A fractional‐order mathematical model for analyzing the pandemic trend of COVID‐19

Abstract

Many countries worldwide have been affected by the outbreak of the novel coronavirus (COVID‐19) that was first reported in China. To understand and forecast the transmission dynamics of this disease, fractional‐order derivative‐based modeling can be beneficial. We propose in this paper a fractional‐order mathematical model to examine the COVID‐19 disease outbreak. This model outlines the multiple mechanisms of transmission within the dynamics of infection. The basic reproduction number and the equilibrium points are calculated from the model to assess the transmissibility of the COVID‐19. Sensitivity analysis is discussed to explain the significance of the epidemic parameters. The existence and uniqueness of the solution to the proposed model have been proven using the fixed‐point theorem and by helping the Arzela–Ascoli theorem. Using the predictor–corrector algorithm, we approximated the solution of the proposed model. The results obtained are represented by using figures that illustrate the behavior of the predicted model classes. Finally, the study of the stability of the numerical method is carried out using some results and primary lemmas.

1. INTRODUCTION

The WHO announced COVID‐19 to be a pandemic epidemic on January 22, 2020. Nowadays, the pandemic is spreading across the world and affects almost every aspect of life. In addition to health problems, it undermines the global economic system and restricting people's contact. Researchers from diverse scientific fields have therefore committed themselves to the study of COVID‐19. The objectives were to contribute to the enhancement of the comprehension, forecast, and interpretation from different points of view for this disease. Any new suggestions are a step forward in solving this health crisis. Mathematical models are a significant and efficient way to comprehend epidemic transmission dynamics. ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ , ⁹ These models can be helping predict disease transmission and thus help decision makers in planning and make necessary decisions.

The fractional‐order models provide a more accurate fit than the integer‐order models for the actual data for various diseases and other experiential studies around modeling and simulation. New mathematical models of fractional order are developed that can be used for simulation to forecast the disease outbreaks and flatten the infection and deaths curve. ¹⁰ , ¹¹ , ¹² , ¹³ , ¹⁴ The mathematicians have also put their efforts forward in the analysis of various nonlinear dynamics of problems related to the infection like epidemics. ¹⁵ , ¹⁶ , ¹⁷ , ¹⁸ Khan and Atangana ¹⁹ described the mathematical modeling and dynamics of a novel coronavirus (2019‐nCoV), then formulated a new mathematical model for simulation of the dynamics COVID‐19 with quarantine and isolation. ²⁰ Atangana ²¹ proposed a model for the spread of COVID‐19, with new fractal‐fractional operators and taking into account the potential of transmission of COVID‐19 from dead bodies to humans and the effect of lockdown. Yadav and Verma ²² investigated a fractional model based on Caputo–Fabrizio fractional derivative and developed for simulation the transmission coronavirus (COVID‐19) in Wuhan Chinese.

Baleanu et al ²³ used the homotopy analysis transformation method to solve the COVID‐19 transmission model with Caputo–Fabrizio derivative. Erturk and Kumar ²⁴ used the corrector–predictor algorithm with a new generalized Caputo type derivative to solve the novel coronavirus (COVID‐19) epidemic fractional‐order model. Abdulwasaa et al ²⁵ generalized a mathematical model based on a fractal‐fractional operator to forecast future trends in the behavior of the COVID‐19 pandemics in India for October 2020. Rezapour et al ²⁶ provided the SEIR model of the spreading epidemic of the COVID‐19 globally and Iran for data reported from December 31, 2019, to January 28, 2020. Shaikh et al ²⁷ , ²⁸ analyzed the fractional‐order COVID‐19 model for simulating the potential transmission of epidemic and suggested some possible control strategies. Zhang et al ²⁹ proposed and applied the time‐dependent SEIR model to fit the time series of coronavirus evolution observed recently in China and then predict transmission. Matouk ³⁰ analyzed complex dynamics in a model for COVID‐19 and its fractional‐order counterpart with multidrug resistance. Tang et al ³¹ adopted a deterministic model that was devised based on the clinical improvement to illuminate the dynamics for transmission of the novel coronavirus and determine the effect of the public health measures on the infection.

The paper is arranged as follows: In Section 2, we recalled some basic concepts and results used in the article. In Section 3, we developed the model pandemic trend Covid‐19 with fractional order then calculated the basic reproduction number and equilibrium points along with sensitivity analysis for parameters. Using the fixed‐point theory, the existence and uniqueness analysis of the model was performed in Section 4. In Section 5, we presented a numerical algorithm for solving the model based on the predictor‐corrector method. In Section 6, we discussed the stability of the numerical method. In Section 7, we have displayed the numerical simulation results for the real data. A conclusion completes the paper.

2. BASIC CONCEPT

In this section, we present some of the fundamental definitions and rand results of fractional order derivatives which, can be used throughout this manuscript.

Definition 1

(Samko et al. and Miller and Ross ³² , ³³ ) For an integrable function f(t), the Caputo derivative of fractional order α ∈ (0, 1) is given by

$${}^C{D}_t^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}{\displaystyle \underset{0}{\overset{t}{\int }}}\frac{f^{(n)}(\tau )}{{\left(t-\tau \right)}^{\alpha -n+1}}d\tau ,n=[\alpha ]+1.$$ (1)

The corresponding fractional integer of order α with Re(α) > 0 is given by

$${}^C{I}_t^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{0}{\overset{t}{\int }}}\frac{f(\tau )}{{\left(t-\tau \right)}^{1-\alpha }}d\tau .$$ (2)

Lemma 1

(Li and Zeng ³⁴ ) If 0 <α < 1 and n≥0 is an integer, then there exists the positive constants ${\complement }_{\alpha ,1}$ and ${\complement }_{\alpha ,2}$ only dependent on α, such that

$${(n+1)}^{\alpha }-n^{\alpha }\le {\complement }_{\alpha ,1}{(n+1)}^{\alpha -1},$$ (3)

and

$${(n+2)}^{\alpha +1}-2{(n+1)}^{\alpha +1}+n^{\alpha +1}\le {\complement }_{\alpha ,2}{(n+1)}^{\alpha -1}.$$ (4)

Lemma 2

(Li and Zeng ³⁴ ) Assume that $a_{k,n}={\left(n-k\right)}^{\alpha -1},k=\mathrm{1,2},\dots ,n-1$ and $a_{k,n}=0$ for k≥n,  α, h, L, T > 0,  τh ≤ T where τ is a positive integer. Let ${\sum }_{k=m}^n{a}_{k,n}\left|e_k\right|=0$ for m > n≥1. If

$$\left|e_n\right|\le L{h}^{\alpha }{\displaystyle \sum _{k=1}^{n-1}}{a}_{k,n}\left|e_k\right|+\left|{\zeta }_0\right|,n=\mathrm{1,2},\dots ,\tau ,$$ (5)

then,

$$\left|e_n\right|\le \complement \left|{\zeta }_0\right|,\tau =\mathrm{1,2},\dots ,$$ (6)

where $\complement$ is appositive constant independent of h and τ.

Lemma 3

(Diethelm ³⁵ ) The function y ∈ C[0, T] is a solution of the following fractional differential equation:

$${}^C{D}_t^{\alpha }y(t)=f\left(t,y(t)\right),y(0)=y_0,$$ (7)

if and only if it is a solution of the nonlinear Volterra integral equation of the second kind

$$y(t)=y(0)+\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}f\left(\tau ,y(\tau )\right)d\tau .$$ (8)

3. THEORETICAL APPROACH

In this section, we are considered to examine a classical model developed by Fatima et al ¹ for the pandemic trend of 2019 coronavirus, which was first identified in the Chinese city of Wuhan in December 2019 and then spread quickly across the world. The population is divided into six subcategories: susceptible people denoted by S, exposed people E, infected people I, asymptomatic infectious people A, isolated or hospitalized people H, recovered or dead people R, and the reservoir for COVID‐19 is denoted by W. The pandemic trend model of COVID‐19 is given by the system of ordinary differential equations as follows:

$$\begin{array}{ll}\dot{S}& =\mathrm{{\rm Y}}-{\xi }_{1}S\left(I+\sigma A+\psi H\right)-{\xi }_{2}WS-\delta S,\\ Ė& ={\xi }_1\left(I+\sigma A+\psi H\right)+{\xi }_{2}WS-\left(\nu +\delta \right)E,\\ İ& =\nu \eta E-\left({\beta }_1+{\beta }_2+\delta \right)I,\\ \dot{A}& =\nu \left(1-\eta \right)E-\left(\rho +\delta \right)A,\\ \dot{H}& ={\beta }_{1}I+\rho A-\left({\beta }_3+\delta \right)H,\\ \dot{R}& ={\beta }_{2}I+{\beta }_{3}H-\delta R,\\ \dot{W}& ={\theta }_{1}I+{\theta }_{2}A-\lambda W.\end{array}$$ (9)

Subject to non‐negative initial conditions.

$$S(0)=S_0,E(0)=E_0,I(0)=I_0,A(0)=A_0,H(0)=H_0,R(0)=R_0,W(0)=W_0,$$ (10)

where $\mathrm{{\rm Y}}=b\times N,b$ is the birth rate, and N is the total number of people; δ the death rate of the population; ξ ₁ and ξ ₂ denote the rate of transmission infected of the susceptible people through sufficient contact with I and W, respectively. The parameters σ and ψ denote the approximate transmissibility from A and H to I, respectively. (ν)⁻¹ represented the transmission rate after the quarantine period and joined the class I and A. The parameters η and (1 − η) are the moving rate from E to I and A, respectively. The rate of infected people who are hospitalized is β ₁, and β ₂ is the recovery rate. β ₃ denotes the recovery rate of hospitalized patient. θ ₁ and θ ₂ are the infected and asymptomatically infected populations contributing to the reservoirs coronavirus, respectively. λ is the lifetime of the virus.

Now, we modify Equations (9) and (10) by putting the Caputo fractional derivatives instead of the time derivatives; with this move, the dimensions on both sides can differ. To avoid this trouble, we use an auxiliary parameter κ, which has a sec. dimension to modify the fractional operator to have the same dimension for the sides. Equations (9) and (10) become

$$\begin{array}{ll}{\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }S(t)& =\mathrm{{\rm Y}}-{\xi }_{1}S(t)\left(I(t)+\sigma A(t)+\psi H(t)\right)-{\xi }_{2}W(t)S(t)-\delta S(t),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }E(t)& ={\xi }_1\left(I(t)+\sigma A(t)+\psi H(t)\right)+{\xi }_{2}W(t)S(t)-\left(\nu +\delta \right)E(t),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }I(t)& =\nu \eta E(t)-\left({\beta }_1+{\beta }_2+\delta \right)I(t),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }A(t)& =\nu \left(1-\eta \right)E(t)-\left(\rho +\delta \right)A(t),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }H(t)& ={\beta }_{1}I(t)+\rho A(t)-\left({\beta }_3+\delta \right)H(t),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }R(t)& ={\beta }_{2}I(t)+{\beta }_{3}H(t)-\delta R(t),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }W(t)& ={\theta }_{1}I(t)+{\theta }_{2}A(t)-\lambda W(t).\end{array}$$ (11)

Subject to non‐negative initial conditions

$$S(0)=S_0,E(0)=E_0,I(0)=I_0,A(0)=A_0,H(0)=H_0,R(0)=R_0,W(0)=W_0.$$ (12)

3.1. Equilibrium points and basic reproduction number

To find the disease‐free equilibrium of the suggested fractional‐order model (11), we solve the following system.

$$\begin{array}{ll}0& =\mathrm{{\rm Y}}-{\xi }_{1}S(t)\left(I(t)+\sigma A(t)+\psi H(t)\right)-{\xi }_{2}W(t)S(t)-\delta S(t),\\ 0& ={\xi }_1\left(I(t)+\sigma A(t)+\psi H(t)\right)+{\xi }_{2}W(t)S(t)-\left(\nu +\delta \right)E(t),\\ 0& =\nu \eta E(t)-\left({\beta }_1+{\beta }_2+\delta \right)I(t),\\ 0& =\nu \left(1-\eta \right)E(t)-\left(\rho +\delta \right)A(t),\\ 0& ={\beta }_{1}I(t)+\rho A(t)-\left({\beta }_3+\delta \right)H(t),\\ 0& ={\beta }_{2}I(t)+{\beta }_{3}H(t)-\delta R(t),\\ 0& ={\theta }_{1}I(t)+{\theta }_{2}A(t)-\lambda W(t).\end{array}$$ (13)

We get $P_0=\left(\mathrm{{\rm Y}}/\delta ,\mathrm{0,0},\mathrm{0,0},\mathrm{0,0}\right)$, which is the point where there is no disease. If the basic reproduction number ${\mathcal{R}}_0>0$, then the system (13) has a positive endemic equilibrium point, which is represented by

$$\begin{array}{cc}\hfill S^{\ast }=& -\frac{uxyz\lambda }{\nu (\lambda (a\delta +bc{\beta }_2+(b\delta \sigma -z\eta ){\beta }_3+{\beta }_1(b\delta \sigma -d\psi +b\sigma {\beta }_3)){\xi }_1-u(z\eta {\theta }_1-by{\theta }_2){\xi }_2)},\hfill \\ \hfill E^{\ast }=& \frac{xz{\delta }^3\lambda +xz\delta \lambda (eu+\delta {\beta }_3)+\lambda \nu \mathrm{{\rm Y}}(a\delta +bc{\beta }_2+(b\delta \sigma -z\eta ){\beta }_3}{x\nu (\lambda (a\delta +bc{\beta }_2+(b\delta \sigma -z\eta ){\beta }_3+{\beta }_1(b\delta \sigma -d\psi +b\sigma {\beta }_3)){\xi }_1-u(z\eta {\theta }_1-by{\theta }_2){\xi }_2)},\hfill \\ \hfill & +\frac{{\beta }_1(b\delta \sigma -d\psi +b\sigma {\beta }_3)){\xi }_1+u\nu \mathrm{{\rm Y}}(by{\theta }_2-z\eta {\theta }_1){\xi }_2}{x\nu (\lambda (a\delta +bc{\beta }_2+(b\delta \sigma -z\eta ){\beta }_3+{\beta }_1(b\delta \sigma -d\psi +b\sigma {\beta }_3)){\xi }_1-u(z\eta {\theta }_1-by{\theta }_2){\xi }_2)},\hfill \\ \hfill I^{\ast }=& \frac{\eta (xz{\delta }^3\lambda +xz\delta \lambda (eu+\delta {\beta }_3)+\lambda \nu \mathrm{{\rm Y}}(a\delta +bc{\beta }_2+(b\delta \sigma -z\eta ){\beta }_3}{xy(\lambda (v\delta +gu{\beta }_2+(u\delta \sigma -z\eta ){\beta }_3+{\beta }_1(u\delta \sigma -f\psi +u\sigma {\beta }_3)){\xi }_1-t(z\eta {\theta }_1-uy{\theta }_2){\xi }_2)},\hfill \\ \hfill & +\frac{{\beta }_1(b\delta \sigma -d\psi +b\sigma {\beta }_3)){\xi }_1+u\nu \mathrm{{\rm Y}}(by{\theta }_2-z\eta {\theta }_1){\xi }_2)}{xy(\lambda (v\delta +gu{\beta }_2+(u\delta \sigma -z\eta ){\beta }_3+{\beta }_1(u\delta \sigma -f\psi +u\sigma {\beta }_3)){\xi }_1-t(z\eta {\theta }_1-uy{\theta }_2){\xi }_2)},\hfill \\ \hfill A^{\ast }=& \frac{b\nu \mathrm{{\rm Y}}}{xz}-\frac{buy\delta \lambda }{\lambda (a\delta +bc{\beta }_2+(b\delta \sigma -z\eta ){\beta }_3+{\beta }_1(b\delta \sigma -d\psi +b\sigma {\beta }_3)){\xi }_1-u(z\eta {\theta }_1-by{\theta }_2){\xi }_2},\hfill \\ \hfill H^{\ast }=& \frac{\nu \mathrm{{\rm Y}}(d{\beta }_1-b\rho (\delta +{\beta }_2))}{uxyz}+\frac{\delta \lambda (d{\beta }_1-b\rho (\delta +{\beta }_2))}{\lambda (a\delta +bc{\beta }_2+(b\delta \sigma -z\eta ){\beta }_3+{\beta }_1(b\delta \sigma -d\psi +b\sigma {\beta }_3)){\xi }_1-u(z\eta {\theta }_1-by{\theta }_2){\xi }_2},\hfill \\ \hfill R^{\ast }=& \frac{\nu \mathrm{{\rm Y}}(z\delta \eta {\beta }_2+(de-b\delta \rho ){\beta }_3)}{uxyz\delta }+\frac{\lambda ((-b\delta \rho +d{\beta }_1){\beta }_3+{\beta }_2(z\delta \eta +d{\beta }_3))}{\lambda (a\delta +bc{\beta }_2+(b\delta \sigma -z\eta ){\beta }_3+{\beta }_1(b\delta \sigma -d\psi +b\sigma {\beta }_3)){\xi }_1-u(z\eta {\theta }_1-by{\theta }_2){\xi }_2},\hfill \\ \hfill W^{\ast }=& \frac{\nu \mathrm{{\rm Y}}(z\eta {\theta }_1-by{\theta }_2)}{xyz\lambda }+\frac{u\delta (z\eta {\theta }_1-by{\theta }_2)}{\lambda (a\delta +bc{\beta }_2+(b\delta \sigma -z\eta ){\beta }_3+{\beta }_1(b\delta \sigma -d\psi +b\sigma {\beta }_3)){\xi }_1-u(z\eta {\theta }_1-by{\theta }_2){\xi }_2},\hfill \end{array}$$

where $x=\delta +\nu ,y=\delta +{\beta }_1+{\beta }_2,z=\delta +\rho ,u=\delta +{\beta }_3,a=-\eta z+b\delta \sigma +b\rho \psi ,b=\eta -1,c=\delta \sigma +\rho \psi +\sigma {\beta }_3,d=\delta \eta +\rho ,e={\beta }_1+{\beta }_2$.

A vital indicator of the spread of infectious disease in the population is the basic reproduction number ${\mathcal{R}}_0$, defined as the number of new infections caused by a typical infectious person in a disease‐free equilibrium population. An epidemic will spread if ${\mathcal{R}}_0>1$. While that an outbreak will most likely not accrue if ${\mathcal{R}}_0<1$. To obtain ${\mathcal{R}}_0$ for the system (13), we use the generation method. ³⁶ The infection components in the model system are E,  I,  A,  H, and W, so we consider the following fractional system:

$${\kappa }^{\alpha -1}{}_{}^C{D}_t^{\alpha }\mathrm{\Phi }(t)=F\left(\mathrm{\Phi }(t)\right)-V\left(\mathrm{\Phi }(t)\right),$$ (14)

where

$$F\left(\mathrm{\Phi }(t)\right)={\kappa }^{1-\alpha }\left[\begin{array}{c}{\xi }_1\left(I(t)+\sigma A(t)+\psi H(t)\right)S(t)+{\xi }_{2}W(t)S(t)\\ 0\\ 0\\ 0\\ 0\end{array}\right],$$

and

$$V\left(\mathrm{\Phi }(t)\right)={\kappa }^{1-\alpha }\left[\begin{array}{c}\left(\nu +\delta \right)E(t)\\ -\nu \eta E(t)+\left({\beta }_1+{\beta }_2+\delta \right)I(t)\\ -\nu \left(1-\eta \right)E(t)+\left(\rho +\delta \right)A(t)\\ -{\beta }_{1}I(t)-\rho A(t)+\left({\beta }_3+\delta \right)H(t)\\ -{\theta }_{1}I(t)-{\theta }_{2}A(t)+\lambda W(t)\end{array}\right].$$

At point P ₀, the Jacobian matrix for F and V is given by

$$J_F\left(P_0\right)=\frac{{\kappa }^{1-\alpha }\mathrm{{\rm Y}}}{\delta }\left[\begin{array}{ccccc}0& {\xi }_1& {\xi }_1\sigma & {\xi }_1\psi & {\xi }_2\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right],$$

and

$$J_V\left(P_0\right)={\kappa }^{1-\alpha }\left[\begin{array}{ccccc}\nu +\delta & 0& 0& 0& 0\\ -\nu \eta & {\beta }_1+{\beta }_2+\delta & 0& 0& 0\\ \nu (\eta -1)& 0& \rho +\delta & 0& 0\\ 0& -{\beta }_1& -\rho & {\beta }_3+\delta & 0\\ 0& -{\theta }_1& -{\theta }_2& 0& \lambda \end{array}\right].$$

The basic reproduction number is given by the spectral radius of $J_F\left(P_0\right){\left[J_V\left(P_0\right)\right]}^{-1}$ as follows:

$$\begin{array}{ll}{\mathcal{R}}_0=\frac{\eta \nu {\xi }_1\mathrm{{\rm Y}}}{\delta \left(\delta +\nu \right)\left(\delta +{\beta }_1+{\beta }_2\right)}& +\frac{\mathrm{{\rm Y}}{\xi }_2\left(\eta \nu {\theta }_1+\left(1-\eta \right)\nu \left(\delta +{\beta }_1+{\beta }_2\right){\theta }_2\right)}{\delta \lambda \left(\delta +\nu \right)\left(\delta +{\beta }_1+{\beta }_2\right)}\\ & +\frac{\mathrm{{\rm Y}}\left(1-\eta \right)\nu \sigma {\xi }_1}{\delta \left(\delta +\nu \right)\left(\delta +\rho \right)}+\frac{\psi \mathrm{{\rm Y}}(\left(\delta \eta +\rho \right){\beta }_1+(1-\eta )\rho (\delta +{\beta }_2)){\xi }_1}{\delta (\delta +\nu )(\delta +\rho )(\delta +{\beta }_1+{\beta }_2)(\delta +{\beta }_3)}.\end{array}$$ (15)

3.2. Sensitivity analysis

The sensitivity analysis of a few parameters used in the proposed model (11) is discuss in this section. That will make it easier for us to identify the parameters that favorably impact the basic reproduction number. To do this, we apply the technic given in Tuan et al. and Rezapour et al. ¹¹ , ²⁶ Using ${\mathcal{R}}_0$, we have

$$\begin{array}{cc}\hfill {\partial }_{{\theta }_1}{\mathcal{R}}_0& =\frac{\eta \nu \mathrm{{\rm Y}}{\xi }_2}{\delta \lambda (\delta +\nu )(\delta +{\beta }_1+{\beta }_2)},{\partial }_{\mathrm{{\rm Y}}}{\mathcal{R}}_0=\frac{{\mathcal{R}}_0}{\mathrm{{\rm Y}}},\hfill \\ \hfill {\partial }_{{\theta }_2}{\mathcal{R}}_0& =\frac{(1-\eta )\nu \mathrm{{\rm Y}}{\xi }_2}{\delta \lambda (\delta +\nu )(\delta +\rho )},\hfill \\ \hfill {\partial }_{{\xi }_1}{\mathcal{R}}_0& =\frac{(1-\eta )\nu \sigma \mathrm{{\rm Y}}}{\delta (\delta +\nu )(\delta +\rho )}+\frac{\nu \mathrm{{\rm Y}}{S}_0\left(\eta \left(\delta +\rho \right)\left(\delta +{\beta }_3\right)+\rho \psi \left(1-\eta \right)\left(\delta +{\beta }_2\right)+\psi {\beta }_1\left(\delta \eta +\rho \right)\right)}{\delta (\delta +\nu )(\delta +\rho )(\delta +{\beta }_1+{\beta }_2)(\delta +{\beta }_3)},\hfill \\ \hfill {\partial }_{{\xi }_2}{\mathcal{R}}_0& =\frac{\eta \nu \mathrm{{\rm Y}}{\theta }_1}{\lambda \delta (\delta +\nu )(\delta +{\beta }_1+{\beta }_2)}+\frac{(1-\eta )\nu \mathrm{{\rm Y}}{\theta }_2}{\lambda \delta (\delta +\nu )(\delta +\rho )},\hfill \\ \hfill {\partial }_{\sigma }{\mathcal{R}}_0& =\frac{(1-\eta )\nu \mathrm{{\rm Y}}{\xi }_1}{\delta (\delta +\nu )(\delta +\rho )},\hfill \\ \hfill {\partial }_{\psi }{\mathcal{R}}_0& =\frac{\nu \mathrm{{\rm Y}}(\left(\delta \eta +\rho \right){\beta }_1+(1-\eta )\rho (\delta +{\beta }_2)){\xi }_1}{\delta (\delta +\nu )(\delta +\rho )(\delta +{\beta }_1+{\beta }_2)(\delta +{\beta }_3)},\hfill \\ \hfill {\partial }_{\nu }{\mathcal{R}}_0& =\frac{\mathrm{{\rm Y}}{\xi }_1\lambda \psi \left(\rho \left(1-\eta \right)\left(\delta +{\beta }_2\right)+{\beta }_1\left(\delta \eta +\rho \right)\right)+\mathrm{{\rm Y}}\eta \left(\delta +\rho \right)\left(\delta +{\beta }_3\right)\left(\lambda {\xi }_1+{\theta }_1{\xi }_2\right)}{\lambda {(\delta +\nu )}^2(\delta +\rho )(\delta +{\beta }_1+{\beta }_2)(\delta +{\beta }_3)}+\frac{(1-\eta )\mathrm{{\rm Y}}(\lambda \sigma {\xi }_1+{\theta }_2{\xi }_2)}{\lambda {(\delta +\nu )}^2(\delta +\rho )},\hfill \\ \hfill {\partial }_{\lambda }{\mathcal{R}}_0& =-\frac{\eta \nu \mathrm{{\rm Y}}{\theta }_1{\xi }_2}{\delta {\lambda }^2\left(\delta +\nu \right)\left(\delta +{\beta }_1+{\beta }_2\right)}-\frac{\left(1-\eta \right)\nu \mathrm{{\rm Y}}{\theta }_2{\xi }_2}{\delta {\lambda }^2\left(\delta +\nu \right)\left(\delta +\rho \right)},\hfill \\ \hfill {\partial }_{\rho }{\mathcal{R}}_0& =-\frac{\left(1-\eta \right)\nu \mathrm{{\rm Y}}}{\lambda \delta \left(\delta +\nu \right){\left(\delta +\rho \right)}^2}\left(\frac{{\theta }_2{\xi }_2\left(\delta +{\beta }_3\right)+{\beta }_3\sigma {\xi }_1\lambda +\lambda \delta {\xi }_1\left(\sigma -\psi \right)}{\lambda \left(\delta +{\beta }_3\right)}\right),\hfill \\ \hfill {\partial }_{{\beta }_1}{\mathcal{R}}_0& =-\frac{\eta \nu \mathrm{{\rm Y}}\left(\lambda {\xi }_1\left(\delta \left(1-\psi \right)-\psi {\beta }_2+{\beta }_3\right)+{\theta }_1{\xi }_2\left(\delta +{\beta }_3\right)\right)}{\delta \lambda (\delta +\nu ){(\delta +{\beta }_1+{\beta }_2)}^2(\delta +{\beta }_3)},\hfill \\ \hfill {\partial }_{{\beta }_2}{\mathcal{R}}_0& =-\frac{\eta \nu \mathrm{{\rm Y}}\left(\lambda {\xi }_1\left(\delta +\psi {\beta }_1+{\beta }_3\right)+{\theta }_1{\xi }_2\left(\delta +{\beta }_3\right)\right)}{\lambda \delta (\delta +\nu ){(\delta +{\beta }_1+{\beta }_2)}^2(\delta +{\beta }_3)},\hfill \\ \hfill {\partial }_{{\beta }_3}{\mathcal{R}}_0& =-\frac{\nu \mathrm{{\rm Y}}\psi {\xi }_1\left(\left(\delta \eta +\rho \right){\beta }_1+\left(1-\eta \right)\rho \left(\delta +{\beta }_2\right)\right)}{\delta (\delta +\nu )(\delta +\rho )(\delta +{\beta }_1+{\beta }_2){(\delta +{\beta }_3)}^2},\hfill \\ \hfill {\partial }_{\eta }{\mathcal{R}}_0& =\frac{\nu \mathrm{{\rm Y}}\left(\delta +{\beta }_3\right)\left(\left(\delta +\rho \right)\left({\theta }_1{\xi }_2+\lambda {\xi }_1\right)-\left(\delta +{\beta }_1+{\beta }_2\right)\left({\theta }_2{\xi }_2+\lambda \sigma {\xi }_1\right)\right)+\psi {\xi }_1\lambda \left(\rho {\beta }_2+\delta \left(\rho -{\beta }_1\right)\right)}{\delta \lambda (\delta +\nu )(\delta +\rho )(\delta +{\beta }_1+{\beta }_2)(\delta +{\beta }_3)},\hfill \\ \hfill {\partial }_{\delta }{\mathcal{R}}_0& =-\frac{\nu \mathrm{{\rm Y}}}{\delta }\left[\frac{\eta \left({\theta }_1{\xi }_2+\lambda {\xi }_1\right)}{\lambda \left(\delta +\nu \right)\left(\delta +{\beta }_1+{\beta }_2\right)}\left(\frac{1}{\delta +\nu }+\frac{1}{\delta +{\beta }_1+{\beta }_2}\right)+\frac{\rho {\xi }_1\psi \left(\left(1-\eta \right)+\left({\beta }_1+{\beta }_2-\delta \right)\right)}{\left(\delta +\nu \right){\left(\delta +\rho \right)}^2\left(\delta +{\beta }_1+{\beta }_2\right)\left(\delta +{\beta }_3\right)}\right.\hfill \\ \hfill & +\frac{\rho {\xi }_1\psi \left(1-\eta \right)+\left({\beta }_1+{\beta }_2-\delta \right)}{\left(\delta +\nu \right){\left(\delta +\rho \right)}^2\left(\delta +{\beta }_1+{\beta }_2\right)\left(\delta +{\beta }_3\right)}\left(\frac{1}{\delta +\nu }+\frac{1}{\delta +{\beta }_3}+\frac{1}{\delta +{\beta }_1+{\beta }_2}\right)\left.+\frac{\left(1-\eta \right)\left({\theta }_2{\xi }_2+\lambda \sigma {\xi }_1\right)}{\lambda \left(\delta +\rho \right)\left(\delta +\nu \right)}\left(\frac{1}{\delta +\nu }+\frac{1}{\delta +\rho }\right)\right].\hfill \end{array}$$

Given all parameters are positive, then ${\partial }_{{\theta }_1}{\mathcal{R}}_0>0$, and if η < 1, we have ${\partial }_{\mathrm{{\rm Y}}}{\mathcal{R}}_0>0,{\partial }_{{\theta }_2}{\mathcal{R}}_0>0,{\partial }_{{\xi }_1}{\mathcal{R}}_0>0,{\partial }_{{\xi }_2}{\mathcal{R}}_0>0,{\partial }_{\sigma }{\mathcal{R}}_0>0,{\partial }_{\psi }{\mathcal{R}}_0>0,{\partial }_{\nu }{\mathcal{R}}_0>0,{\partial }_{\lambda }{\mathcal{R}}_0<0,{\partial }_{{\beta }_2}{\mathcal{R}}_0<0,{\partial }_{{\beta }_3}{\mathcal{R}}_0<0,{\partial }_{\delta }{\mathcal{R}}_0<0,\text{and, if add}\psi <\sigma ,\text{then}{\partial }_{\rho }{\mathcal{R}}_0<0.$

Thus for $\eta <1,{\mathcal{R}}_0$ is decreasing with λ,  β ₂, β ₃, δ, ρ and increasing with θ ₁, θ ₂, ξ ₁, ξ ₂,  σ, ψ, ν,  Υ. Based on the result of derivatives, in general, cannot be commented on ${\mathcal{R}}_0$ sensitivity to other parameters β ₁ and η, but for the parameter values in the proposed model, ${\partial }_{{\beta }_1}{\mathcal{R}}_0<0$ and ${\partial }_{\eta }{\mathcal{R}}_0<0$, thus ${\mathcal{R}}_0$ is decreasing with β ₁ and η.

4. EXISTENCE AND UNIQUENESS ANALYSIS

In this section, we investigate the existence and uniqueness of the proposed model with the assistance of fixed‐point theory. First, let us write Equation (11) as follows:

$$\begin{array}{ll}{\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }S(t)& ={\chi }_1\left(t,S\left(t\right)\right),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }E(t)& ={\chi }_2\left(t,E(t)\right),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }I(t)& ={\chi }_3\left(t,I(t)\right),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }A(t)& ={\chi }_4\left(t,A\left(t\right)\right),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }H(t)& ={\chi }_5\left(t,H(t)\right),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }R(t)& ={\chi }_6\left(t,R(t)\right),\\ {\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }W(t)& ={\chi }_7\left(t,W(t)\right),\end{array}$$ (16)

where

$$\begin{array}{ll}{\chi }_1\left(t,S(t)\right)& =\mathrm{{\rm Y}}-{\xi }_{1}S(t)\left(I(t)+\sigma A(t)+\psi H(t)\right)-{\xi }_{2}W(t)S(t)-\delta S(t),\\ {\chi }_2\left(t,E(t)\right)& ={\xi }_1\left(I(t)+\sigma A(t)+\psi H(t)\right)+{\xi }_{2}W(t)S(t)-\left(\nu +\delta \right)E(t),\\ {\chi }_3\left(t,I(t)\right)& =\nu \eta E(t)-\left({\beta }_1+{\beta }_2+\delta \right)I(t),{\chi }_4\left(t,A\left(t\right)\right)=\nu \left(1-\eta \right)E(t)-\left(\rho +\delta \right)A(t),\\ {\chi }_5\left(t,H(t)\right)& ={\beta }_{1}I(t)+\rho A(t)-\left({\beta }_3+\delta \right)H(t),{\chi }_6\left(t,R(t)\right)={\beta }_{2}I(t)+{\beta }_{3}H(t)-\delta R(t),\\ {\chi }_7\left(t,W(t)\right)& ={\theta }_{1}I(t)+{\theta }_{2}A(t)-\lambda W(t).\end{array}$$

By Lemma 3, the corresponding Volterra integral system of the second kind for Equation (16) is given by

$$\begin{array}{ll}S\left(t\right)& =S(0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{\left(t-\tau \right)}^{\alpha -1}{\chi }_1\left(\tau ,S\left(\tau \right)\right)d\tau ,\\ E\left(t\right)& =E(0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{\left(t-\tau \right)}^{\alpha -1}{\chi }_2\left(\tau ,E\left(\tau \right)\right)d\tau ,\\ I\left(t\right)& =I(0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{\left(t-\tau \right)}^{\alpha -1}{\chi }_3\left(\tau ,I\left(\tau \right)\right)d\tau ,\\ A\left(t\right)& =A(0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{\left(t-\tau \right)}^{\alpha -1}{\chi }_4\left(\tau ,A\left(\tau \right)\right)d\tau ,\\ H\left(t\right)& =H(0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{\left(t-\tau \right)}^{\alpha -1}{\chi }_5\left(\tau ,H\left(\tau \right)\right)d\tau ,\\ R\left(t\right)& =R(0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{\left(t-\tau \right)}^{\alpha -1}{\chi }_6\left(\tau ,R\left(\tau \right)\right)d\tau ,\\ W\left(t\right)& =W(0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{\left(t-\tau \right)}^{\alpha -1}{\chi }_7\left(\tau ,W\left(\tau \right)\right)d\tau .\end{array}$$ (17)

Theorem 1

The kernel of the proposed fractional model will be satisfying the Lipchitz condition if the following inequality hold:

$$0\le L_i<1,i=\mathrm{1,2},\dots ,7,$$

where $L_1={\xi }_1\left(Z_1+\sigma Z_2+\psi Z_3\right)+{\xi }_2{Z}_4+\delta ,L_2=\nu +\delta ,L_3={\beta }_1+{\beta }_2+\delta ,L_4=\rho +\delta ,L_5={\beta }_3+\delta ,L_6=\delta$, and $L_7=\lambda$.

We will prove for the first kernel and similarly for the other. Consider function S(t) and S ₁(t), then

$$\begin{array}{cc}\hfill ‖{\chi }_1(t,S)-{\chi }_1\left(t,S_1\right)‖=& ‖\left({\xi }_1\left(I\left(t\right)+\sigma A\left(t\right)+\psi H\left(t\right)\right)+{\xi }_{2}W\left(t\right)+\delta \right)\left(S\left(t\right)-S\left(t_1\right)\right)‖\hfill \\ \hfill & \le \left({\xi }_1\left(‖I\left(t\right)‖+\sigma ‖A\left(t\right)‖+\psi ‖H\left(t\right)‖\right)+{\xi }_2‖W\left(t\right)‖+\delta \right)‖S\left(t\right)-S\left(t_1\right)‖\hfill \\ \hfill & \le \left({\xi }_1\left(Z_1+\sigma Z_2+\psi Z_3\right)+{\xi }_2{Z}_4+\delta \right)‖S\left(t\right)-S\left(t_1\right)‖=L_1‖S\left(t\right)-S\left(t_1\right)‖,\hfill \end{array}$$

where $Z_1\le ‖I\left(t\right)‖,Z_2\le ‖A\left(t\right)‖,Z_3\le ‖H\left(t\right)‖$, and $Z_4\le ‖W\left(t\right)‖$.

Similarly, we get

$$\begin{array}{cc}\hfill & ‖{\chi }_2(t,E)-{\chi }_2\left(t,E_1\right)‖\le L_2‖E\left(t\right)-E\left(t_1\right)‖\hfill \\ \hfill & ‖{\chi }_3(t,I)-{\chi }_3\left(t,I_1\right)‖\le L_3‖I\left(t\right)-I\left(t_1\right)‖\hfill \\ \hfill & ‖{\chi }_4(t,A)-{\chi }_1\left(t,A_1\right)‖\le L_4‖A\left(t\right)-A\left(t_1\right)‖\hfill \\ \hfill & ‖{\chi }_5(t,H)-{\chi }_5\left(t,H_1\right)‖\le L_5‖H\left(t\right)-H\left(t_1\right)‖\hfill \\ \hfill & ‖{\chi }_6(t,R)-{\chi }_6\left(t,R_1\right)‖\le L_6‖R\left(t\right)-R\left(t_1\right)‖\hfill \\ \hfill & ‖{\chi }_7(t,W)-{\chi }_7\left(t,W_1\right)‖\le L_7‖W\left(t\right)-W\left(t_1\right)‖.\hfill \end{array}$$

□

By helping Equation (16), the proposed model Equation (11) can be written as follows:

$${\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }\chi (t)=\mathrm{\Psi }\left(t,\chi (t)\right),t\in [0,T],\alpha \in (\mathrm{0,1}],$$ (18)

with initial condition    $\chi (0)={\chi }_0$, where $\chi (t)={\left[\begin{array}{ccccccc}S(t)& E(t)& I(t)& A(t)& H(t)& R(t)& W(t)\end{array}\right]}^T,{\chi }_0={\left[\begin{array}{ccccccc}{S}_0& E_0& I_0& A_0& H_0& R_0& W_0\end{array}\right]}^T,$ and $\mathrm{\Psi }\left(t,\chi \right)={\left[\begin{array}{ccccccc}{\chi }_1(t,S)& {\chi }_2(t,E)& {\chi }_3(t,I)& {\chi }_4(t,A)& {\chi }_5(t,H)& {\chi }_6(t,R)& {\chi }_7(t,W)\end{array}\right]}^T$.

Again, by Lemma 3, we have

$$\chi \left(t\right)=\chi (0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)d\tau .$$ (19)

Theorem 2

(Existence) Let $T^{\ast }>0,{\chi }_0\in {ℝ}_{+}^7$ and b > 0. Define

$$D:=\left\{\right.\left.\left(t,\chi \right)\in (\left[0,T^{\ast }\right],{ℝ}_{+}^7):t\in \left[0,T^{\ast }\right],‖\chi -{\chi }_0‖\le b\right\},$$

and assume that $\mathrm{\Psi }:D\to {ℝ}_{+}^7$ be a continuous function; furthermore, define $\zeta :={\sup }_{\left(t,\chi \right)\in D}‖\mathrm{\Psi }\left(t,\chi (t)\right)‖$ then, for α ∈ (0, 1], there exists a function $\chi \in \left([0,T],{ℝ}_{+}^7\right)$, which is a solution of Equation (18), where

$$T=\min \left\{T^{\ast }.{\left(\frac{b\mathrm{\Gamma }\left(\alpha +1\right)}{{\kappa }^{1-\alpha }\zeta }\right)}^{\frac{1}{\alpha }}\right\}.$$

For $\chi \in C\left(\left[0,T\right],{ℝ}_{+}^7\right)$, define norm of ${‖\chi ‖}_{\ast }={\sup }_{t\in [0,T]}‖\chi \left(t\right)‖$, with ${‖·‖}_{\ast }$ a Banach space. Define the set $V:=\left\{\chi \in C\left(\left[0,T\right],{ℝ}_{+}^7\right):{‖\chi -{\chi }_0‖}_{\ast }\le b\right\}$; it is evident that V is bounded, closed, and convex subset of the Banach space of all continuous function on $C\left(\left[0,T\right],{ℝ}_{+}^7\right)$. It is clear that V is a non‐empty set, since χ ₀ ∈ V. We now define an operator $\mathfrak{J}$ on V. For each element χ ∈ V,

$$\left(\mathfrak{J}\chi \right)\left(t\right):=\chi (0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)d\tau .$$ (20)

Using that operator, Equation (19) can be written as $\chi =\mathfrak{J}\chi$; now we must prove that $\mathfrak{J}$ has a fixed‐point. This will be done by using Schuder's second fixed‐point theorem. First, we will show that $\mathfrak{J}\chi \in V$ for χ ∈ V. For any χ ∈ V, we have

$$\begin{array}{ll}‖\left(\mathfrak{J}\chi \right)(t)-\chi (0)‖=& ‖\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)d\tau ‖\\ \le & \frac{{\kappa }^{1-\alpha }\zeta }{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}d\tau =\frac{{\kappa }^{1-\alpha }\zeta }{\mathrm{\Gamma }\left(\alpha +1\right)}{t}^{\alpha }\le \frac{{\kappa }^{1-\alpha }\zeta }{\mathrm{\Gamma }\left(\alpha +1\right)}{T}^{\alpha }=\frac{{\kappa }^{1-\alpha }\zeta }{\mathrm{\Gamma }\left(\alpha +1\right)}\frac{b\mathrm{\Gamma }\left(\alpha +1\right)}{{\kappa }^{1-\alpha }\zeta }=b.\end{array}$$ (21)

Hence, ${‖\mathfrak{J}\chi -{\chi }_0‖}_{\ast }\le b$, so $\mathfrak{J}\chi \in V$ for χ ∈ V.

Second, we show that $\mathfrak{J}$ is continuous. For every ${\chi }_n,\chi \in V,n=\mathrm{1,2},\dots$, with ${\lim }_{n\to \infty }{‖{\chi }_n-\chi ‖}_{\ast }=0$, we have ${\lim }_{n\to \infty }{\chi }_n(t)=\chi (t),t\in [0,T],$ since all components of $\mathrm{\Psi }\left(t,\chi (t)\right)$ are continuous on $ℝ$, thus Ψ is continuous on $\left\{\chi \in C\left(\left[0,T\right],{ℝ}_{+}^7\right):‖\chi -{\chi }_0‖\le b\right\}$, consequently,   ${\lim }_{n\to \infty }\mathrm{\Psi }\left(t,{\chi }_n(t)\right)=\mathrm{\Psi }\left(t,\chi (t)\right)$, so

$$\underset{t\in \left[0,T\right]}{\sup }‖\mathrm{\Psi }\left(t,{\chi }_n(t)\right)-\mathrm{\Psi }\left(t,\chi (t)\right)‖\to 0\text{as}n\to \infty .$$ (22)

On the other hand,

$$\begin{array}{ll}‖\left(\mathfrak{J}{\chi }_n\right)(t)-\left(\mathfrak{J}\chi \right)(t)‖& \le \frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}‖\mathrm{\Psi }\left(\tau ,{\chi }_n\left(\tau \right)\right)-\mathrm{\Psi }\left(\tau ,\chi (\tau )\right)‖d\tau \\ & \le \frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}\underset{t\in \left[0,T\right]}{\sup }‖\mathrm{\Psi }\left(\tau ,{\chi }_n\left(\tau \right)\right)-\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)‖{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}d\tau \\ & \le \frac{{\kappa }^{1-\alpha }{T}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\underset{t\in [0,T]}{\sup }‖\mathrm{\Psi }\left(\tau ,{\chi }_n\left(\tau \right)\right)-\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)‖\to 0asn\to \infty .\end{array}$$ (23)

Hence, ${‖\mathfrak{J}{\chi }_n-\mathfrak{J}\chi ‖}_{\ast }\to 0asn\to \infty$. Thus, $\mathfrak{J}$ is continuous.

Finally, we have shown that $\mathfrak{J}(V)=\left\{\mathfrak{J}\chi :\chi \in V\right\}$ is relatively compact. By the Arzela–Ascoli theorem, it is enough to demonstrate that $\mathfrak{J}(V)$ is uniformly bounded and equicontinuous. Let $y\in \mathfrak{J}(V)$, for all t ∈ [0, T], we have

$$\begin{array}{ll}‖y(t)‖=‖\left(\mathfrak{J}\chi \right)(t)‖& \le ‖{\chi }_0‖+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}‖{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)d\tau ‖\\ & =‖{\chi }_0‖+\frac{{\kappa }^{1-\alpha }\zeta }{\mathrm{\Gamma }\left(\alpha +1\right)}{t}^{\alpha }\le ‖{\chi }_0‖+\frac{{\kappa }^{1-\alpha }\zeta }{\mathrm{\Gamma }\left(\alpha +1\right)}{T}^{\alpha }=‖{\chi }_0‖+b.\end{array}$$ (24)

Thus, ${‖\left(\mathfrak{J}\chi \right)(t)‖}_{\ast }\le ‖{\chi }_0‖+b$, this means that $\mathfrak{J}(V)$ is uniformly bounded. For any 0 ≤ t ₁ ≤ t ₂ ≤ T, we have

$$\begin{array}{cc}\hfill ‖\left(\mathfrak{J}\chi \right)\left(t_1\right)-\left(\mathfrak{J}\chi \right)\left(t_2\right)‖& \le \frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}‖{\displaystyle \underset{0}{\overset{t_1}{\int }}}{\left(t_1-\tau \right)}^{\alpha -1}\mathrm{\Psi }\left(\tau ,\chi (\tau )\right)d\tau -{\displaystyle \underset{0}{\overset{t_2}{\int }}}{\left(t_2-\tau \right)}^{\alpha -1}\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)d\tau ‖\hfill \\ \hfill & =\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}‖{\displaystyle \underset{0}{\overset{t_1}{\int }}}{(\left(t_1-\tau \right)}^{\alpha -1}-{\left(t_2-\tau \right)}^{\alpha -1})\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)d\tau -{\displaystyle \underset{t_1}{\overset{t_2}{\int }}}{\left(t_2-\tau \right)}^{\alpha -1}\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)d\tau ‖\hfill \\ \hfill & \le \frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}\left[{\displaystyle \underset{0}{\overset{t_1}{\int }}}‖{\left(t_1-\tau \right)}^{\alpha -1}-{\left(t_2-\tau \right)}^{\alpha -1}‖‖\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)‖d\tau +{\displaystyle \underset{t_1}{\overset{t_2}{\int }}}{\left(t_2-\tau \right)}^{\alpha -1}‖\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)‖d\tau \right]\hfill \\ \hfill & \le \frac{{\kappa }^{1-\alpha }\zeta }{\mathrm{\Gamma }\left(\alpha \right)}\left[{\displaystyle \underset{0}{\overset{t_1}{\int }}}‖{\left(t_1-\tau \right)}^{\alpha -1}-{\left(t_2-\tau \right)}^{\alpha -1}‖d\tau +{\displaystyle \underset{t_1}{\overset{t_2}{\int }}}{\left(t_2-\tau \right)}^{\alpha -1}d\tau \right].\hfill \end{array}$$

Since α ≤ 1 and t ₁ ≤ t ₂, then ${\left(t_1-\tau \right)}^{\alpha -1}\ge {\left(t_2-\tau \right)}^{\alpha -1}$; therefore,

$$\begin{array}{ll}‖\left(\mathfrak{J}\chi \right)\left(t_1\right)-\left(\mathfrak{J}\chi \right)\left(t_2\right)‖& \le \frac{{\kappa }^{1-\alpha }\zeta }{\mathrm{\Gamma }(\alpha )}\left[{\displaystyle \underset{0}{\overset{t_1}{\int }}}{\left(t_1-\tau \right)}^{\alpha -1}-{\left(t_2-\tau \right)}^{\alpha -1}d\tau +\frac{{\left(t_2-t_1\right)}^{\alpha }}{\alpha }\right]\\ & =\frac{{\kappa }^{1-\alpha }\zeta }{\mathrm{\Gamma }\left(\alpha +1\right)}\left[{\left(t_2-t_1\right)}^{\alpha }+t_1^{\alpha }-t_2^{\alpha }+{\left(t_2-t_1\right)}^{\alpha }\right]\le \frac{2{\kappa }^{1-\alpha }\zeta }{\mathrm{\Gamma }\left(\alpha +1\right)}{\left(t_2-t_1\right)}^{\alpha }\to 0as{t}_2\to t_1.\end{array}$$ (25)

Hence, ${‖\left(\mathfrak{J}\chi \right)\left(t_1\right)-\left(\mathfrak{J}\chi \right)\left(t_2\right)‖}_{\ast }\to 0as{t}_2\to t_1$, and $\mathfrak{J}(V)$ is equicontinuous on [0, T]. Thus, $\mathfrak{J}(V)$ is relatively compact. Moreover, $\mathfrak{J}$ has a fixed‐point, which is the required solution of Equation (18). This completes the proof. □

Remark 1

Assume hypotheses Theorem 1, the kernels ${\chi }_i,i=\mathrm{1,2},\dots ,7$ satisfy Lipchitz condition; thus,

$$\begin{array}{cc}\hfill ‖\mathrm{\Psi }\left(t,\chi (t)\right)-\mathrm{\Psi }\left(t,\bar{\chi }(t)\right)‖=& \underset{t\in [0,T]}{\max }\left\{‖{\chi }_i(t)-{\bar{\chi }}_i(t)‖\right\}\hfill \\ \hfill \le & \underset{t\in [0,T]}{\max }\left\{L_1‖S(t)-\overline{S}(t)‖,L_2‖E(t)-Ē(t)‖,L_3‖I(t)-Ī(t)‖,\right.\hfill \\ \hfill & \left.L_4‖A(t)-Ā(t)‖,L_5‖H(t)-\overline{H}(t)‖,L_6‖R(t)-\overline{R}(t)‖,L_7‖W(t)-\overline{W}(t)‖\right\}\hfill \\ \hfill \le & \max \left\{L_1,L_2,\dots ,L_7\right\}\underset{t\in [0,T]}{\max }\left\{‖S(t)-\overline{S}(t)‖,‖E(t)-Ē(t)‖,‖I(t)-Ī(t)‖,\right.\hfill \\ \hfill & \left.‖A(t)-Ā(t)‖,‖H(t)-\overline{H}(t)‖,‖R(t)-\overline{R}(t)‖,‖W(t)-\overline{W}(t)‖\right\}\hfill \\ \hfill =& L_{\mathrm{\Psi }}‖\chi (t)-\bar{\chi }(t)‖,\hfill \end{array}$$

where $L_{\mathrm{\Psi }}=\max \left\{L_1,L_2,L_3,L_4,L_5,L_6,L_7\right\}$ and $L_i,i=\mathrm{1,2},\dots ,7$ as defined in Theorem 1. Thus, Ψ satisfies Lipchitz's condition.

Theorem 3

(Uniqueness) Assume all hypotheses in Theorems 1 and 2, the solution $\chi \in C\left([0,T],{ℝ}_{+}^7\right)$ is unique, where

$$T<\min \left\{T^{\ast }.{\left(\frac{b\mathrm{\Gamma }\left(\alpha +1\right)}{{\kappa }^{1-\alpha }\zeta L_{\mathrm{\Psi }}}\right)}^{\frac{1}{\alpha }}\right\}.$$

Suppose that $\chi (t)and\bar{\chi }(t)$ are solutions of Equation (18) on $C\left(\left[0,T\right],{ℝ}_{+}^7\right)$, then

$$\begin{array}{ll}‖\chi (t)-\bar{\chi }(t)‖& =\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}‖{\int }_0^t{\left(t-\tau \right)}^{\alpha -1}\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)d\tau -{\int }_0^t{\left(t-\tau \right)}^{\alpha -1}\mathrm{\Psi }\left(\tau ,\overline{\chi }(\tau )\right)d\tau ‖\\ & \le \frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{\left(t-\tau \right)}^{\alpha -1}‖\mathrm{\Psi }\left(\tau ,\chi \left(\tau \right)\right)-\mathrm{\Psi }\left(\tau ,\overline{\chi }(\tau )\right)‖d\tau .\end{array}$$ (26)

By Lipchitz condition, we get

$$\begin{array}{ll}‖\chi (t)-\bar{\chi }(t)‖& \le \frac{{\kappa }^{1-\alpha }{L}_{\mathrm{\Psi }}}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}‖\chi \left(\tau \right)-\bar{\chi }\left(\tau \right)‖d\tau \le \frac{{\kappa }^{1-\alpha }{L}_{\mathrm{\Psi }}}{\mathrm{\Gamma }\left(\alpha \right)}\underset{t\in [0,T]}{\max }‖\chi \left(t\right)-\bar{\chi }\left(t\right)‖{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}d\tau \\ & =\frac{{\kappa }^{1-\alpha }{L}_{\mathrm{\Psi }}{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\underset{t\in [0,T]}{\max }‖\chi \left(t\right)-\bar{\chi }\left(t\right)‖\le \frac{{\kappa }^{1-\alpha }{L}_{\mathrm{\Psi }}{T}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\underset{t\in [0,T]}{\max }‖\chi \left(t\right)-\bar{\chi }\left(t\right)‖=C\underset{t\in [0,T]}{\max }‖\chi \left(t\right)-\bar{\chi }\left(t\right)‖,\end{array}$$ (27)

where $C=\frac{{\kappa }^{1-\alpha }{L}_{\mathrm{\Psi }}{T}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\in (\mathrm{0,1})$.

On the other hand,

$$‖\chi (t)-\bar{\chi }(t)‖=\underset{t\in [0,T]}{\max }‖\chi (t)-\bar{\chi }(t)‖,$$ (28)

this means that

$$\underset{t\in [0,T]}{\max }‖\chi (t)-\bar{\chi }(t)‖\le C\underset{t\in [0,T]}{\max }‖\chi (t)-\bar{\chi }(t)‖.$$ (29)

Thus, $‖\chi \left(t\right)-\bar{\chi }\left(t\right)‖=0\Rightarrow \chi \left(t\right)=\bar{\chi }\left(t\right)$. The solution is unique. □

5. NUMERICAL ALGORITHM

This section presents the numerical algorithm based on the predictor–corrector method. ³⁷ Under the hypotheses of Theorem 2, there is a unique solution on [0, T]. Let $t_0=0<t_1<\dots <t_N=T$ be a uniformly divide of the interval [0, T] where $t_n=nh,n=\mathrm{1,2},\dots ,N$ and $h=T/N$. By Lemma 3, the solution of

$${\kappa }^{\alpha -1}{}^C{D}_t^{\alpha }S(t)={\chi }_1\left(t,S(t)\right),$$

is equivalent to

$$S(t)=S(0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{0}{\overset{t}{\int }}}{\left(t-\tau \right)}^{\alpha -1}{\chi }_1\left(\tau ,S(\tau )\right)d\tau .$$ (30)

By Equation (30), we have

$$S\left(t_{n+1}\right)=S(0)+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{0}{\overset{t_{n+1}}{\int }}}{\left(t_{n+1}-\tau \right)}^{\alpha -1}{\chi }_1\left(\tau ,S(\tau )\right)d\tau .$$ (31)

Let $S_{n+1}\simeq S\left(t_{n+1}\right),n=\mathrm{0,1},2,\dots ,N-1$, by Lagrange interpolation, we approximate the kernel ${\chi }_1\left(t,S(t)\right)$ over $\left[t_k,t_{k+1}\right]$ as follows:

$${\chi }_1\left(\tau ,S(\tau )\right)\simeq \frac{\tau -t_k}{t_{k+1}-t_k}{\chi }_{\mathrm{1}}\left(t_{k+1},S_{k+1}\right)+\frac{\tau -t_{k+1}}{t_k-t_{k+1}}{\chi }_{\mathrm{1}}\left(t_k,S_{\mathrm{k}}\right).$$ (32)

Substituting Equation (32) into Equation (31), we get

$$\begin{array}{ll}{S}_{n+1}=& S_0+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}\left[\frac{{\chi }_{\mathrm{1}}\left(t_{k+1},S_{k+1}\right)}{h}{\displaystyle \underset{t_k}{\overset{t_{k+1}}{\int }}}\left(\tau -t_k\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}d\tau -\frac{{\chi }_{\mathrm{1}}\left(t_k,S_k\right)}{h}{\displaystyle \underset{t_k}{\overset{t_{k+1}}{\int }}}\left(\tau -t_{k+1}\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}d\tau \right]\\ =& S_0+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}\left[\frac{{\chi }_{\mathrm{1}}\left(t_{k+1},S_{k+1}\right)h^{\alpha }}{\alpha \left(\alpha +1\right)}\left({\left(n+1-k\right)}^{\alpha +1}-{\left(n-k\right)}^{\alpha }\left(n-k+1+\alpha \right)\right)\right.\\ & \left.-\frac{{\chi }_{\mathrm{1}}\left(t_k,S_k\right)h^{\alpha }}{\alpha \left(\alpha +1\right)}\left({\left(n-k\right)}^{\alpha +1}-{\left(n+1-k\right)}^{\alpha }\left(n-k-\alpha \right)\right)\right].\end{array}$$ (33)

After the rearrangement of the summation on the right‐hand side of Equation (33), we get

$$S_{n+1}=S_0+\frac{{\kappa }^{1-\alpha }{h}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\left[{\chi }_{\mathrm{1}}\left(t_{n+1},S_{\mathrm{n}\mathrm{+}\mathrm{1}}\right)+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}{\chi }_{\mathrm{1}}\left(t_k,S_k\right)\right],$$ (34)

where

$$C_{k,n+1}=\left\{\begin{array}{cc}{n}^{\alpha +1}-{(n+1)}^{\alpha }\left(n-\alpha \right),& k=0,\\ {\left(n-k+2\right)}^{\alpha +1}+{\left(n-k\right)}^{\alpha }-2{\left(n-k+1\right)}^{\alpha +1},& 1\le k\le n.\end{array}\right.$$

The quantity S _n + 1 on the right‐hand side of Equation (34) is predicted by $S_{n+1}^P$ applying the one‐step Adams–Bashforth method to Equation (31) by replacing the function ${\chi }_1\left(\tau ,S(\tau )\right)$ with the quantity ${\chi }_1\left(t_k,S_k\right)$ as follows:

$$S_{k\mathrm{+}\mathrm{1}}^P=S_0\mathrm{+}\frac{{\kappa }^{\mathrm{1}\mathrm{-}\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}{\chi }_1\left(t_k,S_k\right){\displaystyle \underset{t_k}{\overset{t_{k+1}}{\int }}}{\left(t_{n+1}-\tau \right)}^{\alpha -1}d\tau =S_0+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}{\chi }_1\left(t_k,S_k\right)\left({\left(n-k+1\right)}^{\alpha }-{\left(n-k\right)}^{\alpha }\right).$$ (35)

Similarly, we can obtain the numerical algorithm of the other equation of system Equation (16). Thus, the approximate solution is given by

$$\begin{array}{ll}{S}_{n+1}& =S_0+\frac{{\kappa }^{1-\alpha }{h}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\left[{\chi }_1\left(t_{n+1},S_{n+1}^P\right)+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}{\chi }_1\left(t_k,S_k\right)\right],\\ E_{n+1}& =E_0+\frac{{\kappa }^{1-\alpha }{h}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\left[{\chi }_2\left(t_{n+1},E_{n+1}^P\right)+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}{\chi }_2\left(t_k,E_k\right)\right],\\ I_{n+1}& =I_0+\frac{{\kappa }^{1-\alpha }{h}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\left[{\chi }_3\left(t_{n+1},I_{n+1}^P\right)+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}{\chi }_3\left(t_k,I_k\right)\right],\\ A_{n+1}& =A_0+\frac{{\kappa }^{1-\alpha }{h}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\left[{\chi }_4\left(t_{n+1},A_{n+1}^P\right)+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}{\chi }_4\left(t_k,A_k\right)\right],\\ H_{n+1}& =H_0+\frac{{\kappa }^{1-\alpha }{h}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\left[{\chi }_5\left(t_{n+1},H_{n+1}^P\right)+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}{\chi }_5\left(t_k,H_k\right)\right],\\ R_{n+1}& =R_0+\frac{{\kappa }^{1-\alpha }{h}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\left[{\chi }_6\left(t_{n+1},R_{n+1}^P\right)+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}{\chi }_6\left(t_k,R_k\right)\right],\\ W_{n+1}& =W_0+\frac{{\kappa }^{1-\alpha }{h}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\left[{\chi }_7\left(t_{n+1},W_{n+1}^P\right)+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}{\chi }_7\left(t_k,W_k\right)\right],\end{array}$$ (36)

and

$$\begin{array}{ll}{S}_{n+1}^P& =S_0+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}{\chi }_1\left(t_k,S_k\right),\\ E_{n+1}^P& =E_0+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}{\chi }_2\left(t_k,E_k\right),\\ I_{n+1}^P& =I_0+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}{\chi }_3\left(t_k,I_k\right),\\ A_{n+1}^P& =A_0+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}{\chi }_4\left(t_k,A_k\right),\\ H_{n+1}^P& =H_0+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}{\chi }_5\left(t_k,H_k\right),\\ R_{n+1}^P& =R_0+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}{\chi }_6\left(t_k,R_k\right),\\ W_{n+1}^P& =W_0+\frac{{\kappa }^{1-\alpha }}{\mathrm{\Gamma }(\alpha )}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}{\chi }_7\left(t_k,W_k\right),\end{array}$$ (37)

where $d_{k,n+1}=\left({\left(n-k+1\right)}^{\alpha }-{\left(n-k\right)}^{\alpha }\right)$.

6. STABILITY ANALYSIS OF ITERATION METHOD

Theorem 4

Assume the hypotheses of Theorem 1 and $S_k,E_k,I_k,A_k,H_k,R_k,W_k,k=\mathrm{1,2},\dots ,n+1$ are the solutions of systems Equations (36) and (37). Then, the fractional predictor‐corrector method Equations (36),and (37) is conditional stable.

Assume that $S_0,S_{n+1},S_{n+1}^P,n=\mathrm{0,1},2,\dots ,N-1$ have perturbations ${Š}_0,{Š}_{n+1},{Š}_{n+1}^P$, respectively. Let ${\mu }_0={\max }_{0\le n\le N}\left\{\left|{Š}_0\right|+\frac{L_1{h}^{\alpha }{\kappa }^{1-\alpha }{d}_{0,n}}{\mathrm{\Gamma }\left(\alpha +1\right)}\left|{Š}_0\right|\right\}$ and ${\zeta }_0={\max }_{0\le n\le N}\left\{\left|{Š}_0\right|+\frac{L_1{h}^{\alpha }{\kappa }^{1-\alpha }{C}_{0,n}}{\mathrm{\Gamma }\left(\alpha +2\right)}\left|{Š}_0\right|\right\}$. Then,

$$S_{n+1}+{Š}_{n+1}=S_0+{Š}_0+\frac{h^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +2\right)}\left({\chi }_1\left(t_{n+1},S_{n+1}^P+{Š}_{n+1}^P\right)+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}{\chi }_1\left(t_k,S_k+{Š}_k\right)\right),$$ (38)

$$S_{n+1}^P+{Š}_{n+1}^P=S_0+{Š}_0+\frac{h^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}{\chi }_1\left(t_k,S_k+{Š}_k\right).$$ (39)

Subtracting Equations (34) and (35) from Equations (38) and (39), respectively, then

$$\begin{array}{ll}\left|{Š}_{n+1}\right|& =\left|{Š}_0+\frac{h^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +2\right)}\left({\chi }_1\left(t_{n+1},S_{n+1}^P+{Š}_{n+1}^P\right)-{\chi }_1\left(t_{n+1},S_{n+1}^P\right)+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}\left({\chi }_1\left(t_k,S_k+{Š}_k\right)-{\chi }_1\left(t_k,S_k\right)\right)\right)\right|\\ & \le \left|{Š}_0\right|+\frac{h^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +2\right)}\left(\left|{\chi }_1\left(t_{n+1},S_{n+1}^P+{Š}_{n+1}^P\right)-{\chi }_1\left(t_{n+1},S_{n+1}^P\right)\right|+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}\left|{\chi }_1\left(t_k,S_k+{Š}_k\right)-{\chi }_1\left(t_k,S_k\right)\right|\right)\end{array}$$ (40)

$$\begin{array}{ll}\left|{Š}_{n+1}^P\right|& =\left|{Š}_0+\frac{h^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}\left({\chi }_1\left(t_k,S_k+{Š}_k\right)-{\chi }_1\left(t_k,S_k\right)\right)\right|\\ & \le \left|{Š}_0\right|+\frac{h^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}\left|{\chi }_1\left(t_k,S_k+{Š}_k\right)-{\chi }_1\left(t_k,S_k\right)\right|.\end{array}$$ (41)

By Lipchitz condition, we obtain

$$\left|{Š}_{n+1}\right|\le {\zeta }_0+\frac{L_1{h}^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +2\right)}\left(\left|{Š}_{n+1}^P\right|+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}\left|{Š}_k\right|\right),$$ (42)

$$\left|{Š}_{n+1}^P\right|\le {\mu }_0+\frac{h^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}\left|{Š}_k\right|.$$ (43)

Substituting Equation (43) into Equation (42), we get

$$\begin{array}{ll}\left|{Š}_{n+1}\right|& \le {\zeta }_0+\frac{L_1{h}^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +2\right)}{\mu }_0+\frac{L_1{h}^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +2\right)}\left(\frac{h^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}{\displaystyle \sum _{k=0}^n}{d}_{k,n+1}\left|{Š}_k\right|+{\displaystyle \sum _{k=0}^n}{C}_{k,n+1}\left|{Š}_k\right|\right)\\ & ={\eta }_0+\frac{L_1{h}^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +2\right)}{\displaystyle \sum _{k=0}^n}\left(\frac{h^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}{d}_{k,n+1}+C_{k,n+1}\right)\left|{Š}_k\right|,\end{array}$$ (44)

where ${\eta }_0=\left\{{\zeta }_0+\frac{L_1{h}^{\alpha }{\kappa }^{1-\alpha }}{\mathrm{\Gamma }\left(\alpha +2\right)}{\mu }_0\right\}$. Using Lemma 1, we have

$$\left|{Š}_{n+1}\right|\le {\eta }_0+\frac{L_1{h}^{\alpha }{\kappa }^{1-\alpha }{\complement }_{\alpha }}{\mathrm{\Gamma }\left(\alpha +2\right)}{\displaystyle \sum _{k=0}^n}{\left(n-k+1\right)}^{\alpha -1}\left|{Š}_k\right|,$$ (45)

where, ${\complement }_{\alpha }=\max \left\{1,\alpha \left(\alpha +1\right)2^{1-\alpha }\right\}$, and by using Lemma 2, we get

$$\left|{Š}_{n+1}\right|\le C{\eta }_0,$$ (46)

where C is a positive constant. □

7. NUMERICAL RESULTS

7.1. Numerical simulation of the pandemic of COVID‐19 in the world

This subsection presents a computational simulation of the pandemic trend model of COVID‐19 in the world. To achieves that, we consider some of the literature's parameter values, and estimated the other parameter values, as in Table 1. According to the WHO, the birth rate in 2020 for the world was 18.077 per 1000 people, the death rate was 7.612 per 1000 people, and the total population on February 4, 2020, was $N=7610105452$. Thus, we have $\mathrm{{\rm Y}}=\frac{0.018077\times N}{365}=\mathrm{391,347}.066$ and $\delta =\frac{0.007612}{365}=20.8547\times {10}^{-6}$. The initial values of infected people, death or recovery people, and hospitalized people as stated in the report of the WHO on February 4, 2020, are $I_0=24545,R_0=907$, and $H_0=12627$. ³⁸ The initial values of A ₀,  E ₀, and W ₀ assumed as $A_0=20000,E_0=80000$, and $W_0=50000$. Since $N=S_0+E_0+I_0+A_0+H_0+R_0$, then $S_0=7609967373$.

TABLE 1.

The numerical values of parameters

Parameter	Value	Source	Parameter	Value	Source
ξ 1	2.6 ×  10−8	Tuan et al. 11	β 1	0.02	Tuan et al. 11
ξ 2	1 ×  10−9	Tuan et al. 11	β 2	0.009	Tuan et al. 11
σ	0.0001	Estimated	β 3	0.0074	Estimated
ψ	0.00023	Estimated	ρ	0.0075	Estimated
δ	20.85 ×  10−6	Tuan et al. 11	θ 1	1 ×  10−6	Tuan et al. 11
ν	0.000058	Tuan et al. 11	θ 2	1 ×  10−6	Tuan et al. 11
η	0.075	Tuan et al. 11	λ	0.01	Tuan et al. 11

Concerning the parameter values in Table 1, the basic reproduction number ${\mathcal{R}}_0>1$. That means that the pandemic will spread, and the equilibrium point of the model Equation (11) is positive. $P_0^{\ast }=(2.02\times {10}^7,4.957\times {10}^9,7.43\times {10}^5,2.66\times {10}^6,2.7\times {10}^6,1.37\times {10}^{10},3.4\times {10}^2)$. All the approximate solutions are computed by using Wolfram Mathematica software with $\kappa =0.99$. The graphical solutions of fractional system Equation (11) in the interval [0, 250] have been described in Figures 1, 2, 3, where the unit of time is days. Figures 1, 2, 3 show that the results of the model converge to their equilibrium point for different fractional‐order derivatives and stable at that points. These figures indicate that the obtained plots have the same behavior pattern for different values of $\alpha =1,0.9,0.8,0.7,0.6$.

FIGURE 1.

Graphical approximate solutions of S(t) and E(t) for different values of $\alpha =1,0.9,0.8,0.7,0.6$ [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2.

Graphical approximate solutions of I(t) and A(t) for different values of $\alpha =1,0.9,0.8,0.7,0.6$ [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 3.

Graphical approximate solutions of H(t),  R(t), and W(t) for different values of $\alpha =1,0.9,0.8,0.7,0.6$ [Colour figure can be viewed at wileyonlinelibrary.com]

7.2. Effect of parameters on the epidemic spread

The parameter values of the model play a significant role influences the spread of the epidemic. So, the single most effective way to restrict the spread of a disease is to create quarantine to decrease the mobility of individuals. If we choose the same parameter values and the same initial values of the classical model (see Fatima et al. ¹ ) as in Table 2, then the basic reproduction number ${\mathcal{R}}_0<1$ where $\mathrm{{\rm Y}}\mathrm{=}0.00181$. Thus, we get the results shown in Figures 4, 5, 6. Comparing the results with Fatima et al, ¹ we note that the behavior of graphic solutions is the same. Figure 4, 5, 6 show that all variables in the pandemic model will be decreased and hit zero, indicating the system's stability.

TABLE 2.

The numerical values of the parameters

Parameter	Value	Parameter	Value
ξ 1	0.0026	β 1	0.014
ξ 2	0.001	β 2	0.004
σ	0.04	β 3	0.05
ψ	0.023	ρ	0.045
δ	0.09	θ 1	0.001
ν	0.022	θ 2	0.008
η	0.065	λ	0.033

FIGURE 4.

Approximate solutions of S and E with parameters of Table 2 for different values of $\alpha =\mathrm{1,0}.\mathrm{9,0}.\mathrm{8,0}.\mathrm{7,0}.6$ [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 5.

Approximate solutions of I and A with parameters of Table 2 for different values of $\alpha =\mathrm{1,0}.\mathrm{9,0}.\mathrm{8,0}.\mathrm{7,0}.6$ [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 6.

Approximate solutions of H, R, W with parameters of Table 2 for different values of $\alpha =\mathrm{1,0}.\mathrm{9,0}.\mathrm{8,0}.\mathrm{7,0}.6$ [Colour figure can be viewed at wileyonlinelibrary.com]

8. CONCLUSION

In this paper, modeling and studied the pandemic trend of COVID‐19 has been presented with fractional‐order derivative. Using the generation matrix method, the basic reproduction number is calculated that located whether the disease would persist or disappears from the population. The equilibrium points for this system are calculated, where the graphical solutions show that the results of the model converge to their equilibrium points. Based on the derivatives of the basic reproduction number, the sensitivity of the parameters was analyzed. The existence and uniqueness of the solution to the proposed model have been proven using the fixed‐point theorem and by helping of the Arzela–Ascoli theorem. The fractional proposed model is solved using the predictor–corrector method in the sense of Caputo derivative. Using some essential lemmas, we proved that this method is conditionally stable. The results indicate that the disease will continue where that the basic reproduction number ${\mathcal{R}}_0\mathrm{>}\mathrm{1}$. We selected the same parameter values and same initial conditions in the classical model ¹ for comparing the results of the fractional model with the classical model. We noted that both models have the same behavior. Moreover, parameters played a very significant role in limiting disease outbreaks.

CONFLICT OF INTEREST

This work does not have any conflicts of interest

Agarwal P, Ramadan MA, Rageh AAM, Hadhoud AR. A fractional‐order mathematical model for analyzing the pandemic trend of COVID‐19. Math Meth Appl Sci. 2022;45(8):4625–4642. doi: 10.1002/mma.8057