ^[10,11].

Various methodologies have been employed by researchers in this field to devise successful techniques for managing this condition, with recent studies offering additional insights ^[12,13]. For example, a recent investigation utilized a mathematical model to evaluate the impacts of immunization in nursing homes ^[14]. Furthermore, researchers have explored mathematical modeling and effective intervention strategies for controlling the COVID-19 outbreak ^[15]. Additionally, some studies have delved into COVID-19 mathematical models using stochastic differential equations and environmental white noise ^[16].

In 2019, China experienced a notable outbreak of the coronavirus disease 2019 (COVID-19), prompting concerns about its potential to escalate into a worldwide pandemic ^[17]. Researchers from China, particularly Zhao et al., made significant contributions in addressing the challenges posed by COVID-19. This disease is attributed to the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), a viral infection. The initial verified case was documented in Wuhan, China, in December 2019 ^[18]. The infection rapidly spread worldwide, leading to the declaration of a COVID-19 pandemic.

Transmission occurs through various means, including respiratory droplets from coughing, sneezing, close contact, and touching contaminated surfaces. Key preventive measures include consistent mask usage, frequent hand washing, and maintaining safe interpersonal distances ^[19]. Effective interventions and real-time data play a crucial role in managing the coronavirus outbreak ^[20]. Prior studies have employed real-time analysis to comprehend the transmission of the virus among individuals, the severity of the disease, and the early stages of the pathogen, particularly in the initial week of the outbreak ^[21].

In December 2019, an outbreak of pneumonia cases was reported in Wuhan, initially with unidentified origins. Some cases were associated with exposure to wet markets and seafood. Chinese health authorities, in collaboration with the Chinese Center for Disease Control and Prevention (China CDC), initiated an investigation into the cause and spread of the disease on December 31, 2019 ^[22]. We conducted an analysis of temporal changes in the outbreak by examining the time interval between hospital admission dates and fatalities. Clinical studies on COVID-19 have indicated that symptoms typically manifest around 7 days after the onset of illness ^[23]. It is important to consider the duration between hospitalization and death in order to accurately assess the risk of mortality ^[24]. The information regarding the incubation period of COVID-19 and epidemiological data was sourced from publicly available records of confirmed cases ^[25].

An established method for the fractional-order model is elaborated upon in ^[26]. Recent contributions include various fractional models related to COVID-19, such as the analysis by Atangana and Khan focusing on the pandemic's impact on China ^[27]. Additionally, the COVID-19 model's dynamical aspects were explored using fuzzy Caputo and ABC derivatives, as demonstrated in ^[28]. A similar type of approach using fractional operator techniques are given in ^[29,30,31]. Different author's have investigated the transmission of different infectious disease like COVID-19 with symptomatic and asymptomatic effects in the community by using the fractal-fractional definition ^[32,33].

In light of the aforementioned significance, we aim is to address fundamental issues by concentrating on the distinctive challenges posed by the dynamics of COVID-19. To achieve this, we employ a model tailored to accurately capture the characteristics of COVID-19 dynamics and account for the limitations in our response to the pandemic. Initially, we examine the epidemic dynamics within a specific community characterized by a unique social pattern. For this analysis, we adopt a conventional SEIR design that accommodates prolonged incubation periods.

Here, the previous model is given in ^[34] as follows:

$\begin{eqnarray} \frac{dS}{dt}& = &\textbf{a}-\frac{\rho_{1} S I}{1+\gamma I}-(\delta+\rho)S,\\ \frac{dE}{dt}& = &\rho S-\delta E-\rho_{2}\alpha E I,\\ \frac{dI}{dt}& = &\frac{\rho_{1} S I}{1+\gamma I}\,\,+\rho_{2}\,\alpha\, E I-(\delta+\mu_{0}+\omega-\textbf{b})\,I,\\ \frac{dR}{dt}& = &\omega I-\delta R. \end{eqnarray}$

(1.1)

Initial conditions corresponds to the aforementioned system:

$\begin{equation*} S^{0}(t) \, = \, S_{0},\, \, E^{0}(t) \, = \, E_{0}, \, \,I^{0}(t) \, = \, I_{0},\,\, R^{0}(t)\, = \, R_{0}. \end{equation*}$

The primary goal of this research is to employ novel fractional derivatives within mathematical analysis and simulation to enhance the COVID-19 model. COVID-19 is a highly dangerous disease that presents a significant risk to human life. Verification of the existence and distinct characteristics of the solution system is undertaken, coupled with a qualitative assessment of the system. So, we introduce vaccination measures for low immune individuals. We developed new mathematical model by taking vaccination measures which helps to control COVID-19 early which we shall observe on simulation easily. The research involves confirming the presence of a solution system with unique characteristics and conducting a qualitative evaluation of this system. Furthermore, the fractal-fractional derivative is utilized to investigate the real-world behavior of the newly developed mathematical model. Finally, numerical simulations are used to reinforce and authenticate the biological findings.

Definition 1.1. If 0 $< \xi \leq$ 1 and 0 $< \lambda \leq$ 1, then the Riemann-Liouville operator for the fractal-fractional Operator (FFO) with a Mittag-Leffler (ML) kernel is defined as $U(t)$ ^[35].

$\begin{eqnarray} ^{FFM}_{0} D_{t}^{\xi, \lambda} U(t) = \frac{A B(\xi)}{1- \xi} \int^{t}_{0} E_{\xi} \frac{d \,U(\Omega)}{dt^{\lambda}}\left[- \frac{\xi}{1- \xi} (t-\Omega)^{\xi}\right] {d \Omega}\,\,, \end{eqnarray}$

involving $0 \; < \; \xi, \; \lambda \; \leq \; 1$ , and $AB(\xi) = 1 - \xi + \frac{\xi}{\Gamma(\xi)}.$

Therefore, the function $U(t)$ , which has an order of $(\xi, \lambda)$ and a Mittag-Leffler (ML) kernel, is given as follows:

$\begin{eqnarray} ^{FFM}_{0} D_{t}^{\xi, \lambda} U(t) = \frac{\lambda(1-\xi) t^{\lambda -1} U(t)}{A B(\xi)} + \frac{\xi \lambda}{A B(\xi)} \int^{t}_{0} {\Omega}^{\xi -1} (t-\Omega) U(\Omega) {d \Omega} \,\,. \end{eqnarray}$

2. Formulation of the SEVIR model

A newly developed model for SARS-COVID-19 includes the vaccinated effect, whereas the previous model used the SEIR framework. In this new model, we introduce a new variable called "Vaccinated." With the addition of Vaccinated, the new model is referred to as SEVIR, where "S" represents the Susceptible class, "E" represents the Exposed class, "V" represents the Vaccinated class, "I" represents the Infected class, and "R" represents the Recovered class.

We define several parameters in this model: "a" represents the recruitment rate, " $\delta$ " represents the death rate due to natural causes, " $\rho+\rho_{1}$ " represents the contact rate from the Susceptible class to the Exposed class, " $\rho_{2}$ " represents the vaccination rate, " $\alpha$ " represents the rate at which the infection is reducing due to vaccination effects, " $\mu_{0}$ " represents the infection death rate, " $\omega$ " represents the rate at which an individual recovers from vaccination and becomes recovered, " $\textbf{b}$ " represents the recruitment rate to the Vaccinated class, " $\phi$ " represents the contact rate from the Vaccinated class to the Infected class, and " $\psi$ " represents the recovery rate.

We want to investigate spread of the SEIR model for SARS-COVID-19 with the vaccinated effect.

So, the flow chart for newly developed model SEVIR is given as Figure 1.

Figure 1. The flow chart illustrates the newly developed model.

DownLoad: Full-Size Img PowerPoint

The model that was developed based on the generalized hypothesis with the vaccinated effect is presented as follows:

$\begin{eqnarray} \frac{d S}{dt}& = &\textbf{a}-(\delta+\rho+\rho_{1})S,\\ \frac{d E}{dt}& = &(\rho+\rho_{1})S-\delta E-\rho_{2}\alpha E V,\\ \frac{d V}{dt}& = &\rho_{2}\alpha E V-(\delta+\mu_{0}+\omega-\textbf{b}+\phi)V,\\ \frac{d I}{dt}& = &\phi V-(\mu_{0}+\delta+\psi)I,\\ \frac{d R}{dt}& = &\omega V+\psi I -\delta R. \end{eqnarray}$

(2.1)

The following are initial conditions linked with the described system:

$\begin{eqnarray} S^{0}(t) \, = \, S_{0},\, \, E^{0}(t) \, = \, E_{0}, \, \,V^{0}(t) \, = \, V_{0}, \, \,I^{0}(t) \, = \, I_{0},\,\, R^{0}(t)\, = \, R_{0}. \end{eqnarray}$

Using the fractal-fractional order (FFO) with a Mittag-Leffler (ML) definition, the above model becomes

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t}S(t)& = &\textbf{a}-(\delta+\rho+\rho_{1})S,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t}E(t)& = &(\rho+\rho_{1})S-\delta E-\rho_{2}\alpha E V,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t}V(t)& = &\rho_{2}\alpha E V-(\delta+\mu_{0}+\omega-\textbf{b}+\phi)V,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t}I(t)& = &\phi V-(\mu_{0}+\delta+\psi)I,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t}R(t)& = &\omega V+\psi I-\delta R. \end{eqnarray}$

(2.2)

Here, $^{FFM}_{0}D^{\xi, \lambda}_{t}$ is the fractal-fractional operator with Mittag-Leffler (ML), where $0 < \xi \leq 1$ and $0 < \lambda \leq\; 1$ .

The following are initial conditions linked with the described system:

$\begin{eqnarray} S^{0}(t) \, = \, S_{0},\, \, E^{0}(t) \, = \, E_{0}, \, \,V^{0}(t) \, = \, V_{0}, \, \,I^{0}(t) \, = \, I_{0},\,\, R^{0}(t)\, = \, R_{0}. \end{eqnarray}$

Parameter descriptions are given in the following table.

Parameters	Representation	Reference
$\text{a}$	$\text{contact rate from the susceptible to the exposed class}$	^[34]
$\delta$	$\text{death rate due to natural causes}$	^[34]
$\rho+\rho_{1}$	$\text{contact rate from the susceptible to the exposed class}$	^[34]
$\alpha$	$\text{rate at which the infection is reducing due to vaccination}$	^[34]
$\rho_{2}$	$\text{vaccination rate}$	^[34]
$\mu_{0}$	death rate due to infection	^[34]
$\omega$	rate at which an individuals becomes recovered	^[34]
$\text{b}$	recruitment rate to the vaccinated class	^[34]
$\phi$	contact rate from the vaccinated to the infected class	Assumed
$\psi$	recovery rate	Assumed

2.1. Equilibrium point and reproductive number

For this model, the point of equilibrium without disease (disease free) is

$\begin{eqnarray} D_{1}(S,\,E,\,V,\,I,\,R) = \left(\frac{\textbf{a}}{\delta+\rho+\rho_{1}},\frac{\textbf{a}(\rho+\rho_{1})}{\delta(\delta+\rho+\rho_{1})},0,0,0\right), \end{eqnarray}$

as well as the endemic points of equilibrium $D_{2}(S^{*}, E^{*}, V^{*}, I^{*}, R^{*})$ , where

$\begin{eqnarray} S^{*}& = &\frac{\textbf{a}}{\delta+\rho+\rho_{1}},\\ E^{*}& = &\frac{-\textbf{b}+\delta+\phi+\omega+\mu_{0}}{\alpha\rho_{2}},\\ V^{*}& = &\frac{(\delta+\psi+\mu_{0})\textbf{A}-\textbf{b}\delta\rho_{1}+\delta^{2}\rho_{1}+\delta\phi\rho_{1}+\delta\omega\rho_{1}+\delta\mu_{0}\rho_{1}-\textbf{a}\alpha\rho\rho_{2}-\textbf{a}\alpha\rho_{1}\rho_{2}}{\alpha \phi(\textbf{b}-\delta-\phi-\omega-\mu_{0})(\delta+\rho+\rho_{1})\rho_{2}},\\ I^{*}& = &\phi\Bigg(\frac{\textbf{A}-\textbf{b}\delta\rho_{1}+\delta^{2}\rho_{1}+\delta\phi\rho_{1}+\delta\omega\rho_{1}+\delta\mu_{0}\rho_{1}-\textbf{a}\alpha\rho\rho_{2}-\textbf{a}\alpha\rho_{1}\rho_{2}}{\alpha(\textbf{b}-\delta-\phi-\omega-\mu_{0})(\delta+\rho+\rho_{1})\rho_{2}}\Bigg),\\ R^{*}& = &\big(\phi\psi+\delta\omega+\psi\omega+\omega\mu_{0}\big)\times\Bigg(\frac{\textbf{A}-\textbf{b}\delta\rho_{1}+\delta^{2}\rho_{1}+\delta\phi\rho_{1}+\delta\omega\rho_{1}+\delta\mu_{0}\rho_{1}-\textbf{a}\alpha\rho\rho_{2}-\textbf{a}\alpha\rho_{1}\rho_{2}}{\alpha \delta \phi(\delta+\psi+\mu_{0})(\textbf{b}-\delta-\phi-\omega-\mu_{0})(\delta+\rho+\rho_{1})\rho_{2}}\Bigg), \end{eqnarray}$

where

$\begin{eqnarray} \textbf{A} = -\textbf{b}\delta^{2}+\delta^{3}-\textbf{b}\delta\rho+\delta^{2}\rho+\delta^{2}\phi +\delta\rho\phi+\delta^{2}\omega+\delta\rho\omega+\delta^{2}\mu_{0}+\delta\rho\mu_{0}. \end{eqnarray}$

The reproductive number for the newly developed system by using the next generation method is

$\begin{eqnarray} R_{0} = \frac{\textbf{b}\delta(\delta+\rho+\rho_{1})+\textbf{a}\alpha\rho_{2}(\rho+\rho_{1})}{\delta(\delta+\rho+\rho_{1})(\delta+\phi+\omega+\mu_{0})}. \end{eqnarray}$

3. Bounded and positive solutions

In this section, we demonstrate the boundedness and positivity of the developed model.

Theorem 3.1. The considered initial condition is

$\begin{eqnarray} \{S_{0},E_{0}, V_{0},I_{0},R_{0}\}\subset\Upsilon, \end{eqnarray}$

and therefore the solutions $\{ S, E, V, I, R \}$ will be positive $\forall$ t $\geq$ 0.

Proof. We will begin the primary analysis to show the improved quality of the solutions. These solutions effectively address real-world issues and have positive outcomes. We will follow the methodology provided in references ^[36,37,38]. In this segment, we will examine the conditions required to ensure positive outcomes from the proposed model. To accomplish this, we will establish a standard.

$\begin{eqnarray} \parallel \beta \parallel_{\infty} = \sup\limits_{t\in D_{\beta}}\mid\beta(t)\mid, \end{eqnarray}$

where " $D_{\beta}$ " represents the $\beta$ domain. Now, we continue with $S(t)$ .

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t}S(t)& = &\textbf{a}-(\delta+\rho+\rho_{1})S, \quad \forall t\geq0,\\ &\geq&-\left(\delta+\rho+\rho_{1}\right)S, \quad \forall t\geq0. \end{eqnarray}$

This yield

$\begin{eqnarray} S(t)\geq S(0)E_{\xi}\bigg[-\frac{c^{1-\lambda}{\xi}{(\delta+\rho+\rho_{1})}{t^{\xi}}}{{A B}(\xi)-(1-\xi)(\delta+\rho+\rho_{1})}\bigg], \quad \forall t\geq0, \end{eqnarray}$

where "c" represents the time element. This demonstrates that the $S(t)$ individuals must be positive $\forall \, \, t \geq$ 0. Now, we have the $E(t)$ individuals as follows:

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t}E(t)& = &(\rho+\rho_{1})S-\delta E-\rho_{2}\alpha E V, \quad \forall \,\, t\geq0,\\ &\geq&-(\delta+\rho_{2}\alpha\mid V \mid)E, \quad \forall \,\,t\geq0,\\ &\geq&-(\delta+\rho_{2}\alpha\sup\limits_{t\in D_{V}}\mid V \mid )E, \quad \forall \,\,t\geq0,\\ &\geq&-(\delta+\rho_{2}\alpha\parallel V \parallel_{\infty})E, \quad \forall \,\,t\geq 0. \end{eqnarray}$

This yield

$\begin{eqnarray} E(t)\geq E(0)E_{\xi}\bigg[-\frac{c^{1-\lambda}{\xi}{(\delta+\rho_{2}\alpha\parallel V\parallel_{\infty})}{t^{\xi}}}{AB(\xi)-(1-\xi)(\delta+\rho_{2}\alpha\parallel V\parallel_{\infty})}\bigg], \quad \forall \,\,t\geq0, \end{eqnarray}$

where "c" represents the time element. This demonstrates that the $E(t)$ individuals must be positive $\forall \, \, t \geq$ 0. Now, we have the $V(t)$ individuals as follows:

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t}V(t)& = &\rho_{2}\alpha E I-(\delta+\mu_{0}+\omega-\textbf{b}+\phi)V, \quad \forall t\geq0,\\ &\geq&-(-\rho_{2}\alpha \mid E \mid+\mu_{0}+\delta+\omega-\textbf{b}+\phi)V, \quad \forall t\geq0,\\ &\geq&-(-\rho_{2}\alpha \sup\limits_{t\in D_{E}}\mid E \mid+\mu_{0}+\delta+\omega-\textbf{b}+\phi)V, \quad \forall t\geq0,\\ &\geq&-(-\rho_{2}\alpha \parallel E \parallel_{\infty}+\mu_{0}+\delta+\omega-\textbf{b}+\phi)V, \quad \forall t\geq0. \end{eqnarray}$

This yield

$\begin{eqnarray} V(t)\geq V(0)E_{\xi}\bigg[-\frac{c^{1-\lambda{\xi}}{(-\rho_{2}\alpha \parallel E \parallel_{\infty}+\mu_{0}+\delta+\omega-\textbf{b}+\phi)}{t^{\xi}}}{A B(\xi)-(1-\xi)\left(-\rho_{2}\alpha\parallel E \parallel_{\infty}+\mu_{0}+\delta+\omega-\textbf{b}+\phi\right)}\bigg], \quad \forall t\geq0, \end{eqnarray}$

where "c" represents the time element. This demonstrates that the $V(t)$ individuals must be positive $\forall \, \, t \geq$ 0. Now, we have the $I(t)$ individuals as follows:

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t}I(t)& = &\phi V-(\mu_{0}+\delta+\psi)I, \quad \forall t\geq0,\\ &\geq&-(\mu_{0}+\delta+\psi)I(t), \quad \forall t\geq0. \end{eqnarray}$

This yield

$\begin{eqnarray} I(t)\geq I(0)E_{\xi}\bigg[-\frac{c^{1-\lambda}{\xi}{(\mu_{0}+\delta+\psi)}{t^{\xi}}}{A B(\xi)-(1-\xi)(\mu_{0}+\delta+\psi)}\bigg], \quad \forall t\geq0, \end{eqnarray}$

where "c" represents the time element. This demonstrates that the $I(t)$ individuals must be positive $\forall \, \, t \geq$ 0. Now, we have the $R(t)$ individuals as follows:

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t}R(t)& = &\omega V+\psi I-\delta R , \quad \forall \,t\,\geq\,0,\\ &\geq&-(\delta)R(t), \quad \forall \,t\,\geq\,0. \end{eqnarray}$

This yield

$\begin{eqnarray} R(t)\geq R(0){E_{\xi}}\bigg[-\frac{c^{1-\lambda}\xi{(\delta)}{t^{\xi}}}{{A B}(\xi)-(1-\xi)(\delta)}\bigg], \quad \forall t\geq0, \end{eqnarray}$

where "c" represents the time element. This demonstrates that the $R(t)$ individuals must be positive $\forall \, \, t \geq$ 0.

Theorem 3.2. Solutions of our developed model given in Eq (2.2) with positive initial values are all bounded.

Proof. The above theorem demonstrates that the solutions of our developed model must be positive $\forall \, \, t\, \geq\, 0$ , and the strategies are described in ^[39]. Because $X = S\, +\, E\, +\, V$ , then

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t}X(t) = \textbf{a}-\delta X-(\mu_{0}+\omega-\textbf{b}+\phi)V. \end{eqnarray}$

We achieved as follows:

$\begin{eqnarray} \Psi_{p} = \{S,\quad E,\quad V \in R_{+}^{3}\mid S + V\leq X\} \quad \forall \,\,t\geq0. \end{eqnarray}$

Further we have $X_{\upsilon}$ = I+R. So, we have

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t}X_{\upsilon}(t) = (\phi+\omega)V-\mu_{0}I-X_{\upsilon}\delta. \end{eqnarray}$

Upon solving the above equation and taking $t\rightarrow \infty$ , we get

$\begin{eqnarray} X_{\upsilon}\leq\frac{(\phi+\omega)V-\mu_{0}I}{\delta}. \end{eqnarray}$

Thus,

$\begin{eqnarray} \Psi_{\upsilon} = \bigg\{I,\,\,R\in R_{+}^{2}\mid X_{\upsilon}\leq\frac{(\phi+\omega)I-\mu_{0}Q}{\delta}\bigg\} \quad \forall \,\,t \geq 0. \end{eqnarray}$

The model's mathematical solutions (2.2) are confined to the region $\Psi$ .

$\begin{eqnarray} \Psi = \bigg\{S,\,\,E,\,\,V,\,\,I,\,\,R \in R_{+}^{5}\mid S+V\leq X,X_{\upsilon}\leq\frac{(\phi+\omega)V-\mu_{0}I}{\delta}\bigg\} \quad \forall \,\,t\geq0. \end{eqnarray}$

This demonstrates that for every $t \geq$ 0, all solutions remain positive, consistent with the provided initial conditions in the domain $\Psi$ .

Theorem 3.3. The proposed coronavirus model (2.2) in $R_{+}^{5}$ has positive invariant solutions, in addition to the initial conditions.

Proof. In this particular scenario, we applied the procedure described in ^[40]. We have

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t}(S(t))_{S = 0}& = &\textbf{a} \quad \geq0,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t}(E(t))_{E = 0}& = &(\rho+\rho_{1})S \quad \geq0,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t}(V(t))_{V = 0}& = &\rho_{2}\alpha E V+\textbf{b}V \quad \geq0,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t}(I(t))_{I = 0}& = &\phi V \quad \geq0,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t}(R(t))_{R = 0}& = &\omega V+\psi I \quad \geq0. \end{eqnarray}$

(3.1)

If $(S_{0}, E_{0}, V_{0}, I_{0}, R_{0})$ $\in$ $R_{+}^{5}$ , then our obtained solution is unable to escape from the hyperplane, as stated in Eq (3.1). This proves that the $R_{+}^{5}$ domain is positive invariant.

4. Impact of global derivatives for uniqueness and existence of solutions

The Riemann-Stieltjes integral has been widely recognized in the literature as the most commonly used integral. If

$\begin{eqnarray} Y(x) = \int y(x){d}x, \end{eqnarray}$

then the Riemann-Stieltjes integral is given as follows:

$\begin{eqnarray} Y_{w}(x) = \int y(x){d}{w}(x), \end{eqnarray}$

where the $y(x)$ global derivative with respect to $w(x)$ is

$\begin{eqnarray} D_{w}y(x) = \lim\limits_{h\rightarrow0}\frac{y(x+h)-y(x)}{w(x+h)-w(x)}. \end{eqnarray}$

If the above function's numerator and denominator are differentiated, we get

$\begin{eqnarray} D_{w}y(x) = \frac{y^{'}(x)}{w^{'}(x)}, \end{eqnarray}$

assuming that $w^{'}(x)\, \, \neq$ 0, $\forall x\in D_{w^{'}}$ . Now, we will test the impact on the coronavirus by using the global derivative instead of the classical derivative:

$\begin{eqnarray} D_{w}S& = &\textbf{a}-(\delta+\rho+\rho_{1})S,\\ D_{w}E& = &(\rho+\rho_{1})S-\delta E-\rho_{2}\alpha E V,\\ D_{w}V& = &\rho_{2}\alpha E V-(\delta+\mu_{0}+\omega-\textbf{b}+\phi)V,\\ D_{w}I& = &\phi V-(\mu_{0}+\delta+\psi)I,\\ D_{w}R& = &\omega V+\psi I-\delta R. \end{eqnarray}$

For the sake of clean notation, we shall suppose that w is differentiable.

$\begin{eqnarray} S^{'}& = &w^{'}[\textbf{a}-(\delta+\rho+\rho_{1})S],\\ E^{'}& = &w^{'}[(\rho+\rho_{1})S-\delta E-\rho_{2}\alpha E V],\\ V^{'}& = &w^{'}[\rho_{2}\alpha E V-(\delta+\mu_{0}+\omega-\textbf{b}+\phi)V],\\ I^{'}& = &w^{'}[\phi V-(\mu_{0}+\delta+\psi)I],\\ R^{'}& = &w^{'}[\omega V+\psi I-\delta R]. \end{eqnarray}$

An appropriate choice of the function $w(t)$ will lead to a specific outcome. For instance, if $w(t) = t^{\alpha}$ , where $\alpha$ is a real number, we will observe fractal movement. We had to take action due to the circumstances that

$\begin{eqnarray} \parallel w^{'}\parallel_{\infty} = \sup\limits_{t\in D_{w^{'}}}\mid w^{'}(t)\mid < N. \end{eqnarray}$

The below example demonstrates the unique solution for the developed system:

$\begin{eqnarray} S^{'}& = &w^{'}[\textbf{a}-(\delta+\rho+\rho_{1})S] = Z_{1}(t, S, G),\\ E^{'}& = &w^{'}[(\rho+\rho_{1})S-\delta E-\rho_{2}\alpha E V] = Z_{2}(t, S, G),\\ V^{'}& = &w^{'}[\rho_{2}\alpha E V-(\delta+\mu_{0}+\omega- \textbf{b}+\phi)V] = Z_{3}(t, S, G), \\ I^{'}& = &w^{'}[\phi V-(\mu_{0}+\delta+\psi)I] = Z_{4}(t, S, G),\\ R^{'}& = &w^{'}[\omega V+\psi I-\delta R] = Z_{5}(t, S, G), \end{eqnarray}$

where $G = E, V, I, R$ .

We need to confirm the first two requirements as follows:

(1) $\mid Z(t, S, G)\mid^{2} < K(1+\mid S\mid^{2}$ ,

(2) $\forall \quad S_{1}, S_{2}$ , we have, $\parallel Z(t, S_{1}, G)-Z(t, S_{2}, G)\parallel^{2} < \bar{K}\parallel S_{1}-S_{2}\parallel_{\infty}^{2}$ .

Initially,

$\begin{eqnarray} \mid Z_{1}(t, S, G)\mid^{2}& = &\mid w^{'}\big[\textbf{a}-(\delta+\rho+\rho_{1})S\big]\mid^{2},\\ & = &\mid w^{'}\big[\textbf{a}+\left(-\delta-\rho-\rho_{1}\right)S\big]\mid^{2},\\ &\leq&2\mid w^{'}\mid^{2}\left(\textbf{a}^{2}+\mid(-\delta-\rho-\rho_{1})S\mid^{2}\right),\\ &\leq&2\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}\textbf{a}^{2}+6\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}\left(\rho^{2}+\delta^{2}+\rho_{1}\right)\mid S\mid^{2},\\ &\leq&2\parallel w^{'}\parallel^{2}_{\infty}\textbf{a}^{2}+6\parallel w^{'}\parallel^{2}_{\infty}\left(\rho^{2}+\delta^{2}+\rho_{1}^{2}\right)\mid S\mid^{2},\\ &\leq&2\parallel w^{'}\parallel^{2}_{\infty}\textbf{a}^{2}\left(1+\frac{3}{\textbf{a}^{2}}(\rho^{2}+\delta^{2}+\rho_{1}^{2})\mid S\mid^{2}\right),\\ &\leq&K_{1}(1+\mid S\mid^{2}), \end{eqnarray}$

under the condition

$\begin{eqnarray} \frac{3}{\textbf{a}^{2}}(\rho^{2}+\delta^{2}+\rho_{1}^{2}) < 1, \end{eqnarray}$

involving

$\begin{eqnarray} K_{1} = 2\parallel w^{'}\parallel^{2}_{\infty}\textbf{a}^{2}. \end{eqnarray}$

$\begin{eqnarray} \mid Z_{2}(t, S, G)\mid^{2}& = &\mid w^{'}\big[(\rho+\rho_{1})S-\delta E-\rho_{2}\alpha E V\big]\mid^{2},\\ & = &\mid w^{'}\big[(\rho+\rho_{1})S+(-\delta-\rho_{2}\alpha V)E\big]\mid^{2},\\ &\leq&2\mid w^{'}\mid^{2}\left(\mid(\rho+\rho_{1})S\mid^{2}+(-\delta-\rho_{2}\alpha V)E\mid^{2}\right),\\ &\leq&4\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}(\rho^{2}+\rho_{1}^{2})\sup\limits_{t\in D_{S}}\mid S\mid^{2}+4\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}(\delta^{2}+\rho^{2}_{2}\alpha^{2}\sup\limits_{t\in D_{V}}\mid V \mid^{2})\mid E\mid^{2},\\ &\leq&4\parallel w^{'}\parallel^{2}_{\infty}(\rho^{2}+\rho_{1}^{2})\parallel S\parallel^{2}_{\infty}+4\parallel w^{'}\parallel^{2}_{\infty}(\delta^{2}+\rho^{2}_{2}\alpha^{2}\parallel V \parallel^{2}_{\infty})\mid E\mid^{2},\\ &\leq&4\parallel w^{'}\parallel^{2}_{\infty}(\rho^{2}+\rho_{1}^{2})\parallel S\parallel^{2}_{\infty}\left(1+\frac{(\delta^{2}+\rho^{2}_{2}\alpha^{2}\parallel V \parallel^{2}_{\infty})\mid E\mid^{2}}{(\rho^{2}+\rho_{1}^{2})\parallel S\parallel^{2}_{\infty}}\right),\\ &\leq&K_{2}(1+\mid E\mid^{2}), \end{eqnarray}$

under the condition

$\begin{eqnarray} \frac{(\delta^{2}+\rho^{2}_{2}\alpha^{2}\parallel V \parallel^{2}_{\infty})}{(\rho^{2}+\rho_{1}^{2})\parallel S\parallel^{2}_{\infty}} < 1, \end{eqnarray}$

where

$\begin{eqnarray} K_{2} = 2\parallel w^{'}\parallel^{2}_{\infty}(\rho^{2}+\rho_{1}^{2})\parallel S\parallel^{2}_{\infty}. \end{eqnarray}$

$\begin{eqnarray} \mid Z_{3}(t, S, G)\mid^{2}& = &\mid w^{'}\big[\rho_{2}\alpha E V-(\delta+\mu_{0}+\omega-\textbf{b}+\phi)V\big]\mid^{2},\\ & = &\mid w^{'}\big[\rho_{2}\alpha E V+\textbf{b}V+(-\delta-\mu_{0}-\omega-\phi)V\big]\mid^{2},\\ &\leq&2\mid w^{'}\mid^{2}\left(\mid\rho_{2}\alpha E+\textbf{b}\mid^{2}+\mid(-\delta-\mu_{0}-\omega-\phi)\mid^{2}\right)\mid V \mid^{2},\\ &\leq&4\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}\big[(\rho_{2}^{2}\alpha^{2}\sup\limits_{t\in D_{E}}\mid E\mid^{2}+\textbf{b}^{2})+2(\delta^{2}+\mu_{0}^{2}+\omega^{2}+\phi^{2})\big]\mid V \mid^{2},\\ &\leq&4\parallel\mid w^{'}\parallel^{2}_{\infty}\bigg[(\rho_{2}^{2}\alpha^{2}\parallel E\parallel^{2}_{\infty}+\textbf{b}^{2})+2(\delta^{2}+\mu_{0}^{2}+\omega^{2}+\phi^{2})\bigg]\mid V \mid^{2},\\ &\leq&4\parallel\mid w^{'}\parallel^{2}_{\infty}(\rho_{2}^{2}\alpha^{2}\parallel E\parallel^{2}_{\infty}+\textbf{b}^{2})\bigg[1+\frac{2(\delta^{2}+\mu_{0}^{2}+\omega^{2}+\phi^{2})}{\left(\rho_{2}^{2}\alpha^{2}\parallel E\parallel^{2}_{\infty}+\textbf{b}^{2}\right)}\bigg]\mid V \mid^{2},\\ &\leq&K_{3}(2\mid V \mid^{2}), \end{eqnarray}$

under the condition

$\begin{eqnarray} \frac{2(\delta^{2}+\mu_{0}^{2}+\omega^{2}+\phi^{2})}{\left(\rho_{2}^{2}\alpha^{2}\parallel E\parallel^{2}_{\infty}+\textbf{b}^{2}\right)}\leq1, \end{eqnarray}$

where

$\begin{eqnarray} K_{3} = 4\parallel\mid w^{'}\parallel^{2}_{\infty}(\rho_{2}^{2}\alpha^{2}\parallel E\parallel^{2}_{\infty}+\textbf{b}^{2}). \end{eqnarray}$

$\begin{eqnarray} \mid Z_{4}(t, S, G)\mid^{2}& = &\mid w^{'}\big[\phi V-(\mu_{0}+\delta+\psi)I\big]\mid^{2},\\ & = &\mid w^{'}\big[\phi V+(-\mu_{0}-\delta-\psi)I\big]\mid^{2},\\ &\leq&2\mid w^{'}\mid^{2}\left(\mid\phi V \mid^{2}+\mid(-\mu_{0}-\delta-\psi)I\mid^{2}\right),\\ &\leq&2\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}\phi^{2}\sup\limits_{t\in D_{V}}\mid V \mid^{2}+6\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}(\mu_{0}^{2}+\delta^{2}+\psi^{2})\mid I \mid^{2},\\ &\leq&2\parallel w^{'}\parallel^{2}_{\infty}\phi^{2}\parallel V \parallel^{2}_{\infty}+6\parallel w^{'}\parallel^{2}_{\infty}(\mu_{0}^{2}+\delta^{2}+\psi^{2})\mid I\mid^{2},\\ &\leq&2\parallel w^{'}\parallel^{2}_{\infty}\phi^{2}\parallel I\parallel^{2}_{\infty}\left(1+\frac{3(\mu_{0}^{2}+\delta^{2}+\psi^{2})\mid I \mid^{2}}{\phi^{2}\parallel I\parallel^{2}_{\infty}}\right),\\ &\leq&K_{4}(1+\mid I \mid^{2}), \end{eqnarray}$

under the condition

$\begin{eqnarray} \frac{3(\mu_{0}^{2}+\delta^{2}+\psi^{2})}{\phi^{2}\parallel V \parallel^{2}_{\infty}} < 1, \end{eqnarray}$

where

$\begin{eqnarray} K_{4} = 2\parallel w^{'}\parallel^{2}_{\infty}\phi^{2}\parallel V \parallel^{2}_{\infty}. \end{eqnarray}$

$\begin{eqnarray} \mid Z_{5}(t, S, G)\mid^{2}& = &\mid w^{'}\big[\omega V+\psi I-\delta R\big]\mid^{2},\\ & = &\mid w^{'}\big[(\omega V+\psi I)+(-\delta R)\big]\mid^{2},\\ &\leq&2\mid w^{'}\mid^{2}\left(\mid(\omega V+\psi I)\mid^{2}+\mid-\delta R\mid^{2}\right),\\ &\leq&4\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}\left(\omega^{2}\sup\limits_{t\in D_{V}}\mid V \mid^{2}+\psi\sup\limits_{t\in D_{I}}\mid I \mid^{2}\right)+2\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}\delta^{2}\mid R\mid^{2},\\ &\leq&4\parallel w^{'}\parallel^{2}_{\infty}\left(\omega^{2}\parallel V \parallel^{2}_{\infty}+\psi\parallel I \parallel^{2}_{\infty}\right)+2\parallel w^{'}\parallel^{2}_{\infty}\delta^{2}\mid R\mid^{2}),\\ &\leq&4\parallel w^{'}\parallel^{2}_{\infty}(\omega^{2}\parallel V \parallel^{2}_{\infty}+\psi^{2}\parallel I\parallel^{2}_{\infty})\left(1+\frac{\delta^{2}\mid R\mid^{2}}{2(\omega^{2}\parallel V \parallel^{2}_{\infty}+\psi^{2}\parallel I \parallel^{2}_{\infty})}\right),\\ &\leq&K_{5}(1+\mid R\mid^{2}), \end{eqnarray}$

under the condition

$\begin{eqnarray} \frac{\delta^{2}}{2(\omega^{2}\parallel V \parallel^{2}_{\infty}+\psi^{2}\parallel I \parallel^{2}_{\infty})} < 1, \end{eqnarray}$

where

$\begin{eqnarray} K_{5} = 4\parallel w^{'}\parallel^{2}_{\infty}(\omega^{2}\parallel V \parallel^{2}_{\infty}+\psi^{2}\parallel I \parallel^{2}_{\infty}). \end{eqnarray}$

Hence the linear growth condition is satisfied.

Further, we validate the Lipschitz condition.

$\begin{eqnarray} \mid Z_{1}(t, S_{1}, G)-Z_{1}(t, S_{2}, G)\mid^{2}& = &\mid w^{'}(-\delta-\rho-\rho_{1})(S_{1}-S_{2})\mid^{2},\\ \mid Z_{1}(t, S_{1}, G)-Z_{1}(t, S_{2}, G)\mid^{2}&\leq&\mid w^{'}\mid^{2}(3\delta^{2}+3\rho^{2}+3\rho_{1}^{2})\mid S_{1}-S_{2}\mid^{2},\\ \sup\limits_{t\in D_{S}}\mid Z_{1}(t, S_{1}, G)-Z_{1}(t, S_{2}, G)\mid^{2}&\leq&\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}\left(3\delta^{2}+3\rho^{2}+3\rho_{1}^{2}\right)\sup\limits_{t\in D_{S}}\mid S_{1}-S_{2}\mid^{2},\\ \parallel Z_{1}(t, S_{1}, G)-Z_{1}(t, S_{2}, G)\parallel_{\infty}^{2}&\leq&\parallel w^{'}\parallel^{2}_{\infty}(3\delta^{2}+3\rho^{2}+3\rho_{1}^{2})\parallel S_{1}-S_{2}\parallel^{2}_{\infty},\\ \parallel Z_{1}(t, S_{1}, G)-Z_{1}(t, S_{2}, G)\parallel_{\infty}^{2}&\leq&\bar{K_{1}}\parallel S_{1}-S_{2}\parallel^{2}_{\infty}, \end{eqnarray}$

where

$\begin{eqnarray} \bar{K_{1}} = \parallel w^{'}\parallel^{2}_{\infty}\left(3\delta^{2}+3\rho^{2}+3\rho_{1}^{2}\right). \end{eqnarray}$

$\begin{eqnarray} \mid Z_{2}(t,\,\,S,\,\,E_{1},\,\,V,\,\,I,\,\,R)-Z_{2}(t,\,\,S,\,\,E_{2},\,\,V,\,\,I,\,\,R)\mid^{2}& = &\mid w^{'}(-\delta-\rho_{2}\alpha V)(E_{1}-E_{2})\mid^{2},\\ \mid Z_{2}(t,\,\,S,\,\,E_{1},\,\,V,\,\,I,\,\,R)-Z_{2}(t,\,\,S,\,\,E_{2},\,\,V,\,\,I,\,\,R)\mid^{2}&\leq&\mid w^{'}\mid^{2}(2\delta^{2}+2\rho_{2}^{2}\alpha^{2}\mid V \mid^{2})\mid E_{1}-E_{2}\mid^{2},\\ \sup\limits_{t\in D_{E}}\mid Z_{2}(t,\,\,S,\,\,E_{1},\,\,V,\,\,I,\,\,R)-Z_{2}(t,\,\,S,\,\,E_{2},\,\,V,\,\,I,\,\,R)\mid^{2}&\leq&\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}(2\delta^{2}+2\rho_{2}^{2}\alpha^{2}\sup\limits_{t\in D_{V}}\mid V \mid^{2})\sup\limits_{t\in D_{E}}\\&&\times\mid E_{1}-E_{2}\mid^{2},\\ \parallel Z_{2}(t,\,\,S,\,\,E_{1},\,\,V,\,\,I,\,\,R)-Z_{2}(t,\,\,S,\,\,E_{2},\,\,V,\,\,I,\,\,R)\parallel_{\infty}^{2}&\leq&\parallel w^{'}\parallel^{2}_{\infty}(2\delta^{2}+2\rho_{2}^{2}\alpha^{2}\parallel V \parallel^{2}_{\infty}),\parallel E_{1}-E_{2}\parallel^{2}_{\infty},\\ \parallel Z_{2}(t,\,\,S,\,\,E_{1},\,\,V,\,\,I,\,\,R)-Z_{2}(t,\,\,S,\,\,E_{2},\,\,V,\,\,I,\,\,R)\parallel_{\infty}^{2}&\leq&\bar{K_{2}}\parallel E_{1}-E_{2}\parallel^{2}_{\infty}, \end{eqnarray}$

where

$\begin{eqnarray} \bar{K_{2}} = \parallel w^{'}\parallel^{2}_{\infty}(2\delta^{2}+2\rho_{2}^{2}\alpha^{2}\parallel V \parallel^{2}_{\infty}). \end{eqnarray}$

$\begin{eqnarray} &&\mid Z_{3}(t,\,\,S,\,\,E,\,\,V_{1},\,\,I,\,\,R)-Z_{3}(t,\,\,S,\,\,E,\,\,V_{2},\,\,I,\,\,R)\mid^{2}\\& = &\mid w^{'}\left(\rho_{2}\alpha E+\textbf{b}+(-\delta-\mu_{0}-\omega-\phi)\right)(V_{1}-V_{2})\mid^{2}, \end{eqnarray}$

$\begin{eqnarray} &&\mid Z_{3}(t,\,\,S,\,\,E,\,\,V_{1},\,\,I,\,\,R)-Z_{3}(t,\,\,S,\,\,E,\,\,V_{2},\,\,I,\,\,R)\mid^{2}\\&\leq&2\mid w^{'}\mid^{2}\left(\mid\rho_{2}\alpha E+\textbf{b}\mid^{2}+\mid(-\delta-\mu_{0}-\omega-\phi)\mid^{2}\right)\\&&\times\mid V_{1}-V_{2}\mid^{2}, \end{eqnarray}$

$\begin{eqnarray} \sup\limits_{t\in D_{V}}\mid Z_{3}(t,\,\,S,\,\,E,\,\,V_{1},\,\,I,\,\,R)-Z_{3}(t,\,\,S,\,\,E,\,\,V_{2},\,\,I,\,\,R)\mid^{2}&\leq&4\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}\bigg(\rho_{2}^{2}\alpha^{2}\sup\limits_{t\in D_{E}}\mid E\mid^{2}+\textbf{b}^{2}\\&&+2(\delta^{2}+\mu_{0}^{2}+\omega^{2}+\phi^{2})\bigg)\sup\limits_{t\in D_{V}}\mid V_{1}-V_{2}\mid^{2},\\ \parallel Z_{3}(t,\,\,S,\,\,E,\,\,V_{1},\,\,I,\,\,R)-Z_{3}(t,\,\,S,\,\,E,\,\,V_{2},\,\,I,\,\,R)\parallel_{\infty}^{2}&\leq&4\parallel w^{'}\parallel^{2}_{\infty}\bigg(\rho_{2}^{2}\alpha^{2}\parallel E\parallel^{2}_{\infty}+\textbf{b}^{2}\\&&+2(\delta^{2}+\mu_{0}^{2}+\omega^{2}+\phi^{2})\bigg)\parallel V_{1}-V_{2}\parallel^{2}_{\infty},\\ \parallel Z_{3}(t,\,\,S,\,\,E,\,\,V_{1},\,\,I,\,\,R)-Z_{3}(t,\,\,S,\,\,E,\,\,V_{2},\,\,I,\,\,R)\parallel_{\infty}^{2}&\leq&\bar{K_{3}}\parallel V_{1}-V_{2}\parallel^{2}_{\infty}, \end{eqnarray}$

where

$\begin{eqnarray} \bar{K_{3}} = 4\parallel w^{'}\parallel^{2}_{\infty}\bigg(\rho_{2}^{2}\alpha^{2}\parallel E\parallel^{2}_{\infty}+\textbf{b}^{2}+2(\delta^{2}+\mu_{0}^{2}+\omega^{2}+\phi^{2})\bigg). \end{eqnarray}$

$\begin{eqnarray} \mid Z_{4}(t,\,\,S,\,\,E,\,\,V,\,\,I_{1},\,\,R)-Z_{4}(t,\,\,S,\,\,E,\,\,V,\,\,I_{2},\,\,R)\mid^{2}& = &\mid w^{'}(-\mu_{0}-\delta-\psi)\big](Q_{1}-Q_{2})\mid^{2},\\ \mid Z_{4}(t,\,\,S,\,\,E,\,\,V,\,\,I_{1},\,\,R)-Z_{4}(t,\,\,S,\,\,E,\,\,V,\,\,I_{2},\,\,R)\mid^{2}& = &\mid w^{'}\mid^{2}(3\mu_{0}^{2}+3\delta^{2}+3\psi^{2})\mid I_{1}-I_{2}\mid^{2},\\ \sup\limits_{t\in D_{I}}Z_{4}(t,\,\,S,\,\,E,\,\,V,\,\,I_{1},\,\,R)-Z_{4}(t,\,\,S,\,\,E,\,\,V,\,\,I_{2},\,\,R)\mid^{2}& = &\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}(3\mu_{0}^{2}+3\delta^{2}+3\psi^{2})\sup\limits_{t\in D_{I}}\mid I_{1}-I_{2}\mid^{2},\\ \parallel Z_{4}(t,\,\,S,\,\,E,\,\,V,\,\,I_{1},\,\,R)-Z_{4}(t,\,\,S,\,\,E,\,\,V,\,\,I_{2},\,\,R)\parallel_{\infty}^{2}&\leq&\parallel w^{'}\parallel^{2}_{\infty}(3\mu_{0}^{2}+3\delta^{2}+3\psi^{2})\parallel I_{1}-I_{2}\parallel^{2}_{\infty},\\ \parallel Z_{4}(t,\,\,S,\,\,E,\,\,V,\,\,I_{1},\,\,R)-Z_{4}(t,\,\,S,\,\,E,\,\,V,\,\,I_{2},\,\,R)\parallel_{\infty}^{2}&\leq&\bar{K_{4}}\parallel I_{1}-I_{2}\parallel^{2}_{\infty}, \end{eqnarray}$

where

$\begin{eqnarray} \bar{K_{4}} = \parallel w^{'}\parallel^{2}_{\infty}(3\mu_{0}^{2}+3\delta^{2}+3\psi^{2}). \end{eqnarray}$

$\begin{eqnarray} \mid Z_{5}(t,\,\,S,\,\,E,\,\,V,\,\,I,\,\,R_{1})-Z_{5}(t,\,\,S,\,\,E,\,\,V,\,\,I,\,\,R_{2})\mid^{2}& = &\mid w^{'}(-\delta)(R_{1}-R_{2})\mid^{2},\\ \mid Z_{5}(t,\,\,S,\,\,E,\,\,V,\,\,I,\,\,R_{1})-Z_{5}(t,\,\,S,\,\,E,\,\,V,\,\,I,\,\,R_{2})\mid^{2}&\leq&\mid w^{'}\mid^{2}\delta^{2}\mid(R_{1}-R_{2})\mid^{2},\\ \sup\limits_{t\in D_{R}}\mid Z_{5}(t,\,\,S,\,\,E,\,\,V,\,\,I,\,\,R_{1})-Z_{5}(t,\,\,S,\,\,E,\,\,V,\,\,I,\,\,R_{2})\mid^{2}&\leq&\sup\limits_{t\in D_{w^{'}}}\mid w^{'}\mid^{2}\delta^{2}\sup\limits_{t\in D_{R}}\mid R_{1}-R_{2}\mid^{2},\\ \parallel Z_{5}(t,\,\,S,\,\,E,\,\,V,\,\,I,\,\,R_{1})-Z_{5}(t,\,\,S,\,\,E,\,\,V,\,\,I,\,\,R_{2})\parallel_{\infty}^{2}&\leq&\parallel w^{'}\parallel^{2}_{\infty}\delta^{2}\parallel R_{1}-R_{2}\parallel_{\infty}^{2},\\ \parallel Z_{5}(t,\,\,S,\,\,E,\,\,V,\,\,I,\,\,R_{1})-Z_{5}(t,\,\,S,\,\,E,\,\,V,\,\,I,\,\,R_{2})\parallel_{\infty}^{2}&\leq&\bar{K_{5}}\parallel R_{1}-R_{2}\parallel^{2}_{\infty}, \end{eqnarray}$

involving

$\begin{eqnarray} \bar{K_{5}} = \parallel w^{'}\parallel^{2}_{\infty}\delta^{2}. \end{eqnarray}$

Then, given the condition, system (2.2) has a particular solution.

$\begin{eqnarray} \max\Big[\frac{3}{\textbf{a}^{2}}(\rho^{2}+\delta^{2}+\rho_{1}^{2}),\frac{(\delta^{2}+\rho^{2}_{2}\alpha^{2}\parallel V \parallel^{2}_{\infty})}{(\rho^{2}+\rho_{1}^{2})\parallel S\parallel^{2}_{\infty}},\frac{2(\delta^{2}+\mu_{0}^{2}+\omega^{2}+\phi^{2})}{\left(\rho_{2}^{2}\alpha^{2}\parallel E\parallel^{2}_{\infty}+\textbf{b}^{2}\right)},\frac{3(\mu_{0}^{2}+\delta^{2}+\psi^{2})}{\phi^{2}\parallel V \parallel^{2}_{\infty}},\\ \frac{\delta^{2}}{2(\omega^{2}\parallel V \parallel^{2}_{\infty}+\psi^{2}\parallel I \parallel^{2}_{\infty})}\Big] < 1. \end{eqnarray}$

5. Global stability for developed system

We use Lyapunov's approach and LaSalle's concept of invariance to analyze global stability and determine the conditions for eliminating diseases.

5.1. Lyapunov's first derivative

Theorem 5.1. ^[41] When the reproductive number $R_{0} >$ 1, the endemic equilibrium points of the SEVIR model are globally asymptotically stable.

Proof. The Lyapunov function can be expressed in the following manner:

$\begin{eqnarray} L(S^{*},E^{*},V^{*},I^{*},R^{*})& = &\left(S-S^{*}-S^{*}\log\frac{S}{S^{*}}\right)+\left(E-E^{*}-E^{*}\log\frac{E}{E^{*}}\right) +\left(V-V^{*}-V^{*}\log\frac{V}{V^{*}}\right)\\ &&+\left(I-I^{*}-I^{*}\log\frac{I}{I^{*}}\right)+\left(R-R^{*}-R^{*}\log\frac{R}{R^{*}}\right). \end{eqnarray}$

By applying a derivative on both sides,

$\begin{eqnarray} D^{\xi,\lambda}_{t} L = \dot{L} = \left(\frac{S-S^{*}}{S}\right)D^{\xi,\lambda}_{t}S+\left(\frac{E-E^{*}}{E}\right)D^{\xi,\lambda}_{t}E+\left(\frac{V-V^{*}}{V}\right)D^{\xi,\lambda}_{t}V+\left(\frac{I-I^{*}}{I}\right)D^{\xi,\lambda}_{t}I+\left(\frac{R-R^{*}}{R}\right)D^{\xi,\lambda}_{t}R, \end{eqnarray}$

we get

$\begin{eqnarray} D^{\xi,\lambda}_{t} L & = &\left(\frac{S-S^{*}}{S}\right)\left(\textbf{a}-(\delta+\rho+\rho_{1})S\right) +\left(\frac{E-E^{*}}{E}\right)\left((\rho+\rho_{1})S-\delta E-\rho_{2}\alpha E V\right)+\left(\frac{V-V^{*}}{V}\right)\\&&\times\left(\rho_{2}\alpha E V -(\delta+\mu_{0}+\omega-\textbf{b}+\phi)V\right)+\left(\frac{I-I^{*}}{I}\right)\left(\phi V-(\mu_{0}+\delta+\psi)I\right)+\left(\frac{R-R^{*}}{R}\right)\\&&\times\left(\omega V+\psi I-\delta R\right), \end{eqnarray}$

and setting $S = S-S^{*}, E = E-E^{*}, V = V-V^{*}, I = I-I^{*}$ and $R = R-R^{*}$ results in

$\begin{eqnarray} D^{\xi,\lambda}_{t} L& = &\textbf{a}-\textbf{a}\frac{S^{*}}{S}-(\delta+\rho+\rho_{1})\frac{(S-S^{*})^{2}}{S} +(\rho+\rho_{1})S-(\rho+\rho_{1})S^{*}-(\rho+\rho_{1})S\frac{E^{*}}{E}+(\rho+\rho_{1})S^{*}\frac{E^{*}}{E} \\&&-\delta\frac{(E-E^{*})^{2}}{E}-\rho_{2}\alpha V\frac{(E-E^{*})^{2}}{E}+\rho_{2}\alpha V^{*}\frac{(E-E^{*})^{2}}{E}+\rho_{2}\alpha E\frac{(V-V^{*})^{2}}{V}-\rho_{2}\alpha E^{*}\frac{(V-V^{*})^{2}}{V} \\&&+\textbf{b}\frac{(V-V^{*})^{2}}{V}-(\delta+\mu_{0}+\omega+\phi)\frac{(V-V^{*})^{2}}{V}+\phi V-\phi V^{*}-\phi V\frac{I^{*}}{I}+\phi V^{*}\frac{I^{*}}{I}\\&&-(\mu_{0}+\delta+\psi)\frac{(I-I^{*})^{2}}{I}+\omega V -\omega V^{*}-\omega V\frac{R^{*}}{R}+\omega V^{*}\frac{R^{*}}{R}+\psi I-\psi I^{*}\\&&-\psi I\frac{R^{*}}{R}+\psi I^{*}\frac{R^{*}}{R}-\delta\frac{(R-R^{*})^{2}}{R}. \end{eqnarray}$

We can write $D^{\xi, \lambda}_{t} L = \Sigma-\Omega$ , where

$\begin{eqnarray} \Sigma& = &\textbf{a}+(\rho+\rho_{1})S+(\rho+\rho_{1})S^{*}\frac{E^{*}}{E}+\rho_{2}\alpha V^{*}\frac{(E-E^{*})^{2}}{E}+\rho_{2}\alpha E\frac{(V-V^{*})^{2}}{V}+\textbf{b}\frac{(V-V^{*})^{2}}{V} +\phi V\\&&+\phi V^{*}\frac{I^{*}}{I}+\omega V+\omega V^{*}\frac{R^{*}}{R}+\psi I+\psi I^{*}\frac{R^{*}}{R}, \end{eqnarray}$

and

$\begin{eqnarray} \Omega& = &\textbf{a}\frac{S^{*}}{S}+(\delta+\rho+\rho_{1})\frac{(S-S^{*})^{2}}{S}+(\rho+\rho_{1})S^{*} +(\rho+\rho_{1})S\frac{E^{*}}{E}+\delta\frac{(E-E^{*})^{2}}{E}+\rho_{2}\alpha V\frac{(E-E^{*})^{2}}{E}\\&&+\rho_{2}\alpha E^{*}\frac{(V-V^{*})^{2}}{V}+(\delta+\mu_{0}+\omega+\phi)\frac{(V-V^{*})^{2}}{V}+\phi V^{*}+\phi V\frac{I^{*}}{I}+(\mu_{0}+\delta+\psi)\frac{(I-I^{*})^{2}}{I} +\omega V^{*}\\&&+\omega V\frac{R^{*}}{R}+\psi V^{*}+\psi I\frac{R^{*}}{R}+\delta\frac{(R-R^{*})^{2}}{R}. \end{eqnarray}$

We conclude that if $\Sigma < \Omega$ , this yields $D^{\xi, \lambda}_{t} L < 0$ , however when $S = S^{*}, E = E^{*}, V = V^{*}, I = I^{*}$ and $R = R^{*}$ , $\Sigma-\Omega = 0 \quad \Rightarrow D^{\xi, \lambda}_{t} L = 0$ .

We can observe that $\left\{(S^{*}, E^{*}, V^{*}, I^{*}, R^{*})\in\Gamma : D^{\xi, \lambda}_{t} L = 0\right\}$ represents the point $D_2$ for the developed model.

According to Lasalles' concept of invariance, $D_2$ is globally uniformly stable in $\Gamma$ if $\Sigma-\Omega = 0$ .

6. Solutions by fractal-fractional operator

Now, we will develop a solution using a numerical approach for our newly developed model given in Eq (2.2). We use the ML kernel in the current scenario instead of the classical derivative operator.

Furthermore, we will use the variable order version.

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t} S(t)& = &\textbf{a}-(\delta+\rho+\rho_{1})S,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t} E(t)& = &(\rho+\rho_{1})S-\delta E-\rho_{2}\alpha E V,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t} V(t)& = &\rho_{2}\alpha E V-(\delta+\mu_{0}+\omega-\textbf{b}+\phi)V,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t} I(t)& = &\phi V-(\mu_{0}+\delta+\psi)I,\\ ^{FFM}_{0}D^{\xi,\lambda}_{t} R(t)& = &\omega V+\psi I-\delta R. \end{eqnarray}$

For clarity, we express the above equation as follows:

$\begin{eqnarray} ^{FFM}_{0}D^{\xi,\lambda}_{t} S(t)& = &S_{1}(t, S, G),\\ ^{FFM}_{0}D^{\xi,\lambda}_{t} E(t)& = &E_{1}(t, S, G),\\ ^{FFM}_{0}D^{\xi,\lambda}_{t} V(t)& = &V_{1}(t, S, G),\\ ^{FFM}_{0}D^{\xi,\lambda}_{t} I(t)& = &I_{1}(t, S, G),\\ ^{FFM}_{0}D^{\xi,\lambda}_{t} R(t)& = &R_{1}(t, S, G). \end{eqnarray}$

Where

$\begin{eqnarray} S_{1}(t, S, G)& = &\textbf{a}-(\delta+\rho+\rho_{1})S,\\ E_{1}(t, S, G)& = &(\rho+\rho_{1})S-\delta E-\rho_{2}\alpha E V,\\ V_{1}(t, S, G)& = &\rho_{2}\alpha E V-(\delta+\mu_{0}+\omega-\textbf{b}+\phi)V,\\ I_{1}(t, S, G)& = &\phi V-(\mu_{0}+\delta+\psi)I,\\ R_{1}(t, S, G)& = &\omega V+\psi I-\delta R. \end{eqnarray}$

After using the fractal-fractional integral with the ML kernel, we obtain the following results:

$\begin{eqnarray} S(t_{\eta}+1)& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}S_{1}\left(t_{\eta},S(t_{\eta}),G(t_{\eta})\right)\\&&+ \frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\int_{t_{\nu}}^{t_{\nu+1}}S_{1}(t,S,G)\tau^{\xi-1}(t_{\eta+1} -\tau)^{\xi-1}{d}\tau,\\ E(t_{\eta}+1)& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}E_{1}\left(t_{\eta},S(t_{\eta}),G(t_{\eta})\right) \\&&+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\int_{t_{\nu}}^{t_{\nu+1}}E_{1}(t,S,G)\tau^{\xi-1}(t_{\eta+1} -\tau)^{\xi-1}{d}\tau,\\ V(t_{\eta}+1)& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}V_{1}\left(t_{\eta},S(t_{\eta}),G(t_{\eta})\right) \\&&+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\int_{t_{\nu}}^{t_{\nu+1}}V_{1}(t,S,G)\tau^{\xi-1}(t_{\eta+1} -\tau)^{\xi-1}{d}\tau,\\ I(t_{\eta}+1)& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}I_{1}\left(t_{\eta},S(t_{\eta}),G(t_{\eta})\right) \\&&+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\int_{t_{\nu}}^{t_{\nu+1}}I_{1}(t,S,G)\tau^{\xi-1}(t_{\eta+1} -\tau)^{\xi-1}{d}\tau,\\ R(t_{\eta}+1)& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}R_{1}\left(t_{\eta},S(t_{\eta}),G(t_{\eta})\right) \\&&+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\int_{t_{\nu}}^{t_{\nu+1}}R_{1}(t,S,G)\tau^{\xi-1}(t_{\eta+1} -\tau)^{\xi-1}{d}\tau, \end{eqnarray}$

(6.1)

where $G(t_{\eta}) = E(t_{\eta}), \, \, V(t_{\eta}), \, \, I(t_{\eta}), \, \, R(t_{\eta})$ .

Remember that the Newton polynomial can be obtained by using the Newton interpolation formula.

$\begin{eqnarray} N(t, S, G)&\simeq&N(t_{\eta-2},S_{\eta-2},G_{\eta-2})+\frac{1}{\Delta t}\Big[N(t_{\eta-1},S_{\eta-1},G_{\eta-1})\\ &-&N(t_{\eta-2},S_{\eta-2},G_{\eta-2})\Big]\left(\tau-t_{\eta-2}\right)+\frac{1}{2\Delta t^{2}}\Big[N(t_{\eta},S_{\eta},E_{\eta},I_{\eta},Q_{\eta},R_{\eta})\\ &-&2N(t_{\eta-1},S_{\eta-1},G_{\eta-1})-N(t_{\eta-2},S_{\eta-2},G_{\eta-2})\Big] \left(\tau-t_{\eta-2}\right)\left(\tau-t_{\eta-1}\right), \end{eqnarray}$

where $G_{\eta-2} = E_{\eta-2}, V_{\eta-2}, I_{\eta-2}, R_{\eta-2}$ , $G_{\eta-1} = E_{\eta-1}, V_{\eta-1}, I_{\eta-1}, R_{\eta-1}$ .

When we substitute the Newton polynomial into the system of Eqs (6.1), we obtain the following:

$\begin{eqnarray} S^{\eta+1}& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}S_{1}(t_{\eta},S(t_{\eta}),G(t_{\eta}))+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}S_{1} (t_{\nu-2},S^{\nu-2},G^{\nu-2})\\&&\times t^{\lambda-1}_{\nu-2} \int_{t_{\nu}}^{t_{\nu+1}}(t_{\eta+1}-\tau)^{\xi-1}{d}\tau +\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\frac{1}{\Delta t}\Big[t^{\lambda-1}_{\nu-1}S_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1}) \\&&-t^{\lambda-1}_{\nu-2} S_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau\\&&+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\frac{1}{2\Delta t^{2}}\Big[t^{\lambda-1}_{\nu}S_{1}(t_{\nu},S^{\nu},G^{\nu}) -2t^{\lambda-1}_{\nu-1}S_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\&&+t^{\lambda-1}_{\nu-2} S_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(\tau-t_{\nu-1})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau.\\ E^{\eta+1}& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}E_{1}(t_{\eta},S(t_{\eta}),G(t_{\eta}))+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}E_{1} (t_{\nu-2},S^{\nu-2},G^{\nu-2})\\&&\times t^{\lambda-1}_{\nu-2}\int_{t_{\nu}}^{t_{\nu+1}}(t_{\eta+1}-\tau)^{\xi-1}{d}\tau\ +\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\frac{1}{\Delta t}\Big[t^{\lambda-1}_{\nu-1}E_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\ &&-t^{\lambda-1}_{\nu-2} E_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau\\ &&+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\frac{1}{2\Delta t^{2}}\Big[t^{\lambda-1}_{\nu} E_{1}(t_{\nu},S^{\nu},G^{\nu}) -2t^{\lambda-1}_{\nu-1}E_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\&&+t^{\lambda-1}_{\nu-2} E_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(\tau-t_{\nu-1})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau.\\ V^{\eta+1}& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}V_{1}(t_{\eta},S(t_{\eta}),G(t_{\eta}))+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}V_{1} (t_{\nu-2},S^{\nu-2},G^{\nu-2})\\&&\times t^{\lambda-1}_{\nu-2}\int_{t_{\nu}}^{t_{\nu+1}}(t_{\eta+1}-\tau)^{\xi-1}{d}\tau +\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\frac{1}{\Delta t} \Big[t^{\lambda-1}_{\nu-1} V_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\ &&-t^{\lambda-1}_{\nu-2} V_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau\\&&+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\frac{1}{2\Delta t^{2}}\Big[t^{\lambda-1}_{\nu}V_{1}(t_{\nu},S^{\nu},G^{\nu}) -2t^{\lambda-1}_{\nu-1}V_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\&&+t^{\lambda-1}_{\nu-2} V_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(\tau-t_{\nu-1})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau.\\ I^{\eta+1}& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}I_{1}(t_{\eta},S(t_{\eta}),G(t_{\eta}))+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}I_{1} (t_{\nu-2},S^{\nu-2},G^{\nu-2})\\ &&\times t^{\lambda-1}_{\nu-2}\int_{t_{\nu}}^{t_{\nu+1}}(t_{\eta+1}-\tau)^{\xi-1}{d}\tau +\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\frac{1}{\Delta t}\Big[t^{\lambda-1}_{\nu-1} I_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\&&-t^{\lambda-1}_{\nu-2} I_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau\\ &&+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\frac{1}{2\Delta t^{2}}\Big[t^{\lambda-1}_{\nu}I_{1}(t_{\nu},S^{\nu},G^{\nu}) -2t^{\lambda-1}_{\nu-1}Q_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\&&+t^{\lambda-1}_{\nu-2} I_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(\tau-t_{\nu-1})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau.\\ R^{\eta+1}& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}R_{1}(t_{\eta},S(t_{\eta}),G(t_{\eta}))+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}R_{1} (t_{\nu-2},S^{\nu-2},G^{\nu-2})\\&&\times t^{\lambda-1}_{\nu-2}\int_{t_{\nu}}^{t_{\nu+1}}(t_{\eta+1} -\tau)^{\xi-1}{d}\tau\ +\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\frac{1}{\Delta t}\Big[t^{\lambda-1}_{\nu-1}R_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\ &&-t^{\lambda-1}_{\nu-2}R_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau\\ &&+\frac{\xi \lambda}{A B(\xi)\Gamma(\xi)}\sum^{\eta}_{\nu = 2}\frac{1}{2\Delta t^{2}}\Big[t^{\lambda-1}_{\nu}R_{1}(t_{\nu},S^{\nu},G^{\nu}) -2t^{\lambda-1}_{\nu-1}R_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\&&+t^{\lambda-1}_{\nu-2} R_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(\tau-t_{\nu-1})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau, \end{eqnarray}$

(6.2)

where $G^{\nu-2} = E^{\nu-2}, V^{\nu-2}, I^{\nu-2}, R^{\nu-2}$ , $G^{\nu-1} = E^{\nu-1}, V^{\nu-1}, I^{\nu-1}, R^{\nu-1}$ , $G^{\nu} = E^{\nu}, V^{\nu}, I^{\nu}, R^{\nu}$ , and $G(t_{\eta}) = E(t_{\eta}), V(t_{\eta}), I(t_{\eta}), R(t_{\eta})$ .

We can perform the following calculations for the integral in Eq (6.2):

$\begin{eqnarray} \int_{t_{\nu}}^{t_{\nu+1}}(t_{\eta+1}-\tau)^{\xi-1}{d}\tau& = &\frac{(\Delta t)^{\xi}}{\xi}\Big[(\eta-\nu+1)^{\xi}-(\eta-\nu)^{\xi}\Big],\\ \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau& = &\frac{(\Delta t)^{\xi+1}}{\xi(\xi+1)}\Big[(\eta-\nu+1)^{\xi}\\&&\times (\eta-\nu+3+2\xi)-(\eta-\nu)^{\xi}(\eta-\nu+3+3\xi)\Big].\\ \int_{t_{\nu}}^{t_{\nu+1}}(\tau-t_{\nu-2})(\tau-t_{\nu-1})(t_{\eta+1}-\tau)^{\xi-1}{d}\tau& = &\frac{(\Delta t)^{\xi+2}}{\xi(\xi+1)(\xi+2)}\Big[(\eta-\nu+1)^{\xi}\Big\{2(\eta-\nu)^{2}+(3\xi+10)\\&&\times(\eta-\nu)+2\xi^{2}+9\xi+12\Big\}-(\eta-\nu)^{\xi}\Big\{2(\eta-\nu)^{2}\\ &&+(5\xi+10)(\eta-\nu)+6\xi^{2}+18\xi+12\Big\}\Big], \end{eqnarray}$

(6.3)

substituting integral calculation values into Eq (6.2).

We acquire the numerical solutions $S(t), E(t), V(t), I(t)\, \, and \, \, R(t)$ :

$\begin{eqnarray} S^{\eta+1}& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}S_{1}(t_{\eta},S(t_{\eta}),G(t_{\eta})) +\frac{\xi \lambda(\Delta t)^{\xi}}{A B(\xi)\Gamma(\xi+1)}\sum^{\eta}_{\nu = 2} S_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\\&&\times t^{\lambda-1}_{\nu-2} \Big[(\eta-\nu+1)^{\xi}-(\eta-\nu)^{\xi}\Big]+\frac{\xi \lambda(\Delta t)^{\xi}}{A B(\xi)\Gamma(\xi+2)}\sum^{\eta}_{\nu = 2} \Big[t^{\lambda-1}_{\nu-1}S_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\&&- t^{\lambda-1}_{\nu-2}S_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \Big[(\eta-\nu+1)^{\xi}(\eta-\nu+3+2\xi)-(\eta-\nu)^{\xi}(\eta-\nu+3+3\xi)\Big]\\&&+\frac{\xi \lambda(\Delta t)^{\xi}}{2A B(\xi)\Gamma(\xi+3)}\sum^{\eta}_{\nu = 2}\Big[t^{\lambda-1}_{\nu}S_{1} (t_{\nu},S^{\nu},G^{\nu}) -2t^{\lambda-1}_{\nu-1}S_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\ &&+t^{\lambda-1}_{\nu-2}S_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \Big[(\eta-\nu+1)^{\xi}\Big\{2(\eta-\nu)^{2}+(3\xi+10)(\eta-\nu)+2\xi^{2}+9\xi+12\Big\}\\ &&-(\eta-\nu)^{\xi}\times\Big\{2(\eta-\nu)^{2}+(5\xi+10)(\eta-\nu)+6\xi^{2}+18\xi+12\Big\}\Big],\\ E^{\eta+1}& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}E_{1}(t_{\eta},S(t_{\eta}),G(t_{\eta})) +\frac{\xi \lambda(\Delta t)^{\xi}}{A B(\xi)\Gamma(\xi+1)}\sum^{\eta}_{\nu = 2} E_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\\&&\times t^{\lambda-1}_{\nu-2} \Big[(\eta-\nu+1)^{\xi}-(\eta-\nu)^{\xi}\Big]+\frac{\xi \lambda(\Delta t)^{\xi}}{A B(\xi)\Gamma(\xi+2)}\sum^{\eta}_{\nu = 2} \Big[t^{\lambda-1}_{\nu-1}E_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\ &&- t^{\lambda-1}_{\nu-2}E_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \Big[(\eta-\nu+1)^{\xi}(\eta-\nu+3+2\xi)-(\eta-\nu)^{\xi}(\eta-\nu+3+3\xi)\Big]\\ &&+\frac{\xi \lambda(\Delta t)^{\xi}}{2A B(\xi)\Gamma(\xi+3)}\sum^{\eta}_{\nu = 2}\Big[t^{\lambda-1}_{\nu}E_{1} (t_{\nu},S^{\nu},G^{\nu}) -2t^{\lambda-1}_{\nu-1}E_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\ &&+t^{\lambda-1}_{\nu-2}E_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \Big[(\eta-\nu+1)^{\xi}\Big\{2(\eta-\nu)^{2}+(3\xi+10)(\eta-\nu)+2\xi^{2}+9\xi+12\Big\}\\&&- (\eta-\nu)^{\xi}\times\Big\{2(\eta-\nu)^{2}+(5\xi+10)(\eta-\nu)+6\xi^{2}+18\xi+12\Big\}\Big],\\ V^{\eta+1}& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}V_{1}(t_{\eta},S(t_{\eta}),G(t_{\eta}))+\frac{\xi \lambda(\Delta t)^{\xi}}{A B(\xi)\Gamma(\xi+1)}\sum^{\eta}_{\nu = 2} V_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\\&&\times t^{\lambda-1}_{\nu-2} \Big[(\eta-\nu+1)^{\xi}-(\eta-\nu)^{\xi}\Big]+\frac{\xi \lambda(\Delta t)^{\xi}}{A B(\xi)\Gamma(\xi+2)}\sum^{\eta}_{\nu = 2} \Big[t^{\lambda-1}_{\nu-1}V_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\&&- t^{\lambda-1}_{\nu-2}V_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \Big[(\eta-\nu+1)^{\xi}(\eta-\nu+3+2\xi)-(\eta-\nu)^{\xi}(\eta-\nu+3+3\xi)\Big]\\ &&+\frac{\xi \lambda(\Delta t)^{\xi}}{2A B(\xi)\Gamma(\xi+3)}\sum^{\eta}_{\nu = 2}\Big[t^{\lambda-1}_{\nu}V_{1} (t_{\nu},S^{\nu},G^{\nu}) -2t^{\lambda-1}_{\nu-1}V_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\ &&+t^{\lambda-1}_{\nu-2}V_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \Big[(\eta-\nu+1)^{\xi}\Big\{2(\eta-\nu)^{2}+(3\xi+10)(\eta-\nu)+2\xi^{2}+9\xi+12\Big\}\\&&- (\eta-\nu)^{\xi}\times\Big\{2(\eta-\nu)^{2}+(5\xi+10)(\eta-\nu)+6\xi^{2}+18\xi+12\Big\}\Big],\\ I^{\eta+1}& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}I_{1}(t_{\eta},S(t_{\eta}),G(t_{\eta})) +\frac{\xi \lambda(\Delta t)^{\xi}}{A B(\xi)\Gamma(\xi+1)}\sum^{\eta}_{\nu = 2} I_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\\&&\times t^{\lambda-1}_{\nu-2} \Big[(\eta-\nu+1)^{\xi}-(\eta-\nu)^{\xi}\Big]+\frac{\xi \lambda(\Delta t)^{\xi}}{A B(\xi)\Gamma(\xi+2)}\sum^{\eta}_{\nu = 2} \Big[t^{\lambda-1}_{\nu-1}I_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\ &&-t^{\lambda-1}_{\nu-2}I_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \Big[(\eta-\nu+1)^{\xi}(\eta-\nu+3+2\xi)-(\eta-\nu)^{\xi}(\eta-\nu+3+3\xi)\Big]\\&&+\frac{\xi \lambda(\Delta t)^{\xi}}{2A B(\xi)\Gamma(\xi+3)}\sum^{\eta}_{\nu = 2}\Big[t^{\lambda-1}_{\nu}I_{1} (t_{\nu},S^{\nu},G^{\nu}) -2t^{\lambda-1}_{\nu-1}I_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\ &&+t^{\lambda-1}_{\nu-2}I_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \Big[(\eta-\nu+1)^{\xi}\Big\{2(\eta-\nu)^{2}+(3\xi+10)(\eta-\nu)+2\xi^{2}+9\xi+12\Big\}\\ &&-(\eta-\nu)^{\xi}\times\Big\{2(\eta-\nu)^{2}+(5\xi+10)(\eta-\nu)+6\xi^{2}+18\xi+12\Big\}\Big],\\ R^{\eta+1}& = &\frac{\lambda(1-\xi)}{A B(\xi)}t_{\eta}^{\lambda-1}R_{1}(t_{\eta},S(t_{\eta}),G(t_{\eta}))+\frac{\xi \lambda(\Delta t)^{\xi}}{A B(\xi)\Gamma(\xi+1)}\sum^{\eta}_{\nu = 2} R_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\\&&\times t^{\lambda-1}_{\nu-2} \Big[(\eta-\nu+1)^{\xi}-(\eta-\nu)^{\xi}\Big]+\frac{\xi \lambda(\Delta t)^{\xi}}{A B(\xi)\Gamma(\xi+2)}\sum^{\eta}_{\nu = 2} \Big[t^{\lambda-1}_{\nu-1}R_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\&&- t^{\lambda-1}_{\nu-2}R_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \Big[(\eta-\nu+1)^{\xi}(\eta-\nu+3+2\xi)-(\eta-\nu)^{\xi}(\eta-\nu+3+3\xi)\Big]\\ &&+\frac{\xi \lambda(\Delta t)^{\xi}}{2 A B(\xi)\Gamma(\xi+3)}\sum^{\eta}_{\nu = 2}\Big[t^{\lambda-1}_{\nu}R_{1} (t_{\nu},S^{\nu},G^{\nu}) -2t^{\lambda-1}_{\nu-1}R_{1}(t_{\nu-1},S^{\nu-1},G^{\nu-1})\\ &&+t^{\lambda-1}_{\nu-2}R_{1}(t_{\nu-2},S^{\nu-2},G^{\nu-2})\Big] \Big[(\eta-\nu+1)^{\xi}\Big\{2(\eta-\nu)^{2}+(3\xi+10)(\eta-\nu)+2\xi^{2}+9\xi+12\Big\}\\&&-(\eta-\nu)^{\xi}\times\Big\{2(\eta-\nu)^{2}+(5\xi+10)(\eta-\nu)+6\xi^{2}+18\xi+12\Big\}\Big]. \end{eqnarray}$

7. Simulation explanation

In this section, we utilized an advanced technique to obtain theoretical outcomes and assess their effectiveness. The newly developed SEVIR system was analyzed through simulation. By applying non-integer parametric values in the SARS-COVID-19 model, we obtained interesting findings. – display the solutions for $S(t)$ , $E(t)$ , $V(t)$ , $I(t)$ , and $R(t)$ by reducing the fractional values to the desired level. To validate the efficiency of the theoretical outcomes, we provide the following examples. Numerical simulations for the SARS-COVID-19 model were performed using MATLAB. The initial conditions used in the newly developed model are $S(0) = 217.342565, E(0) = 100, V(0) = 1.386348, I(0) = 1.1$ and $R(0) = 1.271087$ . The parameter values used in the developed system are as follows: $\textbf{a} = 1.43$ , $\delta = 0.000065$ , $\rho = 0.45$ , $\rho_{1} = 0.10$ , $\rho_{2} = 0.020$ , $\alpha = 0.0008601$ , $\mu_{0} = 0.19$ , $\omega = 0.98$ , $\textbf{b} = 0.135$ , $\phi = 0.0001$ , and $\psi = 0.0001$ . Figures 2, 4, and 5 illustrate the changes in susceptible, vaccinated, and infected individuals respectively, showing a sharp decrease before reaching a stable position. Meanwhile, Figures 3 and 6 demonstrate the dynamics of exposed and recovered individuals respectively at different fractional orders in which both individuals increases and after certain time the number of individuals approaches stable state using different dimensions. The research predicts future infection rates and suggests ways to decrease the spread of infection units more effectively. By utilizing a fractal-fractional approach, the study yields reliable and accurate results for all compartments at non-integer order derivatives, which are more trustworthy when fractional values are reduced as well a by reducing its dimensions. The findings suggest that the number of infected individuals decreases significantly due to vaccination measures, while the number of recovered individuals increases due to a decline in infected individuals and the effect of vaccination.

Figure 2. The value of

$S(t)$ using the fractal-fractional operator with various fractional values at different dimensions.

DownLoad: Full-Size Img PowerPoint

Figure 3. The value of

$E(t)$ using the fractal-fractional operator with various fractional values at different dimensions.

DownLoad: Full-Size Img PowerPoint

Figure 4. The value of

$V(t)$ using the fractal-fractional operator with various fractional values at different dimensions.

DownLoad: Full-Size Img PowerPoint

Figure 5. The value of

$I(t)$ using the fractal-fractional operator with various fractional values at different dimensions.

DownLoad: Full-Size Img PowerPoint

Figure 6. The value of

$R(t)$ using the fractal-fractional operator with various fractional values at different dimensions.

DownLoad: Full-Size Img PowerPoint

8. Conclusions

This article employs a fractional order SEVIR model for SARS-COVID-19 with vaccinated effects using an FFO to find reliable solutions. We provide advice on controlling this virus to help our community overcome the pandemic by implementing vaccinated measures for low immune individuals. We analyze the dangerous coronavirus disease with the effect of vaccination to understand its real impact on the community. Qualitative and quantitative analyses are conducted to verify its stable position in a continuous dynamical system. We also verify that the fractional order coronavirus disease model has bounded and unique solutions. We examine the impact of global measures to control the spread of the coronavirus disease. Also, analyses are performed to see how the rate of infection changes after the implementation of vaccination measures. We ensure that our findings are reliable and realistic. FFO is used for continuously monitoring the spread as well as control of the disease in society after vaccination measures. It was observed that infected individuals recover quickly due to the vaccinated strategy. The fractal-fractional operator (FFO) is used for continuously monitoring for the spread of the diseases using different fractional values as well as reliable solutions. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease, and fractional represents the fractional ordered derivative operator which provides the real behavior of the spread as well as control of COVID-19 with different dimensions and continuous monitoring, which can be observed in simulation. We conduct numerical simulations to observe how the disease controlled in the community after the implementation of vaccination measures using different fractional values with different dimensions. Additionally, future estimates are provided based on our findings, which can help in mitigating the risk of the disease spreading in the environment.

Use of AI tools declaration

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

Acknowledgments

The authors would like to extend their sincere appreciation to Researchers Supporting Project number (RSP2024R472), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	X. Liu, D. Hou, F. Lin, J. Luo, J. Xie, Y. Wang, et al., The role of neurovascular unit damage in the occurrence and development of Alzheimer's disease, Rev. Neurosci., 30 (2019), 477–484. https://doi.org/10.1515/revneuro-2018-0056 doi: 10.1515/revneuro-2018-0056
[2]	F. Falahati, E. Westman, A. Simmons, Multivariate data analysis and machine learning in Alzheimer's disease with a focus on structural magnetic resonance imaging, J. Alzheimers Dis., 41 (2014), 685–708. https://doi.org/10.3233/JAD-131928 doi: 10.3233/JAD-131928
[3]	A. B. Sallim, A. A. Sayampanathan, A. Cuttilan, R. Chun-Man Ho, Prevalence of mental health disorders among caregivers of patients with Alzheimer disease, J. Am. Med. Dir. Assoc., 16 (2015), 1034–1041. https://doi.org/10.1016/j.jamda.2015.09.007 doi: 10.1016/j.jamda.2015.09.007
[4]	E. L. G. E. Koedam, V. Lauffer, A. E. van der Vlies, W. M. van der Flier, P. Scheltens, Y. A. L. Pijnenburg, Early-versus late-onset Alzheimer's disease: More than age alone, J. Alzheimers Dis., 19 (2010), 1401–1408. https://doi.org/10.3233/JAD-2010-1337 doi: 10.3233/JAD-2010-1337
[5]	Y. Freudenberg-Hua, W. Li, P. Davies, The role of genetics in advancing precision medicine for Alzheimer's disease—a narrative review, Front. Med., 5 (2018), 108. https://doi.org/10.3389/fmed.2018.00108 doi: 10.3389/fmed.2018.00108
[6]	E. Giacobini, G. Gold, Alzheimer disease therapy—moving from amyloid-β to tau, Nat. Rev. Neurol., 9 (2013), 677–686. https://doi.org/10.1038/nrneurol.2013.223 doi: 10.1038/nrneurol.2013.223
[7]	R. J. Jutten, S. A. M. Sikkes, R. E. Amariglio, R. F. Buckley, M. J. Properzi, G. A. Marshall, et al., Identifying sensitive measures of cognitive decline at different clinical stages of Alzheimer's disease, J. Int. Neuropsychol. Soc., 27 (2021), 426–438. https://doi.org/10.1017/S1355617720000934 doi: 10.1017/S1355617720000934
[8]	D. A. McGrowder, F. Miller, K. Vaz, C. Nwokocha, C. Wilson-Clarke, M. Anderson-Cross, et al., Cerebrospinal fluid biomarkers of Alzheimer's disease: Current evidence and future perspectives, Brain Sci., 11 (2021), 215. https://doi.org/10.3390/brainsci11020215 doi: 10.3390/brainsci11020215
[9]	R. L. Cazzato, J. Garnon, B. Shaygi, G. Koch, G. Tsoumakidou, J. Caudrelier, et al., PET/CT-guided interventions: Indications, advantages, disadvantages and the state of the art, Minimally Invasive Ther. Allied Technol., 27 (2018), 27–32. https://doi.org/10.1080/13645706.2017.1399280 doi: 10.1080/13645706.2017.1399280
[10]	M. Amini, M. M. Pedram, A. Moradi, M. Jamshidi, M. Ouchani, Single and combined neuroimaging techniques for Alzheimer's disease detection, Comput. Intell. Neurosci., 2021 (2021), 9523039. https://doi.org/10.1155/2021/9523039 doi: 10.1155/2021/9523039
[11]	C. E. Wierenga, M. W. Bondi, Use of functional magnetic resonance imaging in the early identification of Alzheimer's disease, Neuropsychol. Rev., 17 (2007), 127–143. https://doi.org/10.1007/s11065-007-9025-y doi: 10.1007/s11065-007-9025-y
[12]	N. J. Gong, C. C. Chan, L. M. Leung, C. S. Wong, R. Dibb, C. Liu, Differential microstructural and morphological abnormalities in mild cognitive impairment and Alzheimer's disease: Evidence from cortical and deep gray matter, Hum. Brain Mapp., 38 (2017), 2495–2508. https://doi.org/10.1002/hbm.23535 doi: 10.1002/hbm.23535
[13]	C. Van Cauwenberghe, C. Van Broeckhoven, K. Sleegers, The genetic landscape of Alzheimer disease: Clinical implications and perspectives, Genet. Med., 18 (2016), 421–430. https://doi.org/10.1038/gim.2015.117 doi: 10.1038/gim.2015.117
[14]	B. L. Romero-Rosales, J. G. Tamez-Pena, H. Nicolini, M. G. Moreno-Treviño, V. Trevino, Improving predictive models for Alzheimer's disease using GWAS data by incorporating misclassified samples modeling, PLoS One, 15 (2020). https://doi.org/10.1371/journal.pone.0232103 doi: 10.1371/journal.pone.0232103
[15]	T. Jo, K. Nho, A. J. Saykin, Deep learning in Alzheimer's disease: diagnostic classification and prognostic prediction using neuroimaging data, Front. Aging Neurosci., 11 (2019). https://doi.org/10.3389/fnagi.2019.00220 doi: 10.3389/fnagi.2019.00220
[16]	J. Ha, MDMF: predicting miRNA-disease association based on matrix factorization with disease similarity constraint, J. Pers. Med., 12 (2022). https://doi.org/10.3390/jpm12060885 doi: 10.3390/jpm12060885
[17]	J. Ha, SMAP: Similarity-based matrix factorization framework for inferring miRNA-disease association, Knowl-Based Syst., 263 (2023). https://doi.org/10.1016/j.knosys.2023.110295 doi: 10.1016/j.knosys.2023.110295
[18]	J. De. Velasco Oriol, E. E. Vallejo, K. Estrada, Benchmarking machine learning models for late-onset Alzheimer's disease prediction from genomic data, BMC Bioinf., 20 (2019), 1–17. https://doi.org/10.1186/s12859-019-3158-x doi: 10.1186/s12859-018-2565-8
[19]	L. Xu, G. Liang, C. Liao, G. D. Chen, C. C. Chang, An efficient classifier for Alzheimer's disease genes identification, Molecules, 23 (2018), 3140. https://doi.org/10.3390/molecules23123140 doi: 10.3390/molecules23123140
[20]	D. Castillo-Barnes, L. Su, J. Ramírez, D. Salas-Gonzalez, F. J. Martinez-Murcia, I. A. Illan, et al., Autosomal dominantly inherited Alzheimer disease: Analysis of genetic subgroups by machine learning, Inf. Fusion, 58 (2020), 153–167. https://doi.org/10.1016/j.inffus.2020.01.001 doi: 10.1016/j.inffus.2020.01.001
[21]	N. Voyle, A. Keohane, S. Newhouse, K. Lunnon, C. Johnston, H. Soininen, et al., A pathway based classification method for analyzing gene expression for Alzheimer's disease diagnosis, J. Alzheimers Dis., 49 (2016), 659–669. https://doi.org/10.3233/JAD-150440 doi: 10.3233/JAD-150440
[22]	E. Moradi, M. Marttinen, T. Häkkinen, M. Hiltunen, M. Nykter, Supervised pathway analysis of blood gene expression profiles in Alzheimer's disease, Neurobiol. Aging, 84 (2019), 98–108. https://doi.org/10.1016/j.neurobiolaging.2019.07.004 doi: 10.1016/j.neurobiolaging.2019.07.004
[23]	D. Cheng, M. Liu, Classification of Alzheimer's disease by cascaded convolutional neural networks using PET images, in Machine Leaening in Medical Imaging, Springer, (2017), 106–113. https://doi.org/10.1007/978-3-319-67389-9_13
[24]	M. Grassi, G. Perna, D. Caldirola, K. Schruers, R. Duara, D. A. Loewenstein, A clinically-translatable machine learning algorithm for the prediction of Alzheimer's disease conversion in individuals with mild and premild cognitive impairment, J. Alzheimers Dis., 61 (2018), 1555–1573. https://doi.org/10.3233/JAD-170547 doi: 10.3233/JAD-170547
[25]	S. M. Plis, D. R. Hjelm, R. Salakhutdinov, E. A. Allen, H. J. Bockholt, J. D. Long, et al., Deep learning for neuroimaging: a validation study, Front. Neurosci., 8 (2014). https://doi.org/10.3389/fnins.2014.00229 doi: 10.3389/fnins.2014.00229
[26]	S. Wang, H. Wang, Y. Shen, X. Wang, Automatic recognition of mild cognitive impairment and Alzheimers disease using ensemble based 3D densely connected convolutional networks, in 17th IEEE International Conference on Machine Learning and Applications (ICMLA), (2018), 517–523. https://doi.org/10.1109/icmla.2018.00083
[27]	W. Yu, B. Lei, M. K. Ng, A. C. Cheung, Y. Shen, S. Wang, Tensorizing GAN with high-order pooling for Alzheimer's disease assessment, IEEE Trans. Neural Networks Learn. Syst., 33 (2022), 4945–4959. https://doi.org/10.1109/TNNLS.2021.3063516 doi: 10.1109/TNNLS.2021.3063516
[28]	W. Yu, B. Lei, S. Wang, Y. Liu, Z. Feng, Y. Hu, et al., Morphological feature visualization of Alzheimer's disease via multidirectional perception GAN, IEEE Trans. Neural Networks Learn. Syst., 2022 (2022), 1–15. https://doi.org/10.1109/TNNLS.2021.3118369 doi: 10.1109/TNNLS.2021.3118369
[29]	T. Lee, H. Lee, Prediction of Alzheimer's disease using blood gene expression data, Sci. Rep., 10 (2020), 3485. https://doi.org/10.1038/s41598-020-60595-1 doi: 10.1038/s41598-020-60595-1
[30]	N. Mahendran, P. Durai Raj Vincent, A deep learning framework with an embedded-based feature selection approach for the early detection of the Alzheimer's disease, Comput. Biol. Med., 141 (2022), 105056. https://doi.org/10.1016/j.compbiomed.2021.105056 doi: 10.1016/j.compbiomed.2021.105056
[31]	C. Park, J. Ha, S. Park, Prediction of Alzheimer's disease based on deep neural network by integrating gene expression and DNA methylation dataset, Expert Syst. Appl., 140 (2020). https://doi.org/10.1016/j.eswa.2019.112873 doi: 10.1016/j.eswa.2019.112873
[32]	Y. Liu, Z. Li, Q. Ge, N. Lin, M. Xiong, Deep feature selection and causal analysis of Alzheimer's disease, Front. Neurosci., 13 (2019). https://doi.org/10.3389/fnins.2019.01198 doi: 10.3389/fnins.2019.01198
[33]	S. Spasov, L. Passamonti, A. Duggento, P. Liò, N. Toschi, A parameter-efficient deep learning approach to predict conversion from mild cognitive impairment to Alzheimer's disease, NeuroImage, 189 (2019), 276–287. https://doi.org/10.1016/j.neuroimage.2019.01.031 doi: 10.1016/j.neuroimage.2019.01.031
[34]	S. Gauthier, B. Reisberg, M. Zaudig, R. C. Petersen, K. Ritchie, K. Broich, et al., Mild cognitive impairment, Lancet, 367 (2006), 1262–1270. https://doi.org/10.1016/S0140-6736(06)68542-5 doi: 10.1016/S0140-6736(06)68542-5
[35]	M. Grundman, R. C. Petersen, S. H. Ferris, R. G. Thomas, P. S. Aisen, D. A. Bennett, et al., Mild cognitive impairment can be distinguished from Alzheimer disease and normal aging for clinical trials, Arch. Neurol., 61 (2004), 59–66. https://doi.org/10.1001/archneur.61.1.59 doi: 10.1001/archneur.61.1.59
[36]	A. Kadra, M. Lindauer, F. Hutter, J. Grabocka, Well-tuned simple nets excel on tabular datasets, 2021.
[37]	S. Popov, S. Morozov, A. Babenko, Neural oblivious decision ensembles for deep learning on tabular data, arXiv preprint, 2019, arXiv: 1909.06312. https://doi.org/10.48550/arXiv.1909.06312
[38]	C. Shah, Q. Du, Y. Xu, Enhanced TabNet: attentive interpretable tabular learning for hyperspectral image classification, Remote Sens., 14 (2022), 716. https://doi.org/10.3390/rs14030716 doi: 10.3390/rs14030716
[39]	Y. Y. Song, Y. Lu, Decision tree methods: applications for classification and prediction, Shanghai Arch Psychiatry, 27 (2015), 130–135.
[40]	G. Biau, E. Scornet, A random forest guided tour, TEST, 25 (2016), 197–227. https://doi.org/10.1007/s11749-016-0481-7 doi: 10.1007/s11749-016-0481-7
[41]	C. Zhang, C. Liu, X. Zhang, G. Almpanidis, An up-to-date comparison of state-of-the-art classification algorithms, Expert Syst. Appl., 82 (2017), 128–150. https://doi.org/10.1016/j.eswa.2017.04.003 doi: 10.1016/j.eswa.2017.04.003
[42]	G. Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, et al., Lightgbm: A highly efficient gradient boosting decision tree, 2017.
[43]	M. Pirooznia, J. Y. Yang, M. Q. Yang, Y. Deng, A comparative study of different machine learning methods on microarray gene expression data, BMC Genomics, 9 (2008). https://doi.org/10.1186/1471-2164-9-S1-S13 doi: 10.1186/1471-2164-9-S1-S13
[44]	Q. S. Zhang, S. C. Zhu, Visual interpretability for deep learning: a survey, Front. Inf. Technol. Electron. Eng., 19 (2018), 27–39. https://doi.org/10.1631/FITEE.1700808 doi: 10.1631/FITEE.1700808
[45]	S. Lovestone, P. Francis, I. Kloszewska, P. Mecocci, A. Simmons, H. Soininen, et al., AddNeuroMed-the european collaboration for the discovery of novel biomarkers for Alzheimer's disease, Ann. N. Y. Acad. Sci., 1180 (2009), 36–46. https://doi.org/10.1111/j.1749-6632.2009.05064.x doi: 10.1111/j.1749-6632.2009.05064.x
[46]	S. Sood, I. J. Gallagher, K. Lunnon, E. Rullman, A. Keohane, H. Crossland, et al., A novel multi-tissue RNA diagnostic of healthy ageing relates to cognitive health status, Genome Biol., 16 (2015), 185. https://doi.org/10.1186/s13059-015-0750-x doi: 10.1186/s13059-015-0750-x
[47]	S. Davis, P. S. Meltzer, GEOquery: a bridge between the Gene Expression Omnibus (GEO) and BioConductor, Bioinformatics, 23 (2007), 1846–1847. https://doi.org/10.1093/bioinformatics/btm254 doi: 10.1093/bioinformatics/btm254
[48]	X. Li, H. Wang, J. Long, G. Pan, T. He, O. Anichtchik, et al., Systematic analysis and biomarker study for Alzheimer's disease, Sci. Rep., 8 (2018), 17394. https://doi.org/10.1038/s41598-018-35789-3 doi: 10.1038/s41598-018-35789-3
[49]	A. Antonell, A. Llado, J. Altirriba, T. Botta-Orfila, M. Balasa, M. Fernandez, et al., A preliminary study of the whole-genome expression profile of sporadic and monogenic early-onset Alzheimer's disease, Neurobiol. Aging, 34 (2013), 1772–1778. https://doi.org/10.1016/j.neurobiolaging.2012.12.026 doi: 10.1016/j.neurobiolaging.2012.12.026
[50]	S. Arık, T. Pfister, TabNet: Attentive interpretable tabular learning, in Proceedings of the AAAI Conference on Artificial Intelligence, 35 (2021), 6679–6687. https://doi.org/10.1609/aaai.v35i8.16826
[51]	N. N. Parikshak, M. J. Gandal, D. H. Geschwind, Systems biology and gene networks in neurodevelopmental and neurodegenerative disorders, Nat. Rev. Genet., 16 (2015), 441–458. https://doi.org/10.1038/nrg3934 doi: 10.1038/nrg3934
[52]	G. K. Smyth, limma: Linear models for microarray data, in Bioinformatics and Computational Biology Solutions Using R and Bioconductor, Springer, (2005), 397–420. https://doi.org/10.1007/0-387-29362-0_23
[53]	C. Garbin, X. Zhu, O. Marques, Dropout vs. batch normalization: an empirical study of their impact to deep learning, Multimedia Tools Appl., 79 (2020), 12777–12815. https://doi.org/10.1007/s11042-019-08453-9 doi: 10.1007/s11042-019-08453-9
[54]	Y. N. Dauphin, A. Fan, M. Auli, D. Grangier, Language modeling with gated convolutional networks, in Conference on Machine Learning, (2017), 933–941.
[55]	J. Gehring, M. Auli, D. Grangier, D. Yarats, Y. N. Dauphin, Convolutional sequence to sequence learning, in Proceedings of the 34th International Conference on Machine Learning, 70 (2017), 1243–1252.
[56]	A. Martins, R. Astudillo, From softmax to sparsemax: a sparse model of attention and multi-label classification, in Proceedings of the 33rd International Conference on Machine Learning, 48 (2016), 1614–1623.
[57]	N. Deepa, S. P. Chokkalingam, Optimization of VGG16 utilizing the Arithmetic Optimization Algorithm for early detection of Alzheimer's disease, Biomed. Signal Process. Control, 74 (2022), 103455. https://doi.org/10.1016/j.bspc.2021.103455 doi: 10.1016/j.bspc.2021.103455
[58]	M. B. Kursa, W. R. Rudnicki, Feature selection with the Boruta package, J. Stat. Software, 36 (2010), 1–13. https://doi.org/10.18637/jss.v036.i11 doi: 10.18637/jss.v036.i11
[59]	A. Kulshrestha, O. Farooq, Seizure prediction using fybrid features, in IEEE 7th Uttar Pradesh Section International Conference on Electrical, Electronics and Computer Engineering (UPCON), (2020), 1–6. https://doi.org/10.1109/upcon50219.2020.9376552
[60]	R. Martinez-Cantin, Bayesian optimization with adaptive kernels for robot control, in IEEE International Conference on Robotics and Automation (ICRA), (2017), 3350–3356. https://doi.org/10.1109/ICRA.2017.7989380
[61]	T. Wu, E. Hu, S. Xu, M. Chen, P. Guo, Z. Dai, et al., ClusterProfiler 4.0: A universal enrichment tool for interpreting omics data, Innovation, 2 (2021), 100141. https://doi.org/10.1016/j.xinn.2021.100141 doi: 10.1016/j.xinn.2021.100141
[62]	The Gene Ontology Consortium, The gene ontology resource: 20 years and still GOing strong, Nucleic Acids Res., 47 (2019), 330–338. https://doi.org/10.1093/nar/gky1055 doi: 10.1093/nar/gky1055
[63]	J. Krawczuk, T. Łukaszuk, The feature selection bias problem in relation to high-dimensional gene data, Artif. Intell. Med., 66 (2016), 63–71. https://doi.org/10.1016/j.artmed.2015.11.001 doi: 10.1016/j.artmed.2015.11.001
[64]	S. S. Mehta, N. S. Lingayat, Development of SVM based classification techniques for the delineation of wave components in 12-lead electrocardiogram, Biomed. Signal Process. Control, 3 (2008), 341–349. https://doi.org/10.1016/j.bspc.2008.04.002 doi: 10.1016/j.bspc.2008.04.002
[65]	Z. Tümer, P. J. P. Croucher, L. R. Jensen, J. Hampe, C. Hansen, V. Kalscheuer, et al., Genomic structure, chromosome mapping and expression analysis of the human AVIL gene, and its exclusion as a candidate for locus for inflammatory bowel disease at 12q13–14 (IBD2), Gene, 288 (2002), 179–185. https://doi.org/10.1016/S0378-1119(02)00478-X doi: 10.1016/S0378-1119(02)00478-X
[66]	S. Hong, V. F. Beja-Glasser, B. M. Nfonoyim, A. Frouin, S. Li, S. Ramakrishnan, et al., Complement and microglia mediate early synapse loss in Alzheimer mouse models, Science, 352 (2016), 712–716. https://doi.org/10.1126/science.aad8373 doi: 10.1126/science.aad8373
[67]	A. Quintana, S. E. Kruse, R. P. Kapur, E. Sanz, R. D. Palmiter, Complex I deficiency due to loss of Ndufs4 in the brain results in progressive encephalopathy resembling Leigh syndrome, Proc. Natl. Acad. Sci., 107 (2010), 10996–11001. https://doi.org/10.1073/pnas.1006214107 doi: 10.1073/pnas.1006214107
[68]	D. F. F. Silva, A. R. Esteves, C. R. Oliveira, S. M. Cardoso, Mitochondria: the common upstream driver of amyloid-β and tau pathology in Alzheimer's disease, Curr. Alzheimer Res., 8 (2011), 563–572. https://doi.org/10.2174/156720511796391872 doi: 10.2174/156720511796391872
[69]	M. Calabrò, C. Rinaldi, G. Santoro, C. Crisafulli, The biological pathways of Alzheimer disease: a review, AIMS Neurosci., 8 (2021), 86–132. https://doi.org/10.3934/Neuroscience.2021005 doi: 10.3934/Neuroscience.2021005

This article has been cited by:

1.	Kottakkaran Sooppy Nisar, Muhammad Owais Kulachi, Aqeel Ahmad, Muhammad Farman, Muhammad Saqib, Muhammad Umer Saleem, Fractional order cancer model infection in human with CD8+ T cells and anti-PD-L1 therapy: simulations and control strategy, 2024, 14, 2045-2322, 10.1038/s41598-024-66593-x
2.	Muhammad Owais Kulachi, Aqeel Ahmad, Evren Hincal, Ali Hasan Ali, Muhammad Farman, Muhammad Taimoor, Control of conjunctivitis virus with and without treatment measures: A bifurcation analysis, 2024, 36, 10183647, 103273, 10.1016/j.jksus.2024.103273
3.	Faisal Javed, Aqeel Ahmad, Ali Hasan Ali, Evren Hincal, Ayesha Amjad, Investigation of conjunctivitis adenovirus spread in human eyes by using bifurcation tool and numerical treatment approach, 2024, 99, 0031-8949, 085253, 10.1088/1402-4896/ad62a5
4.	Kottakkaran Sooppy Nisar, Aqeel Ahmad, Muhammad Farman, Evren Hincal, Anum Zehra, Modeling and mathematical analysis of fractional order Eye infection (conjunctivitis) virus model with treatment impact: Prelicence and dynamical transmission, 2024, 107, 11100168, 33, 10.1016/j.aej.2024.07.020
5.	Khadija Khatun, Md. Mamun-Ur-Rashid Khan, Jun Tanimoto, Aspiration can decline epidemic disease, 2025, 112, 11100168, 151, 10.1016/j.aej.2024.10.087
6.	Aqeel Ahmad, Muhammad Owais Kulachi, Ayman A. Aly, Mustafa Inc, M.O. Ahmad, Shahram Rezapour, Flip bifurcation analysis and investigation of conjunctivitis virus by using sustainable control approach, 2025, 100, 17468094, 106956, 10.1016/j.bspc.2024.106956

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)