Study of fractional order dynamics of nonlinear mathematical model

Abstract

This manuscript is devoted to establishing some theoretical and numerical results for a nonlinear dynamical system under Caputo fractional order derivative. Further, the said system addresses an infectious disease like COVID-19. The proposed system involves natural death rates of susceptible, infected and recovered classes respectively. By using nonlinear analysis feasible region and boundedness have been established first in this study. Global and Local stability analysis along with basic reproduction number have also addressed by using the next generation matrix method. Upon using the fixed point approach, existence and uniqueness of the approximate solution for the mentioned problem has also investigated. Some stability results of Hyers-Ulam (H-U) type have also discussed. Further for numerical treatment, we have exercised two numerical schemes including modified Euler method (MEM) and nonstandard finite difference (NSFD) method. Further the two numerical schemes have also compared with respect to CPU time. Graphical presentations have been displayed corresponding to different fractional order by using some real data.

1. Introduction

Fractional calculus has got much attention in recent times. The foundation of the said area had been provided by Newton and Lebnitz during seventeenth century. Later on Reimann, Liouville, Fourier, Abel and Euler developed this branch very well [1]. The first notable definition had been given by Reimann-Liouville in 1832. Also the mentioned definition has been modified by Caputo in 1967 to elaborate the concept more clearly in geometry [2], [3]. Since fractional derivative infact is a definite integrals which can be defined in numbers of ways. Geometrically the said operator gives accumulation of a function which includes its integer counter part as a special case. Therefore, various researchers have given different definitions for fractional differential operators [4]. The said operators have numerous application in modeling of various real world problems. As in most of the situations where memory terms or index laws are involved cannot be well explained via classical operators. For comprehensive descriptions and well explanations fractional calculus is powerful tool to use (see [5], [6]). Researchers have used the concept of fractional calculus for detailed explanations of many real world problems including physical, biological, dynamical and chemical phenomenons (we refer few article as [7], [8], [9], [10], [11]).

Recently various authors have used other kids of fractional differential operators like Hilfer, nonsingular kernel type, Caputo-Fabrizio operator to investigate various models. Here we state that each operator has own merits and de-merits. Here we remark that using various fractional differential operators numerous results relating controllability, existence theory, optimization and numerical analysis have been published in last few years. The concept of Hilfer derivative of fractional order has been applied to investigate some controllability results for delay type problems in [12], [13]. In same line authors [14] have established existence and controllability results for nonlocal mixed Volterra–Fredholm type fractional delay integro-differential equations. Some advanced results in controllability using fractional order derivatives have been established in [15], [16]. In same line authors have used neutral fractional order derivative to investigate a class of partial differential equations in [17]. Also authors have developed very applicable results for controllability and existence theory for neutral fractional integro-differential equations. For detail work, we refer [18], [19], [20]. Various results regarding analysis of various problems by using generalized proportional fractional integral operators, Atangana-Baleanue-Caputo operators have been established in last few years. Some recent contribution can be read as [21], [22], [23], [24], [25]. The analysis in aforesaid work has also applicable and can be further extended to COVID-19 or other infectious disease model. Here we remark that nonsingular kernel type and Caputo-Fabrizio operators suffer from initialization effect and extra assumptions need to be imposed on right hand side of a differential equations involving such like operators. This is not good effect and therefore now reducing the use of the said operators in applications. Although significant work has been done by using the aforesaid operators in epidemiology. But to the best of our knowledge in epidemiological models, the most powerful operators are those introduced by Reimann-Liouville and Caputo. Because they have clear geometrical interpretation like integer order differential equations and upon using integer value, the concerned equations reduce to their corresponding classical form.

The research area devoted to mathematical models for analysis of various physical phenomenons in real life is very interesting and has received wide attention among researchers in last several years. The idea of mathematical model had been originated by Bernoulli in 1776 (see [26]). Later on the concept adopted by Mekendric and Kermark in 1927 to describe an infectious disease model which is known as $\mathcal{SIR}$. The said model then further modified and extended to form new models for various infectious disease like TB, Cholerea, Typhoid, Dengue, HIV and AIDs, etc (see [27], [28]). Recently the Coronavirous disease (COVID-19) which has been originated from China and transmitted all over the world. The aforesaid disease which reported in an outbreak in 2019 in Wuhan, Hubei province, China, is caused by the SARS-CoV-2 virus (for further etiological information see [29]). According to researchers of virology, the said virus belongs to the class of beta-Coronavirus and like Middle East Respiratory Syndrome Coronavirus and Severe Acute Respiratory Syndrome Coronavirus. The novel virus began to cause pneumonia, and was named as Coronavirus disease (2019) on 11 February 2020 by WHO (see detail in [30]).

Also many researchers have formulated various mathematical models for COVID-19. This disease has greatly affected the whole globe. Due to COVID-19, more than fifty millions people have been died. Nearly five hundred million people have got infectious throughout the world (see some detail in [31], [32]). The said disease has greatly destroyed the life style and economical situation of many countries around the globe. Recently some countries have worked on to prepare vaccine for permanent cure of the disease in which they have got some success like UK, USA, China and Germany, etc (see [33], [34]). Researchers from bioengineering side, virology, epidemiology and those working on mathematical biology have worked significantly in last two years to investigate various procedure how to eradicate or control the disease from further spreading in community. Those who are working on computational biology have developed several mathematical models to understand the transmission dynamics of the said disease. Therefore large numbers of mathematical models have been formulated for COVID-19. Authors [42] have formulated the following SIRD type model to demonstrate the diffusion of the infection in various communities

$$\left\{\begin{array}{c}\dot{\mathcal{S}}(t)=\gamma \mathcal{SI}\hfill \\ \dot{\mathcal{I}}(t)=\gamma \mathcal{SI}-(\alpha +d)\mathcal{I}\hfill \\ \dot{\mathcal{R}}(t)=\alpha \mathcal{I}\hfill \\ \dot{\mathcal{D}}(t)=d\mathcal{I},\hfill \\ \mathcal{S}(0)={\mathcal{S}}_0>0,\mathcal{I}(0)={\mathcal{I}}_0⩾0,\hfill \\ \mathcal{R}(0)={\mathcal{R}}_0⩾0,\mathcal{D}(0)={\mathcal{D}}_0⩾0,\hfill \end{array}\right)$$ (1)

where $\mathcal{S}$ stand for susceptible, $\mathcal{I}$ for infected, $\mathcal{R}$ for recovered and $\mathcal{D}$ for death class. Further, $d$ denote death rate, $\gamma$ is used for infection rates and $\alpha$ represents rate of conversion to susceptible. Recently various analysis, investigations and procedure have been conducted to deal the disease from various aspects. The contribution done in this regards, we refer few article as [35], [36], [37], [38], [39], [40], [41]. Since the concept of fractional calculus has got significant attention during last few decades. Therefore mathematician as well as researchers in biomathematics have given much attention to study said models for the mentioned infection under various type derivatives of arbitrary orders (see [43], [44], [45], [46], [47], [48], [49]). The investigation of biological model of infection disease under fractional derivatives is an interesting study as compared to classical order derivative. Because fractional calculus provides dynamical interpretations of real world phenomenon with more degree of freedom (see few as [50], [51], [52], [53]).

Therefore inspired from the mentioned applications and importance of fractional calculus and biological models, we update model (1) by incorporating natural death rates of various compartments as well as recruitment rate to form the following model

$$\left\{\begin{array}{c}{}_0{}^{C}D_t^{\beta }[\mathcal{S}(t)]=\lambda -\gamma \mathcal{SI}-{\delta }_1\mathcal{S}+\alpha \mathcal{R},\hfill \\ {}_0{}^{C}D_t^{\beta }[\mathcal{I}(t)]=\gamma \mathcal{SI}-({\delta }_2+\mu +d)\mathcal{I},\hfill \\ {}_0{}^{C}D_t^{\beta }[\mathcal{R}(t)]=\mu \mathcal{I}-(\alpha +{\delta }_3)\mathcal{R},\hfill \\ {}_0{}^{C}D_t^{\beta }[\mathcal{D}(t)]=d\mathcal{I},\hfill \\ \mathcal{S}(0)={\mathcal{S}}_0>0,\mathcal{I}(0)={\mathcal{I}}_0⩾0,\hfill \\ \mathcal{R}(0)={\mathcal{R}}_0⩾0,\mathcal{D}(0)={\mathcal{D}}_0⩾0,\hfill \end{array}\right)$$ (2)

where $\lambda$ is recruitment rate, ${\delta }_i,i=1,2,3$ denote natural death of susceptible, infected and recovered class respectively. Also the notation ${}_0{}^{C}D_t^{\beta }$ stands for standard Caputo derivative of fractional order $\beta \in (0,1]$. Also $\mu$ is recovery rate from disease. Here we present the flow chart of the model (2) in Fig. 1 as

Fig. 1.

Flow chart of the proposed model (2).

Here we will establish feasible region and verify boundedness of the solution for the corresponding model (2). Further using the next generation matrix method, we will develop global and local stability results also. Keeping the importance and real approach analysis of fractional calculus, we will investigate model(2) under formal arbitrary order derivative of Caputo type numerically. We establish some sufficient results for existence and uniqueness of solution to the model(2) by using some fixed point theorem due to Banach and Schauder. Some stability results about H-U type are also established for our problem. Further we will establish numerical algorithm based on NSFD scheme to simulate the results for various fractional order against the available real initial data. The concerned NSFD scheme has already used to handle many problems (see [54], [55], [56], [57], [58]). Also we will use modified Euler method (MEM) to describe the numerical results. The said method has also used in many papers for instance [50], [51], [52]. A comparison between both classical and fractional order model to conclude which one is more better for numerical interpretation of the proposed problem is given. We will use Matlab for our numerical analysis in this research work.

Here we organized our work as: In first part we give introduction. Then we give some elementary results in Section second. Third section is devoted to some fundamental results about the model under consideration. Fourth section is devoted to qualitative analysis. Fifth section is related to numerical analysis which is further divided into four subsections. Sixth section is devoted to comparison of both schemes. Further last section seven is related to a brief conclusion.

2. Fundamental result

We recall some basic definition of fractional calculus [1], [2], [3].

Definition 2.1

Arbitrary order integration of $\mathcal{F}\in L[0,T]$ for $\beta >0$ is define as

$$I_{+0}^{\beta }[\mathcal{F}(t)]=\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}\mathcal{F}(\rho )d\rho$$

provided that integral on right exists pointwise on $(0,\infty )$.

Definition 2.2

If $\mathcal{F}\in C[0,T]$, then Caputo derivative of arbitrary order with $0<\alpha \le 1$ is recalled

$${}_0{}^{C}D_t^{\beta }\mathcal{F}(t)=\left\{\begin{array}{c}\frac{1}{\mathrm{\Gamma }(1-\beta )}{\int }_0^t{(t-\rho )}^{-\beta }\mathcal{F}\prime (\rho )d\rho ,0<\beta <1,\hfill \\ \frac{d\mathcal{F}}{\mathit{dt}},\beta =1,\hfill \end{array}\right)$$

provided that integral on right exists pointwise on $(0,\infty )$.

Lemma 2.3

The solution of FDE

$${}_0{}^{C}D_t^{\beta }\mathcal{F}(t)=\mathcal{G}(t),0<\beta \le 1,$$

is given by

$$\mathcal{F}(t)=C_0+I_{+0}^{\beta }\mathcal{G}(t),C_0\in R\text{.}$$

3. Feasible region and stability theory

Here we derive some conditions for feasible region as well as for stability theory using the next generation matrix method.

Lemma 3.1

The solution$(\mathcal{S},\mathcal{I},\mathcal{R},\mathcal{D})\in R_{+}^4$is bounded and attracted towards the feasible region defined by

$$\mathcal{T}=\left\{(\mathcal{S},\mathcal{I},\mathcal{R},\mathcal{D})\in R_{+}^4:0<\mathcal{N}(t)⩽\frac{\lambda }{({\delta }_1+{\delta }_2+{\delta }_3)}\right\}\text{.}$$

Proof

Summing all the equations of the Model (2) yields

$$\begin{array}{c}\frac{d\mathcal{N}}{\mathit{dt}}=\lambda -{\delta }_1\mathcal{S}-{\delta }_2\mathcal{I}-{\delta }_3\mathcal{R}-\alpha \mathcal{R}\hfill \\ \frac{d\mathcal{N}}{\mathit{dt}}⩽\lambda \hfill \\ \frac{d\mathcal{N}}{\mathit{dt}}⩽\lambda +({\delta }_1+{\delta }_2+{\delta }_3)\mathcal{N}\hfill \\ \frac{d\mathcal{N}}{\mathit{dt}}-({\delta }_1+{\delta }_2+{\delta }_3)\mathcal{N}⩽\lambda \text{.}\hfill \end{array}$$ (3)

By solving(3), we get

$$\mathcal{N}(t)⩽\frac{\lambda }{({\delta }_1+{\delta }_2+{\delta }_3)}+C\exp (-({\delta }_1+{\delta }_2+{\delta }_3)t)\text{.}$$ (4)

when $t\to \infty$, then one has $\mathcal{N}(t)⩽\frac{\lambda }{({\delta }_1+{\delta }_2+{\delta }_3)}$. Hence the required result.  □

Theorem 3.2

The disease free equilibrium and pandemic equilibrium points of the model(2)are given by${\mathcal{E}}_0=(\frac{\lambda }{{\delta }_1},0,0,0)$and

$${\mathcal{E}}^{\ast }=({\mathcal{S}}^{\ast },{\mathcal{I}}^{\ast },{\mathcal{R}}^{\ast },{\mathcal{D}}^{\ast }),$$ (5)

such that

$$\begin{array}{cc}\hfill {\mathcal{S}}^{\ast }=& \frac{{\delta }_2+\mu +d}{\gamma },\hfill \\ \hfill {\mathcal{I}}^{\ast }=& \frac{(\alpha +{\delta }_3)({\delta }_1\gamma -\lambda \gamma ({\delta }_2+\mu +d))}{\gamma (\alpha +{\delta }_3)+\alpha },\hfill \\ \hfill {\mathcal{R}}^{\ast }=& \frac{\mu {\delta }_1\gamma -\mu \lambda \gamma ({\delta }_3+\mu +d)}{\gamma (\alpha +{\delta }_3)+\alpha },\hfill \\ \hfill {\mathcal{D}}^{\ast }=& d(\alpha +{\delta }_3)[{\delta }_1\gamma -\lambda \gamma ({\delta }_2+\mu +d)]\text{.}\hfill \end{array}$$ (6)

Proof

For local and global stability analysis, the equilibrium points are important and can be computed from model (2) as

$$\begin{array}{c}{}_0{}^{C}D_t^{\beta }\mathcal{S}(t)=0\hfill \\ {}_0{}^{C}D_t^{\beta }\mathcal{I}(t)=0\hfill \\ {}_0{}^{C}D_t^{\beta }\mathcal{R}(t)=0\hfill \\ {}_0{}^{C}D_t^{\beta }\mathcal{D}(t)=0\text{.}\hfill \end{array}$$ (7)

Hence using (7) in model (2), one can easily computed the equilibrium points which are described in (5), (6) respectively.  □

Theorem 3.3

The basic Reproduction number corresponding to the proposed model(2)is computed as

$${\mathcal{R}}_0=\frac{\gamma \lambda }{{\delta }_1({\delta }_2+\mu +d)}\text{.}$$

Proof

Let $X=\mathcal{I}$ and considering second equation of (2) as

$$\begin{array}{cc}\hfill {}_0{}^{C}D_t^{\beta }(X)=& {}_0{}^{C}D_t^{\beta }(\mathcal{I})=\gamma \mathcal{SI}-({\delta }_2+\mu +d)\mathcal{I}\hfill \\ \hfill {}_0{}^{C}D_t^{\beta }(X)=& F-V,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill F=& \gamma \mathcal{SI},\hfill \\ \hfill V=& ({\delta }_2+\mu +d)\mathcal{I}\hfill \\ \hfill F=& [\frac{{\delta }_1(\gamma \mathcal{SI})}{{\delta }_1\mathcal{I}}=[\gamma \mathcal{S}]\hfill \\ \hfill V=& \left[\frac{{\delta }_1({\delta }_2+\mu +d)I}{{\delta }_1\mathcal{I}}\right]=[({\delta }_2+\mu +d)]\hfill \\ \hfill V^{-1}=& [\frac{1}{({\delta }_2+\mu +d)}]\hfill \\ \hfill {\mathit{FV}}^{-1}=& [\frac{\gamma \lambda }{{\delta }_1({\delta }_2+\mu +d)}]\text{.}\hfill \end{array}$$

Since ${\mathcal{R}}_0$ is equivalent to the maximum eigenvalue of ${\mathit{FV}}^{-1}$ at disease free equilibrium point ${\mathcal{E}}_0=(\frac{\lambda }{{\delta }_1},0,0,0)$. Therefore

$$\rho ({\mathcal{FV}}^{-1})=\left[\frac{\gamma \lambda }{{\delta }_1({\delta }_2+\mu +d)}\right]\text{.}$$ (8)

Hence the required quantity is given by

$${\mathcal{R}}_0=\frac{\gamma \lambda }{{\delta }_1({\delta }_2+\mu +d)}\text{.}$$ (9)

Thus we arrived at our intended result.

Theorem 3.4

The proposed model(2)is locally asymptotically stable at${\mathcal{E}}_0$if${\mathcal{R}}_0<1$, while the the model(2)is locally asymptotically stable at${\mathcal{E}}^{\ast }$if${\mathcal{R}}_0>1$.

Proof

Since the jacobian matrix is computed from all four equations of the model (2), but here we take the first three equations of the model as these are independent of $\mathcal{D}$. Therefore the Jacobian matrix for the model (2) can be computed as

$$\mathcal{J}=\left[\begin{array}{ccc}\hfill \frac{\partial {\mathrm{f}}_1}{\partial \mathcal{S}}\hfill & \hfill \frac{\partial {\mathrm{f}}_1}{\partial \mathcal{I}}\hfill & \hfill \frac{\partial {\mathrm{f}}_1}{\partial \mathcal{R}}\hfill \\ \hfill \frac{\partial {\mathrm{f}}_2}{\partial \mathcal{S}}\hfill & \hfill \frac{\partial {\mathrm{f}}_2}{\partial \mathcal{I}}\hfill & \hfill \frac{\partial {\mathrm{f}}_2}{\partial \mathcal{R}}({\mathrm{f}}_2)\hfill \\ \hfill \frac{\partial {\mathrm{f}}_3}{\partial \mathcal{S}}\hfill & \hfill \frac{\partial {\mathrm{f}}_3}{\partial \mathcal{I}}\hfill & \hfill \frac{\partial {\mathrm{f}}_3}{\partial \mathcal{R}}\hfill \end{array}\right]\text{.}$$

On setting the corresponding values, one has

$$\mathcal{J}=\left[\begin{array}{ccc}\hfill -{\delta }_1-\gamma \mathcal{I}\hfill & \hfill -\mathcal{S}\gamma \hfill & \hfill \alpha \hfill \\ \hfill \gamma \mathcal{I}\hfill & \hfill \gamma \mathcal{S}-{\delta }_2-\mu -d\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mu \hfill & \hfill -\alpha -{\delta }_3\hfill \end{array}\right]\text{.}$$ (10)

After putting the value of ${\mathcal{E}}_0$ in (10), we have

$$\mathcal{J}=\left[\begin{array}{ccc}\hfill -{\delta }_1\hfill & \hfill \frac{-\gamma \lambda }{{\delta }_1}\hfill & \hfill -\alpha \hfill \\ \hfill 0\hfill & \hfill \frac{\gamma \lambda }{{\delta }_1}-(\partial +\mu +d)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mu \hfill & \hfill -(\alpha +{\delta }_3)\hfill \end{array}\right]\text{.}$$

Now the characteristics equation can be computed as

$$\det (J-\eta \mathcal{I})=\left|\begin{array}{ccc}\hfill -{\delta }_1-\eta \hfill & \hfill \frac{-\gamma \lambda }{{\delta }_1}\hfill & \hfill -\alpha \hfill \\ \hfill 0\hfill & \hfill \frac{\gamma \lambda }{{\delta }_1}-({\delta }_2+\mu +d)-\eta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mu \hfill & \hfill -(\alpha +{\delta }_3)-\eta \hfill \end{array}\right|=0\text{.}$$

Thus the eigen values are given by

$$\begin{array}{cc}\hfill {\eta }_1=& -{\delta }_1\hfill \\ \hfill {\eta }_2=& \frac{\gamma \lambda }{{\delta }_1}-({\delta }_2+\mu +d)\hfill \\ \hfill {\eta }_3=& -(\alpha +{\delta }_3)\text{.}\hfill \end{array}$$

Further ${\eta }_2$ can be written as

$$\begin{array}{cc}\hfill {\eta }_2=& \frac{\gamma \lambda -{\delta }_1({\delta }_2+\mu +d)}{{\delta }_1}\hfill \\ \hfill {\eta }_2=& \frac{\gamma \lambda }{{\delta }_1}\left[1-\frac{1}{{\mathcal{R}}_0}\right]\text{.}\hfill \end{array}$$

Since we see that ${\eta }_2<0$, if ${\mathcal{R}}_0<1$. Hence the needful results recived.  □

4. Theoretical analysis

Here in this part of our work, fixed point theory is used to establish sufficient conditions for existence and puniness of solution to the considered model. Therefore, the existence of approximate solution and its uniqueness are investigated by using Schauder and Banach fixed point results [59], [60]. We write our proposed model (2) with $0<\beta \le 1$ as

$$\left\{\begin{array}{c}{}_0{}^{C}D_t^{\beta }[\mathcal{S}(t)]={\mathrm{f}}_1(t,\mathcal{S},\mathcal{I},\mathcal{R},\mathcal{D}),\hfill \\ {}_0{}^{C}D_t^{\beta }[\mathcal{I}(t)]={\mathrm{f}}_2(t,\mathcal{S},\mathcal{I},\mathcal{R},\mathcal{D}),\hfill \\ {}_0{}^{C}D_t^{\beta }[\mathcal{R}(t)]={\mathrm{f}}_3(t,\mathcal{S},\mathcal{I},\mathcal{R},\mathcal{D}),\hfill \\ {}_0{}^{C}D_t^{\beta }[\mathcal{D}(t)]={\mathrm{f}}_4(t,\mathcal{S},\mathcal{I},\mathcal{R},\mathcal{D}),\hfill \\ \mathcal{S}(0)={\mathcal{S}}_0,\mathcal{I}(0)={\mathcal{I}}_0,\hfill \\ \mathcal{R}(0)={\mathcal{I}}_0,\mathcal{D}(0)={\mathcal{I}}_0\text{.}\hfill \end{array}\right)$$ (11)

Thank to Lemma 2.3, the system (11) is equivalent to the given system of nonlinear integral equations as

$$\left\{\begin{array}{c}\mathcal{S}(t)={\mathcal{S}}_0+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}{\mathrm{f}}_1(\rho ,\mathcal{S}(\rho ),\mathcal{I}(\rho ),\mathcal{R}(\rho ),\mathcal{D}(\rho ))d\rho ,\hfill \\ \mathcal{I}(t)={\mathcal{I}}_0+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}{\mathrm{f}}_2(\rho ,\mathcal{S}(\rho ),\mathcal{I}(\rho ),\mathcal{R}(\rho ),\mathcal{D}(\rho ))d\rho ,\hfill \\ \mathcal{R}(t)={\mathcal{R}}_0+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}{\mathrm{f}}_3(\rho ,\mathcal{S}(\rho ),\mathcal{I}(\rho ),\mathcal{R}(\rho ),\mathcal{D}(\rho ))d\rho ,\hfill \\ \mathcal{D}(t)={\mathcal{D}}_0+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}{\mathrm{f}}_1(\rho ,\mathcal{S}(\rho ),\mathcal{I}(\rho ),\mathcal{R}(\rho ),\mathcal{D}(\rho ))d\rho \text{.}\hfill \end{array}\right)$$ (12)

Also $t\in [0,T]$ with $T<\infty$, then ${\mathcal{E}}_1=C([0,T]\times R^4+,R_{+})$ is the Banach space. Further $\mathcal{H}={\mathcal{E}}_1\times {\mathcal{E}}_2\times {\mathcal{E}}_3\times {\mathcal{E}}_4\times$ is also complete norm space endowed with norm

$$\Vert \mathbb{U}\Vert ={\displaystyle \underset{t\in [0,T]}{\sup }}|\mathbb{U}(t)|={\displaystyle \underset{t\in [0,T]}{\sup }}[|\mathcal{S}(t)|+|\mathcal{I}(t)|+|\mathcal{R}(t)|+|\mathcal{D}(t)|]\text{.}$$

Writing (12) as

$$\mathbb{U}(t)={\mathbb{U}}_0(t)+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}\mathcal{G}(\rho ,\mathbb{U}(\rho ))d\rho ,$$ (13)

where

$$\mathbb{U}(t)=\left\{\begin{array}{c}\mathcal{S}(t)\hfill \\ \mathcal{I}(t)\hfill \\ \mathcal{R}(t)\hfill \\ \mathcal{D}(t),\hfill \end{array}\right){\mathbb{U}}_0(t)=\left\{\begin{array}{c}{\mathcal{S}}_0(t)\hfill \\ {\mathcal{I}}_0(t)\hfill \\ {\mathcal{R}}_0(t)\hfill \\ {\mathcal{D}}_0(t),\hfill \end{array}\right)\mathcal{G}(t,\mathbb{U}(t))=\left\{\begin{array}{c}{\mathrm{f}}_1(t,\mathcal{S}(t),\mathcal{I}(t),\mathcal{R}(t),\mathcal{D}(t))\hfill \\ {\mathrm{f}}_1(t,\mathcal{S}(t),\mathcal{I}(t),\mathcal{R}(t),\mathcal{D}(t))\hfill \\ {\mathrm{f}}_1(t,\mathcal{S}(t),\mathcal{I}(t),\mathcal{R}(t),\mathcal{D}(t))\hfill \\ {\mathrm{f}}_2(t,\mathcal{S}(t),\mathcal{I}(t),\mathcal{R}(t),\mathcal{D}(t))\text{.}\hfill \end{array}\right)$$ (14)

For establishing theoretical results, we need the given hypothesis to be hold:

• $(E1)$
  There exists constant $L_{\mathcal{G}}>0$, for each $\mathbb{U}(t),\overline{\mathbb{U}(t)}\in R\times R$, such that

  $$|\mathcal{G}(t,\mathbb{U}(t))-\mathcal{G}(t,\overline{\mathbb{U}(t)})|⩽L_{\mathcal{G}}|\mathbb{U}(t)-\overline{\mathbb{U}(t)}|,$$
• $(E2)$
  There exist constants $C_{\mathcal{G}}>0$ and $M_{\mathcal{G}}>0$, with

  $$|\mathcal{G}(t,\mathbb{U}(t))|⩽C_{\mathcal{G}}|\mathbb{U}|+M_{\mathcal{G}}\text{.}$$

Theorem 4.1

Under the hypothesis$(E_2)$and continuity of$\mathcal{G}$, at least one solution will be exists corresponding to the model(2).

Proof

Thank to Schauder fixed point theorem, considering a closed set $\mathcal{A}\subset \mathcal{H}$ with

$$\mathcal{B}=\{\mathbb{U}\in \mathcal{H}:\Vert \mathbb{U}\Vert ⩽\mathbf{r},\mathbf{r}>0\}\text{.}$$

If $\mathbf{B}:\mathcal{A}\to \mathcal{A}$ be the operator, then inview of (13), one has

$$\mathbf{B}(\mathbb{U}(t))={\mathbb{U}}_0(t)+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}\mathcal{G}(\rho ,\mathbb{U}(\rho ))d\rho ,$$ (15)

For any $\mathbb{U}\in \mathcal{A}$, one has

$$\begin{array}{cc}\hfill |\mathbf{B}(\mathbb{U})(t)|⩽& |{\mathbb{U}}_0|+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}|\mathcal{G}(\rho ,\mathbb{U}(\rho ))|d\rho ,\hfill \\ \hfill ⩽& |{\mathbb{U}}_0|+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}[C_{\mathcal{G}}|\mathbb{U}|+M_{\mathcal{G}}]d\rho ,\hfill \\ \hfill ⩽& |{\mathbb{U}}_0|+\frac{T^{\rho }}{\mathrm{\Gamma }(\beta +1)}[C_{\mathcal{G}}\mathbf{r}+M_{\mathcal{G}}],\hfill \end{array}$$

which implies that

$$\Vert \mathbf{B}(\mathbb{U})\Vert ⩽\mathbf{r}\text{.}$$ (16)

Thus $\mathbb{U}\in \mathcal{A}$ which yields $\mathbf{B}(\mathcal{A})\subset \mathcal{A}$ and hence $\mathbf{B}$ is bounded. Let $t_1<t_2\in [0,T]$, taking

$$\begin{array}{cc}\hfill |\mathbf{B}(\mathbb{U})(t_2)-\mathbf{B}(\mathbb{U})(t_1)|=& \left|\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^{t_2}{(t_2-\rho )}^{\beta -1}\mathcal{G}(\rho ,\mathbb{U}(\rho ))d\rho -\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^{t_1}{(t_1-\rho )}^{\beta -1}\mathcal{G}(\rho ,\mathbb{U}(\rho ))d\rho \right|\hfill \\ \hfill ⩽& \frac{1}{\mathrm{\Gamma }(\beta )}\left[{\int }_0^{t_1}[{(t_1-\rho )}^{\beta -1}-{(t_2-\rho )}^{\beta -1}]|\mathcal{G}(\rho ,\mathbb{U}(\rho ))|d\rho \right)\hfill \\ \hfill +& \left({\int }_{t_1}^{t_2}{(t_2-\rho )}^{\beta -1}|\mathcal{G}(\rho ,\mathbb{U}(\rho ))|d\rho \right]\hfill \\ \hfill ⩽& \frac{(C_{\mathcal{G}}\mathbf{r}+M_{\mathcal{G}})}{\mathrm{\Gamma }(\beta +1)}[t_2^{\beta }-t_1^{\beta }+2{(t_2-t_1)}^{\beta }]\to 0\mathrm{as}{t}_1\to t_2\text{.}\hfill \end{array}$$ (17)

Therefore

$$\Vert \mathbf{B}(\mathbb{U})(t_2)-\mathbf{B}(\mathbb{U})(t_1)\Vert \to 0,\mathrm{as}{t}_1\to t_2\text{.}$$

Hence $\mathbf{B}$ is equi- continuous operator. Hence model (2) has at least one solution.

For existence of unique solution., we have the following result.

Theorem 4.2

Under the hypothesis$(E1)$and if$\frac{T^{\beta }}{\mathrm{\Gamma }(\beta +1)}{L}_{\mathcal{G}}<1$, then the model(2)has a unique solution.

Proof

Let $\mathbf{B}:\mathcal{H}\to \mathcal{H}$ be the operator and taking $\mathbb{U},\overline{\mathbb{U}}\in \mathcal{H}$, then one has

$$\begin{array}{cc}\hfill \Vert \mathbf{B}(\mathbb{U})-\mathbf{B}(\overline{\mathbb{U}})\Vert =& {\displaystyle \underset{t\in [0,T]}{\sup }}\left|\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}\mathcal{G}(\rho ,\mathbb{U}(\rho ))d\rho \right)\hfill \\ \hfill -& \left(\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}\mathcal{G}(\rho ,\overline{\mathbb{U}}(\rho ))d\rho \right|,\hfill \\ \hfill ⩽& \frac{T^{\beta }}{\mathrm{\Gamma }(\beta +1)}{L}_{\mathcal{G}}\Vert \mathbb{U}-\overline{\mathbb{U}}\Vert \text{.}\hfill \end{array}$$ (18)

(18) yields

$$\Vert \mathbf{B}(\mathbb{U})-\mathbf{B}(\overline{\mathbb{U}})\Vert ⩽\frac{T^{\beta }}{\mathrm{\Gamma }(\beta +1)}{L}_{\mathcal{G}}\Vert \mathbb{U}-\overline{\mathbb{U}}\Vert \text{.}$$ (19)

Hence $\mathbf{B}$ is a contraction mapping, so by the use of Banach theorem, the considered system has a unique solution. □

Let $\mathrm{f}\in C[0,T]$ with $\mathrm{f}(0)=0$ independent of $\mathbb{U}$ as.

• •$|\mathrm{f}(t)|⩽\varrho ,\mathrm{for}\varrho >0$;
• •${}_0{}^{C}D_t^{\beta }\mathbb{U}(t)=\mathcal{G}(t,\mathbb{U}(t))+\mathrm{f}(t)$.

Lemma 4.3

The solution of nonlinear FDE

$$\begin{array}{cc}\hfill {}_0{}^{C}D_t^{\beta }\mathbb{U}(t)=& \mathcal{G}(t,\mathbb{U}(t))+\mathrm{f}(t),\hfill \\ \hfill \mathbb{U}(0)=& {\mathbb{U}}_0,\hfill \end{array}$$ (20)

satisfies the given relation

$$\begin{array}{cc}\hfill \left|\mathbb{U}(t)\right)-& \left(\left({\mathbb{U}}_0(t)+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}\mathcal{G}(\rho ,\mathbb{U}(\rho ))d\rho \right)\right|⩽\frac{T^{\beta }}{\mathrm{\Gamma }(\beta +1)}\varrho \hfill \\ \hfill =& {\mathrm{\Omega }}_{T,\beta }\varrho \text{.}\hfill \end{array}$$ (21)

Proof

The proof is similar done in [44], [48], [50].  □

Theorem 4.4

Due to hypothesis$(E2)$andLemma 4.3, the solution of model(2)is H-U stable if$\mathrm{\Delta }=\frac{T^{\beta }}{\mathrm{\Gamma }(\beta +1)}{L}_{\mathcal{G}}<1$.

Proof

If $\overline{\mathbb{U}}\in \mathcal{E}$ is the unique solution of (13), then for any other solution $\mathbb{U}\in \mathcal{E}$, we have

$$\begin{array}{cc}\hfill |\mathbb{U}(t)-\overline{\mathbb{U}}(t)|=& \left|\mathbb{U}(t)-\left({\mathbb{U}}_0(t)+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}\mathcal{G}(\rho ,\overline{\mathbb{U}}(\rho ))d\rho \right)\right|,\hfill \\ \hfill ⩽& \left|\mathbb{U}(t)-\left({\mathbb{U}}_0(t)+\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}\mathcal{G}(\rho ,\overline{\mathbb{U}}(\rho ))d\rho \right)\right|\hfill \\ \hfill +& \left|\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}\mathcal{G}(\rho ,\overline{\mathbb{U}}(\rho ))d\rho -\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t{(t-\rho )}^{\beta -1}\mathcal{G}(\rho ,\overline{\mathbb{U}}(\rho ))d\rho \right|,\hfill \\ \hfill ⩽& {\mathrm{\Omega }}_{T,\rho }\varrho +\mathrm{\Delta }\Vert \mathbb{U}-\overline{\mathbb{U}}\Vert \text{.}\hfill \end{array}$$ (22)

From (22), we have

$$\Vert \mathbb{U}-\overline{\mathbb{U}}\Vert ⩽\frac{{\mathrm{\Omega }}_{T,\rho }}{1-\mathrm{\Delta }}\varrho \text{.}$$ (23)

Thus (23) yields that solution of (2) is H-U stable.  □

5. Numerical solution

This part is devoted to establish two numerical schemes. We apply these schemes to our model one by one and compared the results.

5.1. Numerical simulation by MEM

For the model (2), the numerical approximations are performed in this section by using MEM. Let $[0,T]$ be the set of points, on which we must have to evaluate the series solution of the model (2). Upon further subdivision of $[0,T]$ into m sub-intervals $[t_p,t_{p+1}]$ of equal difference $j=\frac{p}{m}$ between consequent points using $t_P=\mathit{pj}$ with $p=0,1,\cdots ,m$, then obviously

$$\begin{array}{c}\hfill \mathcal{S}(t),\mathcal{I}(t),\mathcal{R}(t),\mathcal{D}(t),{}_0{}^{C}D_t^{\beta }\mathcal{S}(t),{}_0{}^{C}D_t^{\beta }[\mathcal{I}(t)],{}_0{}^{C}D_t^{\beta }[\mathcal{R}(t)],\\ \hfill {}_0{}^{C}D_t^{\beta }[\mathcal{D}(t)],{}_0{}^{C}D_t^{2\beta }[\mathcal{S}(t)],{}_0{}^{C}D_t^{2\beta }[\mathcal{I}(t)],{}_0{}^{C}D_t^{2\beta }[\mathcal{R}(t)],{}_0{}^{C}D_t^{2\beta }[\mathcal{D}(t)]\end{array}$$

are continuous on $[0,T]$. Applying the modified Euler’s or Taylor’s method about $t=t_0$ to the considered model expressed in (??) and for each value of t taking $a\in (0,T)$, the expressions for $t_1$ is given as

$$\begin{array}{cc}\hfill \mathcal{S}(t_1)=& \mathcal{S}(t_0)+{\mathrm{f}}_1(t_0,\mathcal{S}(t_0),\mathcal{I}(t_0),\mathcal{R}(t_0),\mathcal{D}(t_0))\frac{t^{\beta }}{\mathrm{\Gamma }(\beta +1)}+{}_0{}^{C}D_t^{2\beta }\mathcal{S}(t){|}_{t=a}\frac{t^{2\beta }}{\mathrm{\Gamma }(2\beta +1)}\cdots ,\hfill \\ \hfill \mathcal{I}(t_1)=& \mathcal{I}(t_0)+{\mathrm{f}}_2(t_0,\mathcal{S}(t_0),\mathcal{I}(t_0),\mathcal{R}(t_0),\mathcal{D}(t_0))\frac{t^{\beta }}{\mathrm{\Gamma }(\beta +1)}+{}_0{}^{C}D_t^{2\beta }\mathcal{I}(t){|}_{t=a}\frac{t^{2\beta }}{\mathrm{\Gamma }(2\beta +1)}\cdots ,\hfill \\ \hfill \mathcal{R}(t_1)=& \mathcal{R}(t_0)+{\mathrm{f}}_1(t_0,\mathcal{S}(t_0),\mathcal{I}(t_0),\mathcal{R}(t_0),\mathcal{D}(t_0))\frac{t^{\beta }}{\mathrm{\Gamma }(\beta +1)}+{}_0{}^{C}D_t^{2\beta }\mathcal{R}(t){|}_{t=a}\frac{t^{2\beta }}{\mathrm{\Gamma }(2\beta +1)}\cdots ,\hfill \\ \hfill \mathcal{D}(t_1)=& \mathcal{D}(t_0)+{\mathrm{f}}_1(t_0,\mathcal{S}(t_0),\mathcal{I}(t_0),\mathcal{R}(t_0),\mathcal{D}(t_0))\frac{t^{\beta }}{\mathrm{\Gamma }(\beta +1)}+{}_0{}^{C}D_t^{2\beta }\mathcal{D}(t){|}_{t=a}\frac{t^{2\beta }}{\mathrm{\Gamma }(2\beta +1)}\cdots ,\text{.}\hfill \end{array}$$ (24)

By taking the difference between consecutive points as j very very small, such that we may ignore terms containing higher-order derivatives to get

$$\begin{array}{cc}\hfill \mathcal{S}(t_1)=& \mathcal{S}(t_0)+{\mathrm{f}}_1(t_0,\mathcal{S}(t_0),\mathcal{I}(t_0),\mathcal{R}(t_0),\mathcal{D}(t_0))\frac{t^{\beta }}{\mathrm{\Gamma }(\beta +1)},\hfill \\ \hfill \mathcal{I}(t_1)=& \mathcal{I}(t_0)+{\mathrm{f}}_1(t_0,\mathcal{S}(t_0),\mathcal{I}(t_0),\mathcal{R}(t_0),\mathcal{D}(t_0))\frac{t^{\beta }}{\mathrm{\Gamma }(\beta +1)},\hfill \\ \hfill \mathcal{R}(t_1)=& \mathcal{R}(t_0)+{\mathrm{f}}_1(t_0,\mathcal{S}(t_0),\mathcal{I}(t_0),\mathcal{R}(t_0),\mathcal{D}(t_0))\frac{t^{\beta }}{\mathrm{\Gamma }(\beta +1)},\hfill \\ \hfill \mathcal{D}(t_1)=& \mathcal{D}(t_0)+{\mathrm{f}}_2(t_0,\mathcal{S}(t_0),\mathcal{I}(t_0)\mathcal{R}(t_0),\mathcal{D}(t_0))\frac{t^{\beta }}{\mathrm{\Gamma }(\beta +1)}\text{.}\hfill \end{array}$$ (25)

On repeating the procedure, we get sequence of point to approximate $(\mathcal{S}(t),\mathcal{I}(t),(\mathcal{R}(t),(\mathcal{D}(t))$ is formed. A generalized formula in this regard at $t_{p+1}=t_p+h$ is given by

$$\begin{array}{c}\hfill \mathcal{S}(t_{r+1})=\mathcal{S}(t_r)+{\mathrm{f}}_1(t_r,\mathcal{S}(t_r),\mathcal{I}(t_r),\mathcal{R}(t_r),\mathcal{D}(t_r))\frac{h^{\beta }}{\mathrm{\Gamma }(\beta +1)},\\ \hfill \mathcal{I}(t_{r+1})=\mathcal{I}(t_r)+{\mathrm{f}}_1(t_r,\mathcal{S}(t_r),\mathcal{I}(t_r),\mathcal{R}(t_r),\mathcal{D}(t_r))\frac{h^{\beta }}{\mathrm{\Gamma }(\beta +1)},\\ \hfill \mathcal{R}(t_{r+1})=\mathcal{R}(t_r)+{\mathrm{f}}_1(t_r,\mathcal{S}(t_r),\mathcal{I}(t_r),\mathcal{R}(t_r),\mathcal{D}(t_r))\frac{h^{\beta }}{\mathrm{\Gamma }(\beta +1)},\\ \hfill \mathcal{D}(t_{r+1})=\mathcal{D}(t_r)+{\mathrm{f}}_2(t_r,\mathcal{S}(t_r),\mathcal{I}(t_r),\mathcal{R}(t_r),\mathcal{D}(t_r))\frac{h^{\beta }}{\mathrm{\Gamma }(\beta +1)},\end{array}$$ (26)

where $r=0,1,2,\cdots ,n-1$.

5.2. Numerical results and Discussion

Here we take real data of Pakistan as the total pollution of the country $N=220$ millions [61] and the other values are given in Table 1 : Here we now present numerical interpretation of the proposed model (2) using MEM in Fig. 2, Fig. 3, Fig. 4, Fig. 5 . From Fig. 2, we see that susceptibility is decreasing at various rate due to different fractional orders. As fractional order tends to integer order the solution converges to the integer order solution. In same line the infected class is increasing in initial 100 days with various rate of transmission due to various fractional order then it turns to decrease with same behaviors as in Fig. 3. Consequently the increase in death class cause increase in recovered class. The recovery class dynamics raises at various fractional order as shown in Fig. 4. In Fig. 5, the dynamics of death class is also increasing until become stable due to faster infection rate.

Table 1.

Interpretation and approximate values of parameters involve in the model (2).

Compartment/Parameters	Description of parameter	Approximate value
${\mathcal{S}}_0$	Initial density of susceptible class	$217.388901$ millions
${\mathcal{I}}_0$	Initial density of infected class	$1.29$ million
${\mathcal{R}}_0$	Initial density of recovered class	$1.256337$ millions
${\mathcal{D}}_0$	Initial density of death class	$0.028921$ million
$\lambda$	Recruitment rate	$0.00009$ assumed
$\alpha$	transmission rate from infection	$0.009978$
$\mu$	Recovery rate	$0.0025$ assumed
d	Death rate due to infection	$0.019$
$\gamma$	rate of infection	$0.0028$ assumed
${\delta }_1$	Natural death rate of Susceptible class	$0.0009$ assumed
${\delta }_2$	Natural death rate of infected class	$0.00056$ assumed
${\delta }_3$	Natural death rate of recovered class	$0.000013$ assumed

Fig. 2.

Transmission dynamics of susceptible class at various fractional order of the proposed model (2).

Fig. 3.

Transmission dynamics of infected class at various fractional order of the proposed model (2).

Fig. 4.

Transmission dynamics of recovered class at various fractional order of the proposed model (2).

Fig. 5.

Transmission dynamics of death class at various fractional order of the proposed model (2).

5.3. Numerical interpretation of the Model (2) by NSFD scheme

Using NSFD scheme under the concept of fractional order derivative, we approximate the proposed model (2). We use Grünwald-Letnikov approximation for Caputo derivative. About some detail for this scheme, we refer [54], [55], [56], [57], [58]. Since for nonlinear systems the investigation of exact solution in most cases is impossible, so we focuss on best approximation of the solution. In this regards, some NSFD schemes have been introduced. The said scheme has the ability to avoid the full implicit scheme and preserve positivity, monotonicity and convergency which make this method popular. For detail applications of the method see [62].

To construct the scheme, consider Grünwald-Letnikov approximation for the fractional order derivative given by

$${}_0{}^{C}D_t^{\beta }[V(t)]={\displaystyle \underset{p\to 0}{\lim }}{p}^{-\beta }{\displaystyle \sum _{i=0}^{\aleph }}{(-1)}^q\left(\begin{array}{c}\hfill \omega \hfill \\ \hfill q\hfill \end{array}\right)V(t-\mathit{qp}),$$ (27)

with $t=\aleph h$, such that h is the step size. Consider the following FDE as

$$\left\{\begin{array}{c}{}_0{}^{C}D_t^{\beta }[V(t)]=\mathcal{F}(t,V(t)),t\in [0,T],T<\infty ,\hfill \\ V(t_0)=V_0\text{.}\hfill \end{array}\right)$$ (28)

Using (27), from (12) one has

$${\displaystyle \sum _{q=0}^{\aleph }}{\mathbf{K}}^{\beta }V(t_{n-i})=\mathcal{F}(t,V(t)),n=1,2,3,\cdots ,$$ (29)

where $t_n=\mathit{nh}$ and ${\mathbf{K}}_i^{\beta }$ are the Grünwald-Letnikov coefficients calculated as

$${\mathbf{K}}_i^{\beta }=\left(1-\frac{1+\beta }{i}\right){\mathbf{K}}_{i-1}^{\beta },i=1,2,3,\cdots$$ (30)

and

$${\mathbf{K}}_0^{\beta }=h^{-\beta }\text{.}$$ (31)

Based on the above definition, our considered model (2) can be discritized as

$$\left\{\begin{array}{c}\mathcal{S}(t_{n+1})=\frac{1}{{\mathbf{K}}_0^{\beta }}\left[-{\displaystyle \sum _{j=1}^{n+1}}{\mathbf{K}}_j^{\beta }\mathcal{S}(t_{n+1-j})+\lambda -\gamma \mathcal{S}(t_n)\mathcal{I}(t_n)-{\delta }_{1}S(t_n)+\alpha \mathcal{R}(t_n)\right]\hfill \\ \mathcal{I}(t_{n+1})=\frac{1}{{\mathbf{K}}_0^{\beta }}\left[-{\displaystyle \sum _{j=1}^{n+1}}{\mathbf{K}}_j^{\beta }\mathcal{I}(t_{n+1-j})+\gamma \mathcal{S}(t_{n+1})\mathcal{I}-({\delta }_2+\mu +d)\mathcal{I}(t_{n+1})\right]\hfill \\ \mathcal{R}(t_{n+1})=\frac{1}{{\mathbf{K}}_0^{\beta }}\left[-{\displaystyle \sum _{j=1}^{n+1}}{\mathbf{K}}_j^{\beta }\mathcal{I}(t_{n+1-j})+\mu \mathcal{I}(t_{n+1})-(\alpha +{\delta }_3)\mathcal{R}(t_{n+1})\right]\hfill \\ \mathcal{D}(t_{n+1})=\frac{1}{{\mathbf{K}}_0^{\beta }}\left[-d{\displaystyle \sum _{j=1}^{n+1}}{\mathbf{K}}_j^{\beta }\mathcal{I}(t_{n+1-j})\right]\text{.}\hfill \end{array}\right)$$ (32)

5.4. Numerical interpretation and explanation

Here, we now present the transmission dynamics of the proposed model (2) using NSFD scheme in Fig. 6, Fig. 7, Fig. 8, Fig. 9 . The dynamical behaviors presented in Fig. 6, Fig. 7, Fig. 8, Fig. 9 is also same as given in Fig. 2, Fig. 5 respectively.

Fig. 6.

Transmission dynamics of susceptible class at various fractional order of the proposed model (2).

Fig. 7.

Transmission dynamics of infected class at various fractional order of the proposed model (2).

Fig. 8.

Transmission dynamics of recovered class at various fractional order of the proposed model (2).

Fig. 9.

Transmission dynamics of death class at various fractional order of the proposed model (2).

6. Comparison of both methods

Here we compare graphs of both methods for fixed time $t=300$ in Fig. 10, Fig. 11, Fig. 12, Fig. 13 as Here in Fig. 10, Fig. 11, Fig. 12, Fig. 13, we compare the numerical solutions of various compartments of the proposed model by using MEM and NSFD schemes respectively. Both have similar results for same range of time. As MEM slightly simpler than NSFD scheme. Here we compare the numerical simulations of both methods with some real data of Pakistan for 200 days [63] in Fig. 14 . We see that our simulated results have close agreement with real data in both cases for the given fractional orders. This shows that both schemes can be use as a powerful tool to investigate fractional order dynamics. Further we compare CPU time of both method by using Matlab 13 and Machine Cori-7 of HP with $8^{\mathit{th}}$ generation in Table 2 .

Fig. 10.

Transmission dynamics of susceptible class at various fractional order using: (a). Modified Euler method. (b). Nonstandard Finite Difference Method.

Fig. 11.

Transmission dynamics of infected class at various fractional order using: (a). Modified Euler method. (b). Nonstandard Finite Difference Method..

Fig. 12.

Transmission dynamics of recovered class at various fractional order using: (a). Modified Euler method. (b). Nonstandard Finite Difference Method.

Fig. 13.

Transmission dynamics of death class at various fractional order using: (a). Modified Euler method. (b). Nonstandard Finite Difference Method.

Fig. 14.

Comparison between real and simulated results by using NSFD schemes and MEM.

Table 2.

Comparison of both methods.

Range of t	CPU time of MEM in seconds	CPU time of NSFD method in seconds
50	4.99	4.66
100	4.75	4.71
150	4.83	4.73
200	4.85	4.76
250	4.90	4.80
300	4.95	4.83

7. Conclusion

In this research work, we have established a modified type COVID-19 model by incorporating natural death rates of susceptible, infected and recovered classes respectively. The model under investigation has been studied under fractional order derivative of Caputo type. Further by using fixed point theory, we have established existence theory for numerical solutions. Also we have developed various results for global and local stability by using the next generation matrix method. The basic reproduction number has been computed. Moreover, the feasible region has also established for the proposed model along with its boundedness. The numerical interpretations have been performed by using two different numerical approaches based on MEM and NSFD methods. Both methods provide nearly same results for our model. Therefore we have compared both procedures in CPU time to see which one is most expensive with respect to time. In this regards NSFD method which is slightly general than MEM. Further MEM is slightly expensive in time than MEM for same number of iteration corresponding to different range of time. Graphical presentations have been provided for taking some real data about COVID −19 transmission in Pakistan. Both scheme have been compared with real data. The concerned graphs have been plotted against different fractional order. Hence we concluded that fractional calculus provides more best explanations to real world problems for understanding their dynamics. In future, we will investigate some mathematical models under fractional order stochastic differential equations. Also piecewise concept will be applied for investigating epidemiological models.

Funding

There does not exist any funding source.

Availability of data

All data used in this paper is included within the article.

Authors contributions

All authors played their role equally. First and second authors designed the models and did theoretical analysis. Third author did numerical. Last two authors edited and drafted the article.

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.