Recent progress on mathematical analysis and numerical simulations for Maxwell's equations in perfectly matched layers and complex media: a review

Jichun Li; Jichun Li

doi:10.3934/era.2024087

Electronic Research Archive

2024, Volume 32, Issue 3: 1901-1922. doi: 10.3934/era.2024087

Previous Article Next Article

Review Special Issues

Recent progress on mathematical analysis and numerical simulations for Maxwell's equations in perfectly matched layers and complex media: a review

Jichun Li ^,

Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, Nevada 89154-4020, USA

Received: 18 December 2023 Revised: 02 February 2024 Accepted: 18 February 2024 Published: 04 March 2024

In this paper, we presented a review on some recent progress achieved for simulating Maxwell's equations in perfectly matched layers and complex media such as metamaterials and graphene. We mainly focused on the stability analysis of the modeling equations and development and analysis of the numerical schemes. Some open issues were pointed out, too.

Keywords:

Citation: Jichun Li. Recent progress on mathematical analysis and numerical simulations for Maxwell's equations in perfectly matched layers and complex media: a review[J]. Electronic Research Archive, 2024, 32(3): 1901-1922. doi: 10.3934/era.2024087

Related Papers:

[1]	Xiao Xu, Hong Liao, Xu Yang . An automatic density peaks clustering based on a density-distance clustering index. AIMS Mathematics, 2023, 8(12): 28926-28950. doi: 10.3934/math.20231482
[2]	Fatimah Alshahrani, Wahiba Bouabsa, Ibrahim M. Almanjahie, Mohammed Kadi Attouch . $k$ NN local linear estimation of the conditional density and mode for functional spatial high dimensional data. AIMS Mathematics, 2023, 8(7): 15844-15875. doi: 10.3934/math.2023809
[3]	Shabbar Naqvi, Muhammad Salman, Muhammad Ehtisham, Muhammad Fazil, Masood Ur Rehman . On the neighbor-distinguishing in generalized Petersen graphs. AIMS Mathematics, 2021, 6(12): 13734-13745. doi: 10.3934/math.2021797
[4]	Yuming Chen, Jie Gao, Luan Teng . Distributed adaptive event-triggered control for general linear singular multi-agent systems. AIMS Mathematics, 2023, 8(7): 15536-15552. doi: 10.3934/math.2023792
[5]	Luca Bisconti, Paolo Maria Mariano . Global existence and regularity for the dynamics of viscous oriented fluids. AIMS Mathematics, 2020, 5(1): 79-95. doi: 10.3934/math.2020006
[6]	Hsin-Lun Li . Leader–follower dynamics: stability and consensus in a socially structured population. AIMS Mathematics, 2025, 10(2): 3652-3671. doi: 10.3934/math.2025169
[7]	Hongjie Li . Event-triggered bipartite consensus of multi-agent systems in signed networks. AIMS Mathematics, 2022, 7(4): 5499-5526. doi: 10.3934/math.2022305
[8]	Neelakandan Subramani, Abbas Mardani, Prakash Mohan, Arunodaya Raj Mishra, Ezhumalai P . A fuzzy logic and DEEC protocol-based clustering routing method for wireless sensor networks. AIMS Mathematics, 2023, 8(4): 8310-8331. doi: 10.3934/math.2023419
[9]	Yinwan Cheng, Chao Yang, Bing Yao, Yaqin Luo . Neighbor full sum distinguishing total coloring of Halin graphs. AIMS Mathematics, 2022, 7(4): 6959-6970. doi: 10.3934/math.2022386
[10]	Jin Cai, Shuangliang Tian, Lizhen Peng . On star and acyclic coloring of generalized lexicographic product of graphs. AIMS Mathematics, 2022, 7(8): 14270-14281. doi: 10.3934/math.2022786

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]	G. Bao, P. Li, Maxwell's Equations in Periodic Structures, Series on Applied Mathematical Sciences, Science Press, Beijing/Springer, Singapore, 2022.
[2]	W. Cai, Computational Methods for Electromagnetic Phenomena Electrostatics in Solvation, Scattering, and Electron Transport, Cambridge University Press, 2013.
[3]	L. Demkowicz, Computing With hp-Adaptive Finite Elements, I: One and Two-Dimensional Elliptic and Maxwell Problems, CRC Press, Taylor and Francis, 2006.
[4]	L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz, A. Zdunek, Computing with hp-Adaptive Finite Elements, II: Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications, CRC Press, Taylor and Francis, 2007.
[5]	J. M. Jin, The Finite Element Method in Electromagnetics, 3rd edition, Piscataway, NJ, USA, IEEE Press, 2014.
[6]	P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, 2003.
[7]	A. Taflove, S. C. Haguess, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edition, Artech House, Norwood, 2005.
[8]	A. Taflove, A. Oskooi, S. G. Johnson, Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, Artech House, Boston, 2013.
[9]	T. J. Cui, D. R. Smith, R. Liu, Metamaterials: Theory, Design, and Applications, Springer, 2010.
[10]	N. Engheta, R. W. Ziolkowski, Electromagnetic Metamaterials: Physics and Engineering Explorations, Wiley-IEEE Press, 2006.
[11]	L. Solymar, E. Shamonina, Waves in Metamaterials, Oxford University Press, 2009.
[12]	Q. Bao, K. Loh, Graphene photonics, plasmonics, and broadband optoelectronic devices, ACS Nano, 6 (2012), 3677–3694. https://doi.org/10.1021/nn300989g doi: 10.1021/nn300989g
[13]	A. K. Geim, K. S. Novoselov, The rise of graphene, Nat. Mater., 6 (2007), 183–191. https://doi.org/10.1038/nmat1849
[14]	V. G. Veselago, The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$ , Sov. Phys. Usp., 47 (1968), 509–514.
[15]	D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett., 84 (2000), 4184–4187. https://doi.org/10.1103/PhysRevLett.84.4184 doi: 10.1103/PhysRevLett.84.4184
[16]	A. Shelby, D. R. Smith, S. Schultz, Experimental verification of a negative index of refraction, Science, 292 (2001), 489–491. https://doi.org/10.1126/science.1058847 doi: 10.1126/science.1058847
[17]	J. B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., 85 (2000), 3966–3969. https://doi.org/10.1103/PhysRevLett.85.3966 doi: 10.1103/PhysRevLett.85.3966
[18]	A. J. Holden, Towards some real applications for negative materials, Photonics Nanostruct. Fundam. Appl., 3 (2005), 96–99. https://doi.org/10.1016/j.photonics.2005.09.014 doi: 10.1016/j.photonics.2005.09.014
[19]	Y. Hao, R. Mittra, FDTD Modeling of Metamaterials: Theory and Applications, Artech House Publishers, 2008.
[20]	D. H. Werner, D. H. Kwon, Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications, Springer, 2013.
[21]	V. A. Bokil, Y. Cheng, Y. Jiang, F. Li, Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media, J. Comput. Phys., 350 (2017), 420–452. https://doi.org/10.1016/j.jcp.2017.08.009 doi: 10.1016/j.jcp.2017.08.009
[22]	N. Schmitt, C. Scheid, S. Lanteri, A. Moreau, J. Viquerat, A DGTD method for the numerical modeling of the interaction of light with nanometer scale metallic structures taking into account non-local dispersion effects, J. Comput. Phys., 316 (2016), 396–415. https://doi.org/10.1016/j.jcp.2016.04.020 doi: 10.1016/j.jcp.2016.04.020
[23]	J. Lin, An adaptive boundary element method for the transmission problem with hyperbolic metamaterials, J. Comput. Phys., 444 (2021), 110573. https://doi.org/10.1016/j.jcp.2021.110573 doi: 10.1016/j.jcp.2021.110573
[24]	B. Donderici, F. L. Teixeira, Mixed finite-element time-domain method for transient Maxwell equations in doubly dispersive media, IEEE Trans. Microwave Theory Tech., 56 (2008), 113–120. https://doi.org/10.1109/TMTT.2007.912217 doi: 10.1109/TMTT.2007.912217
[25]	J. Li, J. S. Hesthaven, Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comput. Phys., 258 (2014), 915–930. https://doi.org/10.1016/j.jcp.2013.11.018 doi: 10.1016/j.jcp.2013.11.018
[26]	C. Scheid, S. Lanteri, Convergence of a Discontinuous Galerkin scheme for the mixed time domain Maxwell's equations in dispersive media, IMA J. Numer. Anal., 33 (2013), 432–459. https://doi.org/10.1093/imanum/drs008 doi: 10.1093/imanum/drs008
[27]	Z. Xie, J. Wang, B. Wang, C. Chen, Solving Maxwell's equation in meta-materials by a CG-DG method, Commun. Comput. Phys., 19 (2016), 1242–1264. https://doi.org/10.4208/cicp.scpde14.35s doi: 10.4208/cicp.scpde14.35s
[28]	W. Li, D. Liang, The spatial fourth-order compact splitting FDTD scheme with modified energy-conserved identity for two-dimensional Lorentz model, J. Comput. Appl. Math., 367 (2020), 112428. https://doi.org/10.1016/j.cam.2019.112428 doi: 10.1016/j.cam.2019.112428
[29]	W. Li, D. Liang, Symmetric energy-conserved S-FDTD scheme for two-dimensional Maxwell's equations in negative index metamaterials, J. Sci. Comput., 69 (2016), 696–735. https://doi.org/10.1007/s10915-016-0214-9 doi: 10.1007/s10915-016-0214-9
[30]	X. Bai, H. Rui, New energy analysis of Yee scheme for metamaterial Maxwell's equations on non-uniform rectangular meshes, Adv. Appl. Math. Mech., 13 (2021), 1355–1383. https://doi.org/10.4208/aamm.OA-2020-0208 doi: 10.4208/aamm.OA-2020-0208
[31]	S. Nicaise, Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations, Math. Control Relat. Fields, 13 (2023), 265–302. https://doi.org/10.3934/mcrf.2021057 doi: 10.3934/mcrf.2021057
[32]	P. Fernandes, M. Raffetto, Well-posedness and finite element approximability of time-harmonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials, Math. Model. Methods Appl. Sci., 19 (2009), 2299–2335. https://doi.org/10.1142/S0218202509004121 doi: 10.1142/S0218202509004121
[33]	P. Fernandes, M. Ottonello, M. Raffetto, Regularity of time-harmonic electromagnetic fields in the interior of bianisotropic materials and metamaterials, IMA J. Appl. Math., 79 (2014), 54–93. https://doi.org/10.1093/imamat/hxs039 doi: 10.1093/imamat/hxs039
[34]	P. Cocquet, P. Mazet, V. Mouysset, On the existence and uniqueness of a solution for some frequency-dependent partial differential equations coming from the modeling of metamaterials, SIAM J. Math. Anal., 44 (2012), 3806–3833. https://doi.org/10.1137/100810071 doi: 10.1137/100810071
[35]	E. T. Chung, P. Ciarlet Jr., A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and metamaterials, J. Comput. Appl. Math., 239 (2013), 189–207. https://doi.org/10.1016/j.cam.2012.09.033 doi: 10.1016/j.cam.2012.09.033
[36]	M. Cassier, P. Joly, M. Kachanovska, Mathematical models for dispersive electromagnetic waves: An overview, Comput. Math. Appl., 74 (2017), 2792–2830. https://doi.org/10.1016/j.camwa.2017.07.025 doi: 10.1016/j.camwa.2017.07.025
[37]	J. Li, A literature survey of mathematical study of metamaterials, Int. J. Numer. Anal. Model., 13 (2016), 230–243.
[38]	U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777–1780. https://doi.org/10.1126/science.1126493
[39]	J. B. Pendry, D. Schurig, D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780–1782. https://doi.org/10.1126/science.1125907 doi: 10.1126/science.1125907
[40]	J. Li, Y. Huang, Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials, Springer Series in Computational Mathematics (vol.43), Springer, 2013.
[41]	H. Ammari, H. Kang, H. Lee, M. Lim, S. Yu, Enhancement of near cloaking for the full Maxwell equations, SIAM J. Appl. Math., 73 (2013), 2055–2076. https://doi.org/10.1137/120903610 doi: 10.1137/120903610
[42]	G. Bao, H. Liu, J. Zou, Nearly cloaking the full Maxwell equations: Cloaking active contents with general conducting layers, J. Math. Pures Appl., 101 (2014), 716–733. https://doi.org/10.1016/j.matpur.2013.10.010 doi: 10.1016/j.matpur.2013.10.010
[43]	H. Ammari, J. Garnier, V. Jugnon, H. Kang, H. Lee, M. Lim, Enhancement of near-cloaking. Part III: Numerical simulations, statistical stability, and related questions, Comtemp. Math., 577 (2012), 1–24.
[44]	H. Ammari, G. Ciraolo, H. Kang, H. Lee, G. W. Milton, Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Ration. Mech. Anal., 208 (2013), 667–692. https://doi.org/10.1007/s00205-012-0605-5 doi: 10.1007/s00205-012-0605-5
[45]	R. V. Kohn, J. Lu, B. Schweizer, M. I. Weinstein, A variational perspective on cloaking by anomalous localized resonance, Commun. Math. Phys., 328 (2014), 1–27. https://doi.org/10.1007/s00220-014-1943-y doi: 10.1007/s00220-014-1943-y
[46]	R. V. Kohn, D. Onofrei, M. S. Vogelius, M. I. Weinstein, Cloaking via change of variables for the Helmholtz equation, Commun. Pure Appl. Math., 63 (2010), 973–1016. https://doi.org/10.1002/cpa.20326 doi: 10.1002/cpa.20326
[47]	A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, Cloaking devices, electromagnetics wormholes and transformation optics, SIAM Rev., 51 (2009), 3–33. https://doi.org/10.1137/080716827 doi: 10.1137/080716827
[48]	F. Guevara Vasquez, G. W. Milton, D. Onofrei, Broadband exterior cloaking, Opt. Express, 17 (2009), 14800–14805. https://doi.org/10.1364/OE.17.014800
[49]	M. Lassas, M. Salo, L. Tzou, Inverse problems and invisibility cloaking for FEM models and resistor networks, Math. Mod. Meth. Appl. Sci., 25 (2015), 309–342. https://doi.org/10.1142/S0218202515500116 doi: 10.1142/S0218202515500116
[50]	S. C. Brenner, J. Gedicke, L. Y. Sung, An adaptive $P_1$ finite element method for two-dimensional transverse magnetic time harmonic Maxwell's equations with general material properties and general boundary conditions, J. Sci. Comput., 68 (2016), 848–863. https://doi.org/10.1007/s10915-015-0161-x doi: 10.1007/s10915-015-0161-x
[51]	J. J. Lee, A mixed method for time-transient acoustic wave propagation in metamaterials, J. Sci. Comput., 84 (2020). https://doi.org/10.1007/s10915-020-01275-0
[52]	S. Nicaise, J. Venel, A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients, J. Comput. Appl. Math., 235 (2011), 4272–4282. https://doi.org/10.1016/j.cam.2011.03.028 doi: 10.1016/j.cam.2011.03.028
[53]	Z. Yang, L. L. Wang, Accurate simulation of ideal circular and elliptic cylindrical invisibility cloaks, Commun. Comput. Phys., 17 (2015), 822–849. https://doi.org/10.4208/cicp.280514.131014a doi: 10.4208/cicp.280514.131014a
[54]	Z. Yang, L. L. Wang, Z. Rong, B. Wang, B. Zhang, Seamless integration of global Dirichlet-to-Neumann boundary condition and spectral elements for transformation electromagnetics, Comput. Methods Appl. Mech. Engrg., 301 (2016), 137–163. https://doi.org/10.1016/j.cma.2015.12.020 doi: 10.1016/j.cma.2015.12.020
[55]	B. Wang, Z. Yang, L. L. Wang, S. Jiang, On time-domain NRBC for Maxwell's equations and its application in accurate simulation of electromagnetic invisibility cloaks, J. Sci. Comput., 86 (2021). https://doi.org/10.1007/s10915-020-01354-2
[56]	U. Leonhardt, T. Tyc, Broadband invisibility by non-Euclidean cloaking, Science, 323 (2009), 110–112. https://doi.org/10.1126/science.1166332 doi: 10.1126/science.1166332
[57]	R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, D. R. Smith, Broadband ground-plane cloak, Science, 323 (2009), 366–369. https://doi.org/10.1126/science.1166949
[58]	J. Li, Two new finite element schemes and their analysis for modeling of wave propagation in graphene, Results Appl. Math., 9 (2021), 100136. https://doi.org/10.1016/j.rinam.2020.100136 doi: 10.1016/j.rinam.2020.100136
[59]	Y. Wu, J. Li, Total reflection and cloaking by zero index metamaterials loaded with rectangular dielectric defects, Appl. Phys. Lett., 102 (2013), 183105. https://doi.org/10.1063/1.4804201 doi: 10.1063/1.4804201
[60]	J. Li, Well-posedness study for a time-domain spherical cloaking model, Comput. Math. Appl., 68 (2014), 1871–1881. https://doi.org/10.1016/j.camwa.2014.10.007 doi: 10.1016/j.camwa.2014.10.007
[61]	J. Li, Y. Huang, W. Yang, Well-posedness study and finite element simulation of time-domain cylindrical and elliptical cloaks, Math. Comput., 84 (2015), 543–562. https://doi.org/10.1090/s0025-5718-2014-02911-6 doi: 10.1090/s0025-5718-2014-02911-6
[62]	W. Yang, J. Li, Y. Huang, Mathematical analysis and finite element time domain simulation of arbitrary star-shaped electromagnetic cloaks, SIAM J. Numer. Anal., 56 (2018), 136–159. https://doi.org/10.1137/16M1093835 doi: 10.1137/16M1093835
[63]	J. Li, J. B. Pendry, Hiding under the carpet: a new strategy for cloaking, Phys. Rev. Lett., 101 (2008), 2039014. https://doi.org/10.1103/PhysRevLett.101.203901 doi: 10.1103/PhysRevLett.101.203901
[64]	J. Li, Y. Huang, W. Yang, A. Wood, Mathematical analysis and time-domain finite element simulation of carpet cloak, SIAM J. Appl. Math., 74 (2014), 1136–1151. https://doi.org/10.1137/140959250 doi: 10.1137/140959250
[65]	J. Li, C. W. Shu, W. Yang, Development and analysis of two new finite element schemes for a time-domain carpet cloak model, Adv. Comput. Math., 48 (2022), 24. https://doi.org/10.1007/s10444-022-09948-0 doi: 10.1007/s10444-022-09948-0
[66]	J. Li, Z. Liang, J. Zhu, X. Zhang, Anisotropic metamaterials for transformation acoustics and imaging, in Acoustic Metamaterials: Negative Refraction, Imaging, Sensing and Cloaking (eds. R. V. Craster and S. Guenneau), Springer Series in Materials Science, 166 (2013), 169–195. https://doi.org/10.1007/978-94-007-4813-2_7
[67]	K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, et al., Electric field effect in atomically thin carbon films, Science, 306 (2004), 666–669. https://doi.org/10.1126/science.1102896 doi: 10.1126/science.1102896
[68]	F. Bonaccorso, Z. Sun, T. Hasan, A. C. Ferrari, Graphene photonics and optoelectronics, Nat. Photonics, 4 (2010), 611–622. https://doi.org/10.1038/nphoton.2010.186 doi: 10.1038/nphoton.2010.186
[69]	F. H. L. Koppens, D. E. Chang, F. J. Garcia de Abajo, Graphene plasmonics: A platform for strong light-matter interactions, Nano Lett., 11 (2011), 3370–3377. https://doi.org/10.1021/nl201771h doi: 10.1021/nl201771h
[70]	S. K. Tiwari, S. Sahoo, N. Wang, A. Huczko, Graphene research and their outputs: Status and prospect, J. Sci.: Adv. Mater. Devices, 5 (2020), 10–29. https://doi.org/10.1016/j.jsamd.2020.01.006 doi: 10.1016/j.jsamd.2020.01.006
[71]	G. Bal, P. Cazeaux, D. Massatt, S. Quinn, Mathematical models of topologically protected transport in twisted bilayer graphene, Multiscale Model. Simul., 21 (2023), 1081–1121. https://doi.org/10.1137/22M1505542 doi: 10.1137/22M1505542
[72]	Y. Hong, D. P. Nicholls, On the consistent choice of effective permittivity and conductivity for modeling graphene, JOSA A, 38 (2021), 1511–1520. https://doi.org/10.1364/JOSAA.430088 doi: 10.1364/JOSAA.430088
[73]	J. P. Lee-Thorp, M. I. Weinstein, Y. Zhu, Elliptic operators with honeycomb symmetry: Dirac points, edge states and applications to photonic graphene, Arch. Ration. Mech. Anal., 232 (2019), 1–63. https://doi.org/10.1007/s00205-018-1315-4 doi: 10.1007/s00205-018-1315-4
[74]	M. Maier, D. Margetis, M. Luskin, Dipole excitation of surface plasmon on a conducting sheet: finite element approximation and validation, J. Comput. Phys., 339 (2017), 126–145. https://doi.org/10.1016/j.jcp.2017.03.014 doi: 10.1016/j.jcp.2017.03.014
[75]	M. Maier, D. Margetis, M. Luskin, Generation of surface plasmon-polaritons by edge effects, Commun. Math. Sci., 16 (2018), 77–95.
[76]	J. H. Song, M. Maier, M. Luskin, Adaptive finite element simulations of waveguide configurations involving parallel 2D material sheets, Comput. Methods Appl. Mech. Eng., 351 (2019), 20–34. https://doi.org/10.1016/j.cma.2019.03.039 doi: 10.1016/j.cma.2019.03.039
[77]	J. Wilson, F. Santosa, P. A. Martin, Temporally manipulated plasmons on graphene, SIAM J. Appl. Math., 79 (2019), 1051–1074. https://doi.org/10.1137/18M1226889 doi: 10.1137/18M1226889
[78]	W. Yang, J. Li, Y. Huang, Time-domain finite element method and analysis for modeling of surface plasmon polaritons in graphene devices, Comput. Methods Appl. Mech. Eng., 372 (2020), 113349. https://doi.org/10.1016/j.cma.2020.113349 doi: 10.1016/j.cma.2020.113349
[79]	J. Li, L. Zhu, T. Arbogast, A new time-domain finite element method for simulation surface plasmon polaritons on graphene sheets, Comput. Math. Appl., 142 (2023), 268–383. https://doi.org/10.1016/j.camwa.2023.05.003 doi: 10.1016/j.camwa.2023.05.003
[80]	Y. Gong, N. Liu, Advanced numerical methods for graphene simulation with equivalent boundary conditions: a review, Photonics, 10 (2023), 712. https://doi.org/10.3390/photonics10070712 doi: 10.3390/photonics10070712
[81]	P. Li, L. J. Jiang, H. Bağci, Discontinuous Galerkin time-domain modeling of graphene nanoribbon incorporating the spatial dispersion effects, IEEE Trans. Antennas Propag., 66 (2018), 3590–3598. https://doi.org/10.1109/TAP.2018.2826567 doi: 10.1109/TAP.2018.2826567
[82]	L. Yang, J. Tian, K. Z. Rajab, Y. Hao, FDTD modeling of nonlinear phenomena in wave transmission through graphene, IEEE Antennas Wirel. Propag. Lett., 17 (2018), 126–129. https://doi.org/10.1109/LAWP.2017.2777530 doi: 10.1109/LAWP.2017.2777530
[83]	B. Alpert, L. Greengard, T. Hagstrom, Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation, SIAM J. Numer. Anal., 37 (2000), 1138–1164. https://doi.org/10.1137/S0036142998336916 doi: 10.1137/S0036142998336916
[84]	B. Engquist, A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1977), 629–651. https://doi.org/10.1073/pnas.74.5.1765 doi: 10.1073/pnas.74.5.1765
[85]	T. Hagstrom, T. Warburton, D. Givoli, Radiation boundary conditions for time-dependent waves based on complete plane wave expansions, J. Comput. Appl. Math., 234 (2010), 1988–1995. https://doi.org/10.1016/j.cam.2009.08.050 doi: 10.1016/j.cam.2009.08.050
[86]	J. P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185–200. https://doi.org/10.1006/jcph.1994.1159 doi: 10.1006/jcph.1994.1159
[87]	R. W. Ziolkowski, Maxwellian material based absorbing boundary conditions, Comput. Methods Appl. Mech. Eng., 169 (1999), 237–262. https://doi.org/10.1016/S0045-7825(98)00156-X doi: 10.1016/S0045-7825(98)00156-X
[88]	Y. Huang, J. Li, Z. Fang, Mathematical analysis of Ziolkowski's PML model with application for wave propagation in metamaterials, J. Comp. Appl. Math., 366 (2020), 112434. https://doi.org/10.1016/j.cam.2019.112434 doi: 10.1016/j.cam.2019.112434
[89]	J. Li, L. Zhu, Analysis and application of two novel finite element methods for solving Ziolkowski's PML model in the integro-differential form, SIAM J. Numer. Anal., 61 (2023), 2209–2236. https://doi.org/10.1137/22M1506936 doi: 10.1137/22M1506936
[90]	G. C. Cohen, P. Monk, Mur-Nédélec finite element schemes for Maxwell's equations, Comput. Methods Appl. Mech. Eng., 169 (1999), 197–217. https://doi.org/10.1016/S0045-7825(98)00154-6 doi: 10.1016/S0045-7825(98)00154-6
[91]	M. Chen, Y. Huang, J. Li, Development and analysis of a new finite element method for the Cohen-Monk PML model, Numer. Math., 147 (2021), 127–155. https://doi.org/10.1007/s00211-020-01166-4 doi: 10.1007/s00211-020-01166-4
[92]	J. L. Lions, J. Métral, O. Vacus, Well-posed absorbing layer for hyperbolic problems, Numer. Math., 92 (2002), 535–562. https://doi.org/10.1007/s002110100263 doi: 10.1007/s002110100263
[93]	G. Bao, H. Wu, Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell's equations, SIAM J. Numer. Anal., 43 (2005), 2121–2143. https://doi.org/10.1137/040604315 doi: 10.1137/040604315
[94]	J. H. Bramble, J. E. Pasciak, Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems, Math. Comput., 76 (2007), 597–614. https://doi.org/10.1090/S0025-5718-06-01930-2 doi: 10.1090/S0025-5718-06-01930-2
[95]	J. Chen, Z. Chen, An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems, Math. Comput., 77 (2007), 673–698. https://doi.org/10.1090/S0025-5718-07-02055-8 doi: 10.1090/S0025-5718-07-02055-8
[96]	Z. Chen, W. Zheng, PML method for electromagnetic scattering problem in a two-layer medium, SIAM J. Numer. Anal., 55 (2017), 2050–2084. https://doi.org/10.1137/16M1091757 doi: 10.1137/16M1091757
[97]	T. Hohage, F. Schmidt, L. Zschiedrich, Solving time-harmonic scattering problems based on the pole condition II: convergence of the PML method, SIAM J. Math. Anal., 35 (2003), 547–560. https://doi.org/10.1137/S0036141002406485 doi: 10.1137/S0036141002406485
[98]	M. Lassas, E. Somersalo, On the existence and convergence of the solution of PML equations, Computing, 60 (1998), 229–241. https://doi.org/10.1007/BF02684334 doi: 10.1007/BF02684334
[99]	J. H. Bramble, J. E. Pasciak, Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem, Math. Comput., 77 (2008), 1–10. https://doi.org/10.1090/S0025-5718-07-02037-6 doi: 10.1090/S0025-5718-07-02037-6
[100]	Z. Chen, W. Zheng, Convergence of the uniaxial perfectly matched layer method for time-harmonic scattering problems in two-layered media, SIAM J. Numer. Anal., 48 (2010), 2158–2185. https://doi.org/10.1137/090750603 doi: 10.1137/090750603
[101]	Z. Chen, T. Cui, L. Zhang, An adaptive anisotropic perfectly matched layer method for 3-D time harmonic electromagnetic scattering problems, Numer. Math., 125 (2013), 639–677. https://doi.org/10.1007/s00211-013-0550-8 doi: 10.1007/s00211-013-0550-8
[102]	Y. Lin, K. Zhang, J. Zou, Studies on some perfectly matched layers for one-dimensional time-dependent systems, Adv. Comput. Math., 30 (2009), 1–35. https://doi.org/10.1007/s10444-007-9055-2 doi: 10.1007/s10444-007-9055-2
[103]	Y. Gao, P. Li, Electromagnetic scattering for time-domain Maxwell's equations in an unbounded structure, Math. Models Methods Appl. Sci., 27 (2017), 1843–1870. https://doi.org/10.1142/S0218202517500336 doi: 10.1142/S0218202517500336
[104]	P. Li, L. Wang, A. Wood, Analysis of transient electromagnetic scattering from a three-dimensional open cavity, SIAM J. Appl. Math., 75 (2015), 1675–1699. https://doi.org/10.1137/140989637 doi: 10.1137/140989637
[105]	C. Wei, J. Yang, B. Zhang, Convergence analysis of the PML method for time-domain electromagnetic scattering problems, SIAM J. Numer. Anal., 58 (2020), 1918–1940. https://doi.org/10.1137/19M126517X doi: 10.1137/19M126517X
[106]	C. Wei, J. Yang, B. Zhang, Convergence of the uniaxial PML method for time-domain electromagnetic scattering problems, ESAIM Math. Model. Numer. Anal., 55 (2021), 2421–2443. https://doi.org/10.1051/m2an/2021064 doi: 10.1051/m2an/2021064
[107]	F. L. Teixeira, W. C. Chew, Advances in the theory of perfectly matched layers, in Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Boston, 7 (2001), 283–346.
[108]	W. C. Chew, W. H. Weedon, A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates, Microwave Opt. Technol. Lett., 7 (1994), 599–604. https://doi.org/10.1002/mop.4650071304 doi: 10.1002/mop.4650071304
[109]	F. Collino, P. Monk, Optimizing the perfectly matched layer, Comput. Methods Appl. Mech. Eng., 164 (1998), 157–171. https://doi.org/10.1016/S0045-7825(98)00052-8 doi: 10.1016/S0045-7825(98)00052-8
[110]	T. Lu, P. Zhang, W. Cai, Discontinuous Galerkin methods for dispersive and lossy Maxwell's equations and PML boundary conditions, J. Comput. Phys., 200 (2004), 549–580.
[111]	J. L. Hong, L. H. Ji, L. H. Kong, Energy-dissipation splitting finite-difference time-domain method for Maxwell equations with perfectly matched layers, J. Comput. Phys., 269 (2014), 201–214. https://doi.org/10.1016/j.jcp.2014.03.025 doi: 10.1016/j.jcp.2014.03.025
[112]	J. S. Hesthaven, On the analysis and construction of perfectly matched layers for the linearized Euler equations, J. Comput. Phys., 142 (1998), 129–147. https://doi.org/10.1006/jcph.1998.5938 doi: 10.1006/jcph.1998.5938
[113]	F. Q. Hu, On absorbing boundary conditions for linearized euler equations by a perfectly matched layer, J. Comput. Phys., 129 (1996), 201–219. https://doi.org/10.1006/jcph.1996.0244 doi: 10.1006/jcph.1996.0244
[114]	F. Pled, C. Desceliers, Review and recent developments on the perfectly matched layer (PML) method for the numerical modeling and simulation of elastic wave propagation in unbounded domains, Arch. Comput. Methods Eng., 29 (2022), 471–518. https://doi.org/10.1007/s11831-021-09581-y doi: 10.1007/s11831-021-09581-y
[115]	D. Appelö, T. Hagstrom, G. Kreiss, Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability, SIAM J. Appl. Math., 67 (2006), 1–23. https://doi.org/10.1137/050639107 doi: 10.1137/050639107
[116]	D. Appelö, T. Hagstrom, A general perfectly matched layer model for hyperbolic-parabolic systems, SIAM J. Sci. Comput., 31 (2009), 3301–23. https://doi.org/10.1137/080713951 doi: 10.1137/080713951
[117]	D. H. Baffet, M. J. Grote, S. Imperiale, M. Kachanovska, Energy decay and stability of a perfectly matched layer for the wave equation, J. Sci. Comput., 81 (2019), 2237–2270. https://doi.org/10.1007/s10915-019-01089-9 doi: 10.1007/s10915-019-01089-9
[118]	Z. Yang, L. L. Wang, Y. Gao, A truly exact perfect absorbing layer for time-harmonic acoustic wave scattering problems, SIAM J. Sci. Comput., 43 (2021), A1027–A1061. https://doi.org/10.1137/19M1294071 doi: 10.1137/19M1294071
[119]	W. C. Chew, Q. H. Liu, Perfectly matched layers for elastodynamics: a new absorbing boundary condition, J. Comput. Acoust., 4 (1996), 341–349. https://doi.org/10.1142/S0218396X96000118 doi: 10.1142/S0218396X96000118
[120]	F. Collino, C. Tsogka, Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media, Geophysics, 88 (2001), 43–73.
[121]	K. Duru, L. Rannabauer, A. A. Gabriel, G. Kreiss, M. Bader, A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form, Numer. Math., 146 (2020), 729–782. https://doi.org/10.1007/s00211-020-01160-w doi: 10.1007/s00211-020-01160-w
[122]	S. Abarbanel, D. Gottlieb, A mathematical analysis of the PML method, J. Comput. Phys., 134 (1997), 357–363. https://doi.org/10.1006/jcph.1997.5717 doi: 10.1006/jcph.1997.5717
[123]	E. Bécache, P. Joly, On the analysis of Bérenger's perfectly matched layers for maxwell's equations, ESAIM: M2AN, 36 (2002), 87–119. https://doi.org/10.1051/m2an:2002004 doi: 10.1051/m2an:2002004
[124]	L. Zhao, A. C. Cangellaris, A general approach for the development of unsplit-field time-domain implementations of perfectly matched layers for FDTD grid truncation, IEEE Microwave Guid Wave Lett., 6 (1996), 209–211. https://doi.org/10.1109/75.491508 doi: 10.1109/75.491508
[125]	Y. Huang, M. Chen, J. Li, Development and analysis of both finite element and fourth-order in space finite difference methods for an equivalent Berenger's PML model, J. Comput. Phys., 405 (2020), 109154. https://doi.org/10.1016/j.jcp.2019.109154 doi: 10.1016/j.jcp.2019.109154
[126]	Y. Huang, J. Li, X. Liu, Developing and analyzing an explicit unconditionally stable finite element scheme for an equivalent Bérenger's PML model, ESAIM: M2AN, 57 (2023), 621–644. https://doi.org/10.1051/m2an/2022086 doi: 10.1051/m2an/2022086
[127]	D. Correia, J. M. Jin, 3D-FDTD-PML analysis of left-handed metamaterials, Microwave Opt. Technol. Lett., 40 (2004), 201–205. https://doi.org/10.1002/mop.11328 doi: 10.1002/mop.11328
[128]	S. A. Cummer, Perfectly matched layer behavior in negative refractive index materials, IEEE Antennas Wirel. Propag. Lett., 3 (2004), 172–175. https://doi.org/10.1109/LAWP.2004.833710 doi: 10.1109/LAWP.2004.833710
[129]	X. T. Dong, X. S. Rao, Y. B. Gan, B. Guo, W. Y. Yin, Perfectly matched layer-absorbing boundary condition for left-handed materials, IEEE Microwave Wireless Compon. Lett., 14 (2004), 301–303. https://doi.org/10.1109/LMWC.2004.827104 doi: 10.1109/LMWC.2004.827104
[130]	P. R. Loh, A. F. Oskooi, M. Ibanescu, M. Skorobogatiy, S. G. Johnson, Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backward-wave structures, Phys. Rev. E, 79 (2009), 065601. https://doi.org/10.1103/PhysRevE.79.065601 doi: 10.1103/PhysRevE.79.065601
[131]	Y. Shi, Y. Li, C. H. Liang, Perfectly matched layer absorbing boundary condition for truncating the boundary of the left-handed medium, Microwave Opt. Technol. Lett., 48 (2006), 57–63. https://doi.org/10.1002/mop.21260 doi: 10.1002/mop.21260
[132]	E. Bécache, P. Joly, M. Kachanovska, V. Vinoles, Perfectly matched layers in negative index metamaterials and plasmas, ESAIM: Proc. Surv., 50 (2015), 113–132. https://doi.org/10.1051/proc/201550006 doi: 10.1051/proc/201550006
[133]	E. Bécache, P. Joly, M. Kachanovska, V. Vinoles, On the analysis of perfectly matched layers for a class of dispersive media and application to negative index metamaterials, Math. Comput., 87 (2018), 2775–2810. https://doi.org/10.1090/mcom/3307 doi: 10.1090/mcom/3307
[134]	J. Li, L. Zhu, Analysis and FDTD simulation of a perfectly matched layer for the Drude metamaterial, Ann. Appl. Math., 38 (2022), 1–23.
[135]	Y. Huang, J. Li, X. Yi, H. Zhao, Analysis and application of a time-domain finite element method for the Drude metamaterial perfectly matched layer model, J. Comput. Appl. Math., 438 (2024), 115575. https://doi.org/10.1016/j.cam.2023.115575 doi: 10.1016/j.cam.2023.115575
[136]	E. Bécache, M. Kachanovska, Stable perfectly matched layers for a class of anisotropic dispersive models, Part I: necessary and sufficient conditions of stability, ESAIM: M2AN, 51 (2017), 2399–2434. https://doi.org/10.1051/m2an/2017019 doi: 10.1051/m2an/2017019
[137]	C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O'Hara, J. Booth, D.R. Smith, An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials, IEEE Antennas Propag. Mag., 54 (2012), 10–35. https://doi.org/10.1109/MAP.2012.6230714 doi: 10.1109/MAP.2012.6230714
[138]	K. Achouri, C. Caloz, Design, concepts, and applications of electromagnetic metasurfaces, Nanophotonics, 7 (2018), 1095–1116. https://doi.org/10.1515/nanoph-2017-0119 doi: 10.1515/nanoph-2017-0119
[139]	S. B. Glybovski, S. A. Tretyakov, P. A. Belov, Y. S. Kivshar, C. R. Simovski, Metasurfaces: From microwaves to visible, Phys. Rep., 634 (2016), 1–72. https://doi.org/10.1016/j.physrep.2016.04.004 doi: 10.1016/j.physrep.2016.04.004
[140]	K. A. Lurie, An Introduction to the Mathematical Theory of Dynamic Materials, Springer, 2017.
[141]	P. A. Huidobro, E. Galiffi, S. Guenneau, R. V. Craster, J. B. Pendry, Fresnel drag in space-time-modulated metamaterials, Proc. Natl. Acad. Sci. U.S.A., 116 (2019), 24943–24948. https://doi.org/10.1073/pnas.1915027116 doi: 10.1073/pnas.1915027116
[142]	C. Caloz, Z. Deck-Léger, Spacetime metamaterials–Part II: theory and applications, IEEE Trans. Antennas Propag., 68 (2020), 1583–1598. https://doi.org/10.1109/TAP.2019.2944216 doi: 10.1109/TAP.2019.2944216
[143]	E. Galiffi, R. Tirole, S. Yin, H. Li, S. Vezzoli, P. A. Huidobro, et al., Photonics of time-varying media, Adv. Photonics, 4 (2022), 014002–014002. https://doi.org/10.1117/1.AP.4.1.014002 doi: 10.1117/1.AP.4.1.014002
[144]	N. Engheta, Four-dimensional optics using time-varying metamaterials, Science, 379 (2023), 1190–1191. https://doi.org/10.1126/science.adf1094 doi: 10.1126/science.adf1094

Reader Comments

Your name:*

Email:*
© 2024 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)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Electronic Research Archive

1.1 1.7

Metrics

Article views(2170) PDF downloads(169) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Electronic Research Archive

Recent progress on mathematical analysis and numerical simulations for Maxwell's equations in perfectly matched layers and complex media: a review

Related Papers:

Abstract

1. Introduction

2. Formulation of the SEVIR model

2.1. Equilibrium point and reproductive number

3. Bounded and positive solutions

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

5. Global stability for developed system

5.1. Lyapunov's first derivative

6. Solutions by fractal-fractional operator

7. Simulation explanation

8. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

Abstract

1. Introduction

2. Formulation of the SEVIR model

2.1. Equilibrium point and reproductive number

3. Bounded and positive solutions

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

5. Global stability for developed system

5.1. Lyapunov's first derivative

6. Solutions by fractal-fractional operator

7. Simulation explanation

8. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

Electronic Research Archive

Recent progress on mathematical analysis and numerical simulations for Maxwell's equations in perfectly matched layers and complex media: a review

Related Papers:

Abstract

1. Introduction

2. Formulation of the SEVIR model

2.1. Equilibrium point and reproductive number

3. Bounded and positive solutions

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

5. Global stability for developed system

5.1. Lyapunov's first derivative

6. Solutions by fractal-fractional operator

7. Simulation explanation

8. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. Formulation of the SEVIR model

2.1. Equilibrium point and reproductive number

3. Bounded and positive solutions

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

5. Global stability for developed system

5.1. Lyapunov's first derivative

6. Solutions by fractal-fractional operator

7. Simulation explanation

8. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References