Stability and numerical analysis via non-standard finite difference scheme of a nonlinear classical and fractional order model

Abstract

In this paper, we develop a new mathematical model for an in-depth understanding of COVID-19 (Omicron variant). The mathematical study of an omicron variant of the corona virus is discussed. In this new Omicron model, we used idea of dividing infected compartment further into more classes i.e asymptomatic, symptomatic and Omicron infected compartment. Model is asymptotically locally stable whenever ${\mathcal{R}}_0<1$ and when ${\mathcal{R}}_0\le 1$ at disease free equilibrium the system is globally asymptotically stable. Local stability is investigated with Jacobian matrix and with Lyapunov function global stability is analyzed. Moreover basic reduction number is calculated through next generation matrix and numerical analysis will be used to verify the model with real data. We consider also the this model under fractional order derivative. We use Grunwald–Letnikov concept to establish a numerical scheme. We use nonstandard finite difference (NSFD) scheme to simulate the results. Graphical presentations are given corresponding to classical and fractional order derivative. According to our graphical results for the model with numerical parameters, the population’s risk of infection can be reduced by adhering to the WHO’s suggestions, which include keeping social distances, wearing facemasks, washing one’s hands, avoiding crowds, etc.

Introduction

The infection casus by Omicron variant of COVID-19 is called Omicron virus. In the last week of November 2021, this new invariant is identify in South Africa. After identification and repaid growth the virus transmits to other countries. As compared to other variant of corona virus omicron is not severe but transmission rate is very high. According to (CDC) Center for Diseases Control says that any one having infected by Omicron virus can transfer the infection to other vaccinated or individuals which do not have any common symptoms [1]. Some known symptoms of Omicron are runny nose, body ache, congestion, fatigue, cough etc. In order to minimize the infection rate other countries of the universe stop their flight to infected country South Africa.

The dynamics of corona virus have been studied by many scientists and researchers to control the spread of disease in the population [2], [3], [4], [5]. To more effectively minimize the infection in population in future spreading [6], [7], [8] the scientist of the filed are tried to produce the immune and vaccinated the majority of population. Although new SARS-CoV-2 varieties have emerged with the passage of time, infections are still a problem in many nations. This paper discusses some integral order mathematical model for investigating SARS-CoV-2 infections. For instance, [9] discusses the early infection of disease in China by using a very comprehensive model. By taking into account the actual SARS-CoV-2 cases, the best elimination and control of the disease in Pakistan has been researched in [10]. The best and most efficient technique to minimize the infection is isolation and quarantine, which has explored by the authors in [11] using a mathematical modeling approach. The SARS-CoV-2 infection can spread to other uninfected persons extremely quickly. An analysis of the lockout and its effects on disease prevention using mathematical modeling can be found in [12]. The authors provided the disease control scenario for a model SEIR approaches utilizing real numerical data from France and Italy [13].

Due to its numerous attributes and practical applications to issues in engineering and physics, fractional calculus is attracting the interest of researchers all over the world. The memory, the crossover behavior, and the hereditary qualities can be shown in a fractional order model. In integral differential equations [14], the construction of the operators and applications to epidemiology [15], [16], [17], applications to follow dynamical equations [18], [19], etc., the fractional calculus with various fractional operators has been found. Researchers [20] explored the SARS-CoV-2 stochastic model along time delay. Furthermore, authors [21] have used a model of fractional order to examine the singular and nonsingular kernels of the SARS-CoV-2 infection. [22] discusses the Caputo fractional model for the investigation of SARS-CoV-2 infection. The authors of [23] developed a mathematical model that takes the media campaign role into account while tackling the SARS-CoV-2 infection. [24] studies the modeling of the SAES-CoV-2 in China. The investigation and analysis of SARS-CoV-2 non reported cases are done in [25]. Authors in [26] examined how to use the new generalized idea of Caputo fractional order differentiation to arrive at the numerical analysis of the SARS-Cov-2 infecting model. [27] has examined the SARS-CoV-2 disease model with integral and non-integral orders. The time-delayed COVID-19 infection model has been proposed in [28]. Researchers [29] have looked at a fractional model to pinpoint when the illness peaked in Brazil. [30] discusses the numerical analysis of the SARS-CoV-2 model. The authors of [31] took into account the affected instances in India, created a fractional model, and then got their results. Authors [32] have discussed a SARS-CoV-2 infection model using immunization. The infection cases in Argentina have been studied using a fractional model proposed in [31]. The authors published their numerical simulations and two vaccine models for the SARS-CoV-2 infection in [32]. To analyze the infection cases in Spain, a fractional-order SARS-CoV-2 model with optimal control strategies was applied [33]. The authors examined the fractional-order SARS-CoV-2 model infection using the idea of the modified Euler approach in [34]. The conditions for the SARS-CoV-2 model’s global stability in the situation of no disease were proposed by the authors [35]. A nonlinear fractional order SARS-CoV-2 model infection has been proposed in [36]. To explore the qualitative analysis of the model, a mathematical model for SARS-CoV-2 has been built in [37]. Moreover authors [38] have investigated a model on the SARS-CoV-2 for global dynamics. In additions, the concepts of fractional calculus have been used very well recently to investigate COVID-19. Additionally, various models involving concepts of mathematical analysis have been utilized very well. In this regards we refer some work, where various analysis and numerical scheme have been used to study different kinds of problems as [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50].

The primary objective of this paper is to use mathematical modeling to comprehend the behavior of the new SARS-CoV-2 type known as the Omicron. We develop the model using the omicron feature and parameterize it using real data from South Africa. In order to explore the potential existence of several layers or waves, a NCOVID-19 mathematical model for new variant (omicron) is taken into consideration. Sensitivity analysis is used to identify the variables that have the greatest potential to improve or reduce the fundamental reproduction number $R_0$.

This study present epidemics model for SARS-CoV-2 regarding focus on strategies, control and specific vaccination of population. Proposed model are solve for Reproduction Number, analysis and model description respectively. Numerical simulation are calculated by NSFD Scheme [51]. A general group of techniques in numerical analysis known as non-finite difference schemes create a discrete model to provide numerical solutions to differential equations. In the last conclude this paper by ‘Conclusion’.

Model formulation

Due mainly to the new SARS-CoV-2 version, also called Omicron virus, which was first discovered in South Africa, people there once more had to adhere to tight SOPs, keep social distances, wear face masks, and other limitations. Because of the new SARS-CoV-2 variant’s rapid proliferation in several countries, these nations prohibited their citizens from traveling to South Africa. We are developed a new model to comprehend the dynamical behavior of the COVID-19 cases from South Africa while keeping in mind the Omicron variation. We take into account the entire population, indicating it by $\mathcal{N}\left(t\right)$, and $\mathcal{I}$ is divided in $\mathcal{I}=\mathcal{A}+\mathcal{B}+\mathcal{C}$. Further classifying it into six distinct epidemiological groups: Those who have been exposed to $S\left(t\right)$, represent the uncontagious population, and $\mathcal{IE}\left(t\right)$ is stand for exposed population, which coming into touch with asymptomatic population, symptomatic population, or omicron variant-infected individuals; asymptomatic population are represent $\mathcal{A}\left(t\right)$; and symptomatic population by $B\left(t\right)$ (infected individuals population although they have no clear symptoms sign), symptomatic people (population who exhibit clinical symptoms consistent with SARS-CoV-2 infections), having Omicron variant infection is represent by $\mathcal{C}\left(t\right)$ (who exhibit clear clinical symptom consistent with omicron infection; and have the ability to spread the disease to others whether or not they have received the vaccine), the recovered population is represent by, $\mathcal{R}\left(t\right)$ (population recovered from $\mathcal{A}\left(t\right)$, $\mathcal{B}\left(t\right)$ and $\mathcal{C}\left(t\right)$). We obtain our model as

$$\frac{d\mathcal{S}}{dt}=\beta -k\mathcal{S}\mathcal{I}-d_0\mathcal{S},$$$$\frac{d\mathcal{E}}{dt}=k\mathcal{S}\mathcal{I}-(\eta +d_0)\mathcal{E},$$$$\frac{d\mathcal{A}}{dt}=a\eta \mathcal{E}-({\gamma }_1+d_0)\mathcal{A},$$$$\frac{d\mathcal{B}}{dt}=(1-a-b)\eta \mathcal{E}-({\gamma }_2+d_0+d_I)\mathcal{B},$$$$\frac{d\mathcal{C}}{dt}=b\eta \mathcal{E}-({\gamma }_3+d_0)\mathcal{C},$$$$\frac{d\mathcal{R}}{dt}={\gamma }_1\mathcal{A}+{\gamma }_2\mathcal{B}+{\gamma }_3\mathcal{C}-d_0\mathcal{R}.$$ (1)

The flow of infection and different rate from one compartment to another is presented as in Fig. 1.

Fig. 1.

Diagrammatical presentation of the model (1).

Symbols involved in the model (1) are described in Table 1.

Table 1.

Nomenclature and discerption of the model (1).

Variables	The physical representation
$\mathcal{S}$	Susceptible class
$\mathcal{E}$	Exposed class
$\mathcal{A}$	Asymptomatic compartment
$\mathcal{B}$	Symptomatic with no visible symptoms compartment
$\mathcal{C}$	Omicron infected compartment
$\mathcal{R}$	Recovered compartment
$\beta$	New emergent population
$d_0$	Natural death rate
${\gamma }_1$	Recovery from Asymptomatic
${\gamma }_2$	Recovery from Symptomatic
${\gamma }_3$	Recovery from omicron infected
$d_I$	Death rate from Symptomatic
$a$	Infection rate of asymptomatic
$\eta$	Exposed rate
$b$	Infection rate of symptomatic
$k$	Contact rate

Feasible region, positivity and boundedness

On addition of all equations of the model (1) yields

$$\mathcal{N}\left(t\right)=\mathcal{S}\left(t\right)+\mathcal{E}\left(t\right)+\mathcal{A}\left(t\right)+\mathcal{B}\left(t\right)+\mathcal{C}\left(t\right)+\mathcal{R}\left(t\right),$$

one has

$$\frac{d\mathcal{N}\left(t\right)}{dt}=\beta -d_0\mathcal{N}-d_I\mathcal{B}.\le \beta -d_0\mathcal{N}$$ (2)

From (2), we have

$$\mathcal{N}\left(t\right)=\frac{\beta }{d_0}+\left({\mathcal{N}}_0-\frac{\beta }{d_0}\right)e^{d_{0}t}.$$ (3)

All values of non-negative for $t$.

Therefore, all solutions began of the system (1) will remain nonnegative for all value of t is equal to zero. So, the system (1) is well-posed mathematically.

The feasible region for the dynamical analysis is following

$$\chi =(\mathcal{S},\mathcal{E},\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{R}):0\le \mathcal{S},\mathcal{S}+\mathcal{E}+\mathcal{A}+\mathcal{B}+\mathcal{C}+\mathcal{R}\le \frac{\beta }{d_0}.$$

Disease-free Equilibrium (DFE)

The DFE of the system (1) is denoted by ${\mathcal{E}}_0=({\mathcal{S}}^0,0,0,0,0,0)$ which is follow

$${\mathcal{E}}^0=\left(\frac{\beta }{d_0},0,0,0,0,0\right).$$ (4)

Endemic equilibrium (EE)

Also the EE is computed as

$${\mathcal{S}}^{\ast }\left(t\right)=\frac{\beta }{k{\mathcal{I}}^{\ast }},$$

$${\mathcal{E}}^{\ast }\left(t\right)=\frac{k\beta {\mathcal{I}}^{\ast }}{(k{\mathcal{I}}^{\ast }+d_0)(\eta +d_0)},$$

$${\mathcal{A}}^{\ast }\left(t\right)=\frac{ka\eta \beta {\mathcal{I}}^{\ast }}{(k{\mathcal{I}}^{\ast }+d_0)(\eta +d_0)({\gamma }_1+d_0)},$$

$${\mathcal{B}}^{\ast }\left(t\right)=\frac{(1-a-b)k\beta {\mathcal{I}}^{\ast }}{(k{\mathcal{I}}^{\ast }+d_0)(\eta +d_0)({\gamma }_1+d_0)({\gamma }_2+d_0+d_I)},$$

$${\mathcal{C}}^{\ast }\left(t\right)=\frac{kb\eta \beta {\mathcal{I}}^{\ast }}{(k{\mathcal{I}}^{\ast }+d_0)(\eta +d_0)({\gamma }_1+d_0)({\gamma }_3+d_0)},$$

$${\mathcal{R}}^{\ast }\left(t\right)=\frac{{\gamma }_1{\mathcal{A}}^{\ast }+{\gamma }_2{\mathcal{B}}^{\ast }{\gamma }_3{\mathcal{C}}^{\ast }}{d_0}.$$

Expression for ${ℛ}_0$

In epidemiology there is a factor, which called the basic reproduction number ${ℛ}_0$ which describe the control and transmission of infection. Furthermore from ${ℛ}_0$, we can determine how the infection of COVID-19 is going in the population and which is the greatest choice to control the infection from the population. The next generation matrix method is used to find ${ℛ}_0$ is given below let $\chi =(\mathcal{E},\mathcal{A},\mathcal{B},\mathcal{C})$, then form system (1), we have

$$\frac{d\chi }{dt}=\mathcal{F}-\mathcal{V}.$$

Where

$$\mathcal{F}=\left(\begin{array}{c}\hfill k\mathcal{S}\mathcal{I}\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right).$$

And

$$\mathcal{V}=\left(\begin{array}{c}\hfill -(\eta +d_0)\mathcal{E}\hfill \\ \hfill -a\eta \mathcal{E}+({\gamma }_1+d_0)\mathcal{A}\hfill \\ \hfill -(1-a-b)\eta \mathcal{E}-({\gamma }_2+d_0+d_I)\mathcal{B}\hfill \\ \hfill b\eta \mathcal{E}-({\gamma }_3+d_0)\mathcal{C}\hfill \end{array}\right).$$

Jacobian of $\mathcal{F}$ at DFE point is

$$F=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill k{\mathcal{S}}^0\hfill & \hfill k{\mathcal{S}}^0\hfill & \hfill k{\mathcal{S}}^0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Also for the DFE, Jacobian of matrix $\mathcal{V}$ is follow

$$V=\left(\begin{array}{cccc}\hfill \eta +d_0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -a\eta \hfill & \hfill {\gamma }_1+d_0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -(1-a-b)\eta \hfill & \hfill 0\hfill & \hfill {\gamma }_2+d_0+d_I\hfill & \hfill 0\hfill \\ \hfill -b\eta \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\gamma }_3{d}_0\hfill \end{array}\right).$$

Hence, ${\mathcal{R}}_0$ is spectral radius of $F{V}^{-1}$ computed as

$${\mathcal{R}}_0=\frac{kb\eta }{({\gamma }_3+d_0)(\eta +d_0)}+\frac{ka\eta }{({\gamma }_1+d_0)(\eta +d_0)}+\frac{k(1-a-b)\eta }{({\gamma }_2+d_0+d_I)(\eta +d_0)}.$$

Hence, ${\mathcal{R}}_0$ has three parts, first part represent contact between infected population with uninfected (healthy) population. While second part represent the contact between uninfected population to asymptomatic and third part show infection of symptomatic population. The infection of SARS-CoV-2 can be control in the population whenever the valve of ${\mathcal{R}}_0<1$, on the other hand the infection will be spread in the community if ${\mathcal{R}}_0>1$. In next theorem, we discus the stability of the system (1) under condition of ${\mathcal{R}}_0>1$ and ${\mathcal{R}}_0<1$ at disease-free equilibrium. Using the numerical value of Table 2, we give a 3D profile of ${\mathcal{R}}_0$ in Fig. 2.

Table 2.

The parameters and their discerption involve in the model (1).

Nomenclature	Numerical values
${\mathcal{S}}_0$	6.0069540 millions
${\mathcal{E}}_0$	0.062000 millions
${\mathcal{A}}_0$	0.008000 millions
${\mathcal{B}}_0$	0.000100 millions
${\mathcal{C}}_0$	0..000360 millions
${\mathcal{R}}_0$	0 millions
$\beta$	0.2553
$d_0$	0.00425
${\gamma }_1$	0.8447
${\gamma }_2$	0.200
${\gamma }_3$	0.6746
$d_I$	0.0015
$b$	0.0101
$\eta$	0.8999
$a$	0.9566
$k$	0.8999

Fig. 2.

3D profile of ${\mathcal{R}}_0$.

Theorem 1

“(i) There does not exits positive equilibrium for the model (1) , if ${ℛ}_1\le 1$ or/and ${ℛ}_2\le 1$ .

(ii) There exists a distinct positive (unique) equilibrium ${\mathcal{E}}^{\ast }=({\mathcal{S}}^{\ast },{\mathcal{E}}^{\ast },{\mathcal{A}}^{\ast },{\mathcal{B}}^{\ast },{\mathcal{C}}^{\ast },{\mathcal{R}}^{\ast })$ which is also called an endemic equilibrium if ${ℛ}_1>1$ or/and ${ℛ}_2>1$ .”

Stability analysis

This section has the study of properties of equilibrium point with local and global stability of the system (1). Also, we will investigate the properties of these equilibrium and global analysis of the system (1).

Local stability

In this section, we present the local stability of the model (1) with condition ${\mathcal{R}}_0<1$.

Theorem 2

“If ${ℛ}_0<1$ , infection model COVID-19 is stable locally asymptotically at DFE ${\mathcal{E}}_0$ .”

Proof

To obtained the stability result, the Jacobian Matrix at ${\mathcal{E}}_0$ is given below

$$\mathcal{M}=\left[\begin{array}{cccccc}\hfill -d_0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -(d_0+\eta )\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill a\eta \hfill & \hfill -({\gamma }_1+d_0)\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill (1-a-b)\eta \hfill & \hfill 0\hfill & \hfill -({\gamma }_2+d_0+d_I)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill b\eta \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -({\gamma }_3+d_0)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\gamma }_1\hfill & \hfill {\gamma }_2\hfill & \hfill {\gamma }_3\hfill & \hfill -d_0\hfill \end{array}\right]$$

The characteristic polynomial of the Jacobian matrix at DFE is given by $det({\mathcal{M}}^0-\lambda I)=0$, where $\lambda$ is the eigenvalue and I is 6 by 6 identity matrix. Thus, $\mathcal{M}$ has eigenvalues given by

$${\lambda }_1=-d_0,$$

$${\lambda }_2=-(\eta +d_0),$$

$${\lambda }_3=-({\gamma }_1+d_0),$$

$${\lambda }_4=-({\gamma }_2+d_0+d_I),$$

$${\lambda }_5=-({\gamma }_3+d_0),$$

$${\lambda }_6=-d_0.$$

${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, ${\lambda }_4$, ${\lambda }_5$ and ${\lambda }_6$ are negative strictly. Thus, Disease free stability (DEF) is stable locally asymptotically. □

Global stability

In this part of our manuscript, we present global stability at DFE of the system (1). We constructed a function called Lyapunov function to show the model (1) is stable asymptotically globally in the next theorem.

Theorem 3

“ If ${\mathcal{R}}_0\le 0$ , then infection of system (1) is stable globally asymptotically at ${\mathcal{E}}_0$ .”

Proof

To prove this theorem result, we constructed a function called Lyapunov function.

$$\mathcal{W}=C_1(\mathcal{S}-{\mathcal{S}}^0)+C_2(\mathcal{E}-{\mathcal{E}}^0)+C_3(\mathcal{A}-{\mathcal{A}}^0)+C_4(\mathcal{B}-{\mathcal{B}}^0)+C_5\mathcal{C}.$$ (5)

Here $C_1$, $C_2$, $C_3$ and $C_4$ are some constants. Now w.r.t “t” take derivative of above equation.

$$\dot{\mathcal{W}}=C_1\dot{\mathcal{S}}+C_2\dot{\mathcal{E}}+C_3\dot{\mathcal{A}}+C_4\dot{\mathcal{B}}+C_5\dot{\mathcal{C}}.$$

Putting the values from (1)

$$\dot{\mathcal{W}}=C_1\dot{\beta -k\mathcal{S}\mathcal{I}-d_0\mathcal{S}}+C_2\dot{k\mathcal{S}\mathcal{I}-(\eta +d_0)\mathcal{E}}+C_3\dot{a\eta \mathcal{E}+({\gamma }_1+d_0)\mathcal{A}}+C_4\dot{(1-a-b)\eta \mathcal{E}+({\gamma }_2+d_0+d_I)\mathcal{B}}+C_{5}b\eta \mathcal{E}+({\gamma }_3+d_0)\mathcal{C}.$$

After some basic calculation

$$\dot{W}=k\mathcal{S}\mathcal{I}(C_2-C_1)+a\eta \mathcal{E}(C_3-C_4)+\eta \mathcal{E}(C_4-C_2)+b\eta \mathcal{E}(C_5-C_4)-(C_1{d}_0\mathcal{S}-C_1\beta )-C_2{d}_0\mathcal{E}-C_3{\gamma }_1\mathcal{A}-c_3{d}_0\mathcal{A}-C_4{\gamma }_2\mathcal{B}-C_4{d}_0\mathcal{B}-C_4{d}_I\mathcal{B}-C_5{\gamma }_3\mathcal{C}-C_5{d}_0\mathcal{C}.$$

Let us choose $C_1=C_2=C_3=C_4=C_{5}1$.

We obtained

$$\dot{W}=-(d_0\mathcal{S}-\beta )-d_0\mathcal{E}-{\gamma }_1\mathcal{A}-d_0\mathcal{A}-{\gamma }_2\mathcal{B}-d_I\mathcal{B}-{\gamma }_3\mathcal{C}-d_0\mathcal{C}<0$$

Hence, ${\mathcal{R}}_0\le 0$ then the system (1) is globally asymptotically stable. □

Numerical results and discussion

We take into account the infected cases that were recorded in South Africa. The infection are evaluated on a regular basis using a unit per day. Calculated natural birth rate and death rate 0.0155432, which is the mean rate of life of South Africans in 2020–21, are two of the 12 parameters in the given model (1). The method $N\left(0\right)=\frac{\beta }{d_0}$ is used to calculate the birth rate, here $N\left(0\right)$ represent total population of South Africa in 2021. About $N\left(0\right)=60140000$ people are projected to live in South Africa in 2021. $\mathcal{S}\left(0\right)=60069540$ is the uninfected population individual in the non-appearance of the disease, and $\mathcal{E}\left(0\right)=0.062$ million, $\mathcal{A}\left(0\right)=0.008$ million, $\mathcal{B}\left(0\right)=100$, and $\mathcal{C}\left(0\right)=360$ are the reported people on 1st November, 2021, with $\mathcal{R}\left(0\right)=0$ being the initial given conditions susceptible to fitting data. Let suppose that recovery from infection is zero yet. In order to suitable the system to the indicated numerical data with the time mentioned early, we used the nonlinear least square method. The experiments were carried out up until the required level of accuracy in the model fitting. Here, we establish NSFD scheme [51] for our considered model. Therefore, consider first equation of (1) as

$$\frac{d\mathcal{S}\left(t\right)}{dt}=\beta -k\mathcal{S}\mathcal{I}-d_0\mathcal{S}.$$ (6)

Which is decomposed by NSFD scheme as

$$\frac{{\mathcal{S}}_{j+1}-{\mathcal{S}}_j}{h}=\beta -k{\mathcal{S}}_j{\mathcal{I}}_j-d_0{\mathcal{S}}_j.$$ (7)

Like (7), by utilizing NSFD method, we present the model (1) as

$${\mathcal{S}}_{j+1}={\mathcal{S}}_j+h\left(\beta -k{\mathcal{S}}_j{\mathcal{I}}_j-d_0{\mathcal{S}}_j\right),$$

$${\mathcal{E}}_{j+1}={\mathcal{E}}_j+h\left(k{\mathcal{S}}_j{\mathcal{I}}_j-(\eta +d_0){\mathcal{E}}_j\right),$$ (8)

$${\mathcal{A}}_{j+1}={\mathcal{I}}_j+h\left(a\eta {\mathcal{E}}_j-({\gamma }_1+d_0){\mathcal{A}}_j\right)$$ (9)

$${\mathcal{B}}_{j+1}={\mathcal{V}}_j+h\left((1-a-b)\eta {\mathcal{E}}_j-({\gamma }_2+d_0+d_I){\mathcal{B}}_j\right),$$ (10)

$${\mathcal{C}}_{j+1}={\mathcal{R}}_j+h\left(b\eta {\mathcal{E}}_j-({\gamma }_3+d_0){\mathcal{C}}_j\right),$$

$${\mathcal{R}}_{j+1}={\mathcal{R}}_j+h\left({\gamma }_1{\mathcal{A}}_j+{\gamma }_2{\mathcal{B}}_j+{\gamma }_3{\mathcal{C}}_j-d_0{\mathcal{R}}_j\right).$$

Developing NSFD scheme [51], we plotted the model (1). To understand the dynamics of the system, we used real data of South Africa to simulate. With using numerical value table, we present graphically the dynamical behaviors of various compartments of our model in Fig. 3, Fig. 4, Fig. 5 using classical order model.

Fig. 3.

Dynamical behaviors of susceptible and exposed classes of the considered model corresponding to classical model.

Fig. 4.

Dynamical behaviors of symptomatic and symptomatic with no visible symptoms classes of the considered model corresponding to classical model.

Fig. 5.

Dynamical behaviors of omicron infected and recovered classes of the considered model corresponding to classical model.

NSFD Scheme for Fractional Order Model (1)

In this part of our paper, we present some fractional concept for model (1), we used Grunwald–Letnikov method for Caputo derivative as used in [41], [42]. Moreover, the numerical method used in this paper has the ability to demonstrate the dynamical properties of model (1). Taking a small interval $h$, our using numerical method preserve the stability of disease-free and endemic equilibrium points. Mostly researchers used different methods like Euler, RK2, and RK4 method but our numerical method have the advantages over it. For instance, the said numerical scheme has less complexity than RK2 and RK4 methods. This numerical scheme has a significant advantage over other scheme that for a large time raising it yield a good numerical results.

Definition 4

“Integral of non-integer order $\beta >0$ of a function $\mathcal{S}:[0,\infty )\to \mathbb{R}$ is defined as

$$I_t^{\phi }\mathcal{X}\left(t\right)=\frac{1}{\chi \left(\alpha \right)}{\int }_0^t\frac{\mathcal{X}\left(\theta \right)}{{(t-\theta )}^{1-\alpha }}d\theta ,$$

provided the integral exists at the right sides”. Moreover, by definition of Caputo, we have

$$D_{0+}^{\alpha }\mathcal{X}\left(t\right)=\left\{\begin{array}{cc}\hfill & \frac{1}{\chi (1-\alpha )}{\int }_0^t{(t-\eta )}^{-\alpha }{\mathcal{X}}^{\prime }\left(\eta \right)d\eta ,0<\alpha \le 1,\hfill \\ \hfill & \frac{d\mathcal{X}}{dt},\alpha =1.\hfill \end{array}\right)$$

From Riemann Liouville operator integral with some fractional order as

$$I^{\alpha }\chi \left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\tau )}^{\alpha -1}\chi \left(\theta \right)d\theta ,\alpha >0.$$ (11)

Following the procedure of [42], we have

$${}^C{\mathrm{D}}^{\alpha }\chi \left(t\right)=\sum _{j=0}^{n+1}{N}_j^{\alpha }{\chi }_{n-j+1},n=0,1,2,\dots ,$$ (12)

where $N_0=\frac{1}{{\mathfrak{h}}^{\alpha }},N_j=\left(\frac{j-1-\alpha }{j}\right)N_{j-1}^{\alpha },j=1,2,3,\dots$. Here it is interesting if $n=0$, we have from (12)

$${}^C{\mathrm{D}}^{\alpha }\chi \left(t\right)=\frac{{\chi }_1-\alpha {\chi }_0}{{\mathfrak{h}}^{\alpha }}.$$

Consider

$$\left\{\begin{array}{cc}\hfill & {}^C{\mathrm{D}}^{\alpha }\chi \left(t\right)=f(t,\chi \left(t\right)),t\in [0,T],0<T<\infty ,\hfill \\ \hfill & \chi \left(t_0\right)={\chi }_0.\hfill \end{array}\right)$$ (13)

From Eq. (12) to discretion (13) as

$$\sum _{j=0}^{n+1}{N}_j^{\alpha }{\chi }_{n-j+1}=f(t_{n+1},\chi \left(t_{n+1}\right)),n=0,1,2,\dots ,$$ (14)

we have

$${\chi }_{n+1}=\frac{1}{N_0^{\alpha }}\left[-\sum _{j=1}^{n+1}{N}_j^{\alpha }{\chi }_{n-j+1}+f(t_{n+1},\chi \left(t_{n+1}\right))\right],\text{where}n=0,1,2,\dots ,$$ (15)

where $N_0^{\alpha }={\left[\frac{1}{\phi (\mathfrak{h},\omega )}\right]}^{\alpha },N_j^{\alpha }=\left[\frac{j-1-\alpha }{j}\right]N_{j-1}^{\alpha },j=1,2,3,\dots$. Let assume $\phi (\mathfrak{h},\omega )$ obtained $\mathfrak{h}+O\left({\mathfrak{h}}^2\right)$. The stability rule is obtained from some functions as $sin\mathfrak{h},cos\mathfrak{h},\mathfrak{h}$, etc. Here, we confer our counsel system (1) to non-integral order $0<\alpha \le 1$ as follow

$$\left\{\begin{array}{cc}\hfill & {}^C{\mathrm{D}}^{\alpha }S\left(t\right)=\beta -k\mathcal{S}\mathcal{I}-d_0\mathcal{S},\hfill \\ \hfill & {}^C{\mathrm{D}}^{\alpha }E\left(t\right)=k\mathcal{S}\mathcal{I}-(\eta +d_0)\mathcal{E},\hfill \\ \hfill & {}^C{\mathrm{D}}^{\alpha }A\left(t\right)=a\eta \mathcal{E}-({\gamma }_1+d_0)\mathcal{A},\hfill \\ \hfill & {}^C{\mathrm{D}}^{\alpha }B\left(t\right)=(1-a-b)\eta \mathcal{E}-({\gamma }_2+d_0+d_I)\mathcal{B},\hfill \\ \hfill & {}^C{\mathrm{D}}^{\alpha }C\left(t\right)=b\eta \mathcal{E}-({\gamma }_3+d_0)\mathcal{C},\hfill \\ \hfill & {}^C{\mathrm{D}}^{\alpha }R\left(t\right)={\gamma }_1\mathcal{A}+{\gamma }_2\mathcal{B}+{\gamma }_3\mathcal{C}-d_0\mathcal{R}.\hfill \end{array}\right)$$ (16)

Considering of (15) and by utilizing Grunwald–Letnikov discrimination method, for system (16) for non-integral order, we present numerical method here.

$$\left\{\begin{array}{cc}\hfill & S\left(t_{n+1}\right)=\frac{1}{N_0^{\alpha }}\left[-\sum _{j=1}^{n+1}{N}_j^{\alpha }S\left(t_{n+1-j}\right)+\beta -k\mathcal{S}\left(t_n\right)\mathcal{I}\left(t_n\right)-d_0\mathcal{S}\left(t_n\right)\right],\hfill \\ \hfill & E\left(t_{n+1}\right)=\frac{1}{N_0^{\alpha }}\left[-\sum _{j=1}^{n+1}{N}_j^{\alpha }E\left(t_{n+1-i}\right)+k\mathcal{S}\left(t_n\right)\mathcal{I}\left(t_n\right)-(\eta +d_0)\mathcal{E}\left(t_n\right)\right],\hfill \\ \hfill & A\left(t_{n+1}\right)=\frac{1}{N_0^{\alpha }}\left[-\sum _{j=1}^{n+1}{N}_j^{\alpha }A\left(t_{n+1-i}\right)+a\eta \mathcal{E}\left(t_n\right)+({\gamma }_1+d_0)\mathcal{A}\left(t_n\right)\right],\hfill \\ \hfill & B\left(t_{n+1}\right)=\frac{1}{N_0^{\alpha }}\left[-\sum _{j=1}^{n+1}{N}_j^{\alpha }B\left(t_{n+1-i}\right)+(1-a-b)\eta \mathcal{E}\left(t_n\right)+({\gamma }_2+d_0+d_I)\mathcal{B}\left(t_n\right)\right],\hfill \\ \hfill & C\left(t_{n+1}\right)=\frac{1}{N_0^{\alpha }}\left[-\sum _{j=1}^{n+1}{N}_j^{\alpha }C\left(t_{n+1-i}\right)+b\eta \mathcal{E}\left(t_n\right)+({\gamma }_3+d_0)\mathcal{C}\left(t_n\right)\right]\hfill \\ \hfill & R\left(t_{n+1}\right)=\frac{1}{N_0^{\alpha }}\left[-\sum _{j=1}^{n+1}{N}_j^{\alpha }R\left(t_{n+1-i}\right)+{\gamma }_1\mathcal{A}\left(t_n\right)+{\gamma }_2\mathcal{B}\left(t_n\right)+{\gamma }_3\mathcal{C}\left(t_n\right)-d_0\mathcal{R}\left(t_n\right)\right].\hfill \end{array}\right)$$ (17)

Here in Fig. 6, Fig. 7, Fig. 8, we present graphically the dynamical behaviors of various classes using different fractional orders.

Fig. 6.

Dynamical behaviors of susceptible and exposed classes of the considered model corresponding to fractional order model.

Fig. 7.

Dynamical behaviors of symptomatic and symptomatic with no visible symptoms classes of the considered model corresponding to fractional order model.

Fig. 8.

Dynamical behaviors of omicron infected and recovered classes of the considered model corresponding to fractional order model.

Conclusion

In this manuscript, we constructed a new mathematical SEIVR model for NCOVID-19, which is base on behavior of virus. Our main goal is to estimate the effect of immunization on population. In the presence of presenting no clear infection or symptoms and loss of immunity are the root measure which differentiate this virus from other modeled infectious diseases. On the bases of analytical expression for ${\mathcal{R}}_0$ which is a main element to determine necessary and sufficient conditions for disease-free and endemic equilibrium. We testify our theoretical results with real data for a case of South Africa.

Our main resulting and finding are follow.

Vaccination helps to overcome the disease, we provide a mathematical proof that transmission rates are reduce with vaccination. We mathematical justified that the increasing in number of infection in South Africa was followed by vaccination campaign. On the bases of collected data from the parameters of model, we proved that the rate of vaccination for SARS-CoV-2 control spread of disease efficiently.

In near past, the researchers [52] of the fields observe that this virus change their behavior each time. As we know that this virus change their behavior time to time. We are trying to develop a new model for observed new pattern as like beta and omicron in future.

CRediT authorship contribution statement

Hussam Alrabaiah: Theoretical part. Rahim Ud Din: Drafted the paper. Khursheed J. Ansari: Done the numerical part. Ateeq ur Rehman Irshad: Edited and revised the last draft. Burhanettin Ozdemir: Edited and revised the last draft.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No data was used for the research described in the article.