This study aims to structure and evaluate a new COVID19 model which predicts vaccination effect in the Kingdom of Saudi Arabia (KSA) under AtanganaBaleanuCaputo (ABC) fractional derivatives. On the statistical aspect, we analyze the collected statistical data of fully vaccinated people from June 01, 2021, to February 15, 2022. Then we apply the Eviews program to find the best model for predicting the vaccination against this pandemic, based on daily series data from February 16, 2022, to April 15, 2022. The results of data analysis show that the appropriate model is autoregressive integrated moving average ARIMA (1, 1, 2), and hence, a forecast about the evolution of theCOVID19 vaccination in 60 days is presented. The theoretical aspect provides equilibrium points, reproduction number
Coronaviruses have three branches known as alpha, beta, and gamma. Recently, various strains of SARSCoV2 have emerged, including the most destructive and most dangerous delta variant, SARSCoV2 [
Despite the different nonpharmaceutical control systems against COVID19, a portion of the vaccines that have acquired crisis use approval (EUA) by the United States Centers for Disease Control and Prevention (CDC) are the PfizerBioNTech with 95% efficacy, the Moderna immunization with 94.5% viability, and the Janssen vaccine with 67% adequacy, and many others, see [
KSA began vaccinating people against the Coronavirus on December 17, 2020. On November 03, 2021, the use of the Pfizer vaccine was approved for the age groups 5–11 years. In the early stages of the vaccine, citizens and residents over 65 were targeted, which ended on February 18, 2021, whereas the second phase was launched on February 18, 2021, it targeted citizens and residents over 50 years old. The third phase targeted all citizens and residents wishing to receive the vaccine.
On the other hand, time series models are employed in the estimate process of variable behavior and other phenomena, as well as their future trends in diseases, epidemiology, climate sciences, economics, management, and other sciences. Lately, mathematical and statistical modeling have been used to forecast the behavior and patterns of some epidemics and diseases. The prediction method for these models includes four fundamental procedures, such as determining the model, estimation of unknown parameters, diagnostic process, and prediction process. The models (MA, AR, ARCH, ARMA, GARCH, ARIMA) are among the most commonly used time series models for forecasting [
In this context, the processes of making a mathematical model of a problem, interpreting the solution, validating the model, then making the model ready for utilization are not processes that can be conquered directly. Particularly these days, large numbers of the problems are complicated, nonlinear, have memory impact, or possess a stochastic construction; therefore, modeling methods and solutions specific to these problems must be developed.
Fractional calculus has shown gigantic improvement in applications to various realworld problems in different fields such as continuum mechanics, electromagnetic theory, and biological mathematics. Fractional calculus has become a substantial mathematical tool for the investigation of nonlinear derivative problems, see [
The literature shows that very few works available on COVID19 models are in the form of ABC fractional derivatives, even though they describe the dynamic behavior accurately compared to the classical derivatives. To our knowledge, there is no available literature on the COVID19 fractional model with an expectation of the vaccination effect on KSA. Therefore, this is the main motivation behind our work. Also, the comparison of the real data with different fractional order simulations is one of our unique aims in this work. In addition, we extend and generalize the model studied by Yavuz et al. [
This work is organized as follows.
In this section, we discuss some empirical results. First, we analyze and present the descriptive data of the vaccinated persons in KSA. Then we predict the number of people who will be fully vaccinated during the next period.
The latest statistics about the vaccination of people against the spread of the Coronavirus epidemic are collected from the official data published by the Saudi Ministry of Health as the fully vaccinated people reached 144,057 during the period January 06, 2021, to June 27, 2021, with an average of 8,401 per day. Then, data is collected on the number of fully vaccinated people daily during the period June 28, 2021, to February 15, 2022.
Months  Vaccinated people  Mean of vaccinated people 

06 Jan to 27 Jun21  1,445,087  8,401 
From 28 Jun21  142,494  47,498 
Jul21  6,493,739  209,476 
Aug21  6,833,828  220,446 
Sep21  3,807,265  126,909 
Oct21  2,789,720  89,991 
Nov21  955,376  30,819 
Dec21  713,370  23,012 
Jan22  519,054  16,744 
Until 15 Feb22  264,633  17,642 
Total  23,964,566  102,852 
Months  New doses  Mean of new doses 

Jan21  57865  1867 
Feb21  114810  4100 
March21  3475774  112122 
Apr21  4911419  163714 
May21  4864301  156913 
Jun21  3672578  122419 
Jul21  8986886  289899 
Aug21  10265432  331143 
Sep21  4976527  165884 
Oct21  3611147  116489 
Nov21  1590443  53015 
Dec21  3406053  109873 
Jan22  5697591  183793 
Until 15 Feb22  2333438  155563 
Total  57964264  248773 
In this portion, we are collecting data on the fully vaccinated people in KSA for the period June 01, 2021 to February 15, 2022. Moreover, the stability tests of these data are examined to use the prediction process as unit root tests and estimation of coefficients (ACF & APCF). PhillipsPeron’s and DickeyFuller’s tests show that the data series is unstable at the level, which means that there is a general trend in the series. As shown in
Test critical values  

Test statistic  tstatistic  Prob.  0.01  0.05  0.10 
Augmented DickeyFuller  −6.293644  0.0000  −3.459362  −2.874200  −2.573594 
PhillipsPerron  −25.09233  0.0000  −3.458594  −2.873863  −2.573413 
We apply ARIMA models for forecasting through the Eviews program on the data series about the number of vaccinated people completely in KSA. As well, we estimate ARIMA models by using the Ordinary Least Squares method, as shown in
MODELS  SIGMASQ  Adjusted R2  AIC  SC 

ARIMA(1, 1, 1)  0.23  0.27  1.44  1.50 
ARIMA(1, 1, 2)  0.23  0.28  1.43  1.49 
ARIMA(1, 1, 7)  0.26  0.18  1.55  1.61 
ARIMA(1, 1, 0)  0.29  0.11  1.63  1.67 
ARIMA(0, 1, 1)  0.24  0.26  1.44  1.49 
ARIMA(0, 1, 7)  0.31  0.05  1.71  1.74 
ARIMA(2, 1, 0)  0.31  0.02  1.72  1.76 
ARIMA(2, 1, 1)  0.23  0.27  1.43  1.49 
ARIMA(2, 1, 7)  0.30  0.07  1.68  1.74 
ARIMA(5, 1, 1)  0.23  0.27  1.44  1.50 
Moreover, we check the ARIMA (1, 1, 2) model by testing residuals and the shape of the autocorrelation and partial autocorrelation coefficients. It follows from
We see through the chosen model a decrease in the number of people who will be fully dosed to (927164) during the predictive period from February 16, 2022, to April 14, 2022. For more details, see
Finally, we estimate a linear model using the leastsquares method to test the predictive ability of the model. Indeed, we take the actual values as a dependent variable and the estimated values as an independent variable. We conclude that the closer the estimated parameter to one, the more closely the estimated values are to the actual values. Through the results of
Dependent variable: Completed vaccinations  

Method: Least squares  
Included observations: 233  
Variable  Coefficient  Std. error  tstatistic  Prob. 
PREDICTED  0.955688  0.018192  52.53  0.0000 
Rsquared  0.847083  
Adjusted Rsquared  0.847083  
F  1287.919  
Prob.  0.000000  
DurbinWatson stat  1.870376 
Date  Forecast  UCL  LCL  Date  Forecast  UCL  LCL 

2/16/2022  11344  26416  3832  3/18/2022  15432  62461  1611 
2/17/2022  15750  38665  4865  3/19/2022  15504  63476  1568 
2/18/2022  14055  34898  4257  3/20/2022  15577  64492  1526 
2/19/2022  13295  33778  3867  3/21/2022  15650  65508  1485 
2/20/2022  13558  35452  3745  3/22/2022  15724  66524  1446 
2/21/2022  13769  36936  3629  3/23/2022  15797  67540  1408 
2/22/2022  13812  37925  3485  3/24/2022  15872  68557  1372 
2/23/2022  13851  38903  3348  3/25/2022  15946  69575  1337 
2/24/2022  13918  39957  3223  3/26/2022  16021  70592  1303 
2/25/2022  13987  41014  3107  3/27/2022  16096  71610  1270 
2/26/2022  14053  42053  2997  3/28/2022  16172  72629  1238 
2/27/2022  14118  43086  2892  3/29/2022  16248  73648  1207 
2/28/2022  14184  44119  2792  3/30/2022  16324  74668  1178 
3/1/2022  14251  45149  2698  3/31/2022  16400  75688  1149 
3/2/2022  14318  46176  2609  4/1/2022  16477  76709  1121 
3/3/2022  14385  47201  2524  4/2/2022  16555  77731  1094 
3/4/2022  14453  48224  2442  4/3/2022  16632  78753  1067 
3/5/2022  14520  49246  2365  4/4/2022  16711  79775  1042 
3/6/2022  14589  50267  2291  4/5/2022  16789  80799  1017 
3/7/2022  14657  51286  2220  4/6/2022  16868  81822  993 
3/8/2022  14726  52304  2153  4/7/2022  16947  82847  970 
3/9/2022  14795  53321  2088  4/8/2022  17026  83872  947 
3/10/2022  14864  54338  2026  4/9/2022  17106  84898  926 
3/11/2022  14934  55354  1967  4/10/2022  17187  85924  904 
3/12/2022  15004  56370  1910  4/11/2022  17267  86951  884 
3/13/2022  15075  57386  1855  4/12/2022  17348  86951  863 
3/14/2022  15146  58401  1802  4/13/2022  17430  89007  844 
3/15/2022  15217  59416  1752  4/14/2022  17512  90035  825 
3/16/2022  15288  60431  1703  4/15/2022  17594  91065  807 
3/17/2022  15360  61446  1656 
The normalization function
Yavuz et al. [
With initial conditions
In the present work, we make extend the model
With initial conditions
Here S(ϰ) is the class of Susceptible Individuals, E(ϰ) is the class of Exposed Individuals, I(ϰ) is the class of Infected individuals, V(ϰ) is the class of Individuals Vaccinated, R(ϰ) is the class of Recovered Individuals. The model parameter values and source are given in
Parameter  Description  Numerical value  Source 

Rate of transmission from all individuals to individuals sensitive to the disease  0.999  Assumed  
Rate of transmission from susceptible individuals to individuals who are exposed to the disease  0.002  [ 

Rate of exposure and those who have not been exposed to the disease are passed on to individuals to be vaccinated  0.4  Estimated  
Rate of vaccinated individuals and their likelihood of contracting the disease due to vaccine failure  0.0001  Estimated  
Transmission rate from symptomatic individuals to the active patient portion  0.008  [ 

The rate of people recovering without symptoms and moving to the recovery part  0.005  Assumed  
Deaths rate among active patients  0.08  [ 

Recovery rate depending on the disease  0.012  [ 

Natural deaths rate in all compartments  0.009  [ 

Initial susceptible population  9,317,558  Assumed  
Initial exposed population  0  Assumed  
Initial infected population  562,300  Assumed  
Initial vaccinated population  24,000,000  Estimated  
Initial recovered population  580,000  Estimated 
Total population at time ϰ, denoted by N(ϰ) and given by N(ϰ) = S(ϰ) + E(ϰ) + I(ϰ) + V(ϰ) + R(ϰ).
In this portion, we show that the nonnegative domain
The norm and all assumptions of the classical results are valid. It follows that:
For
This shows that if
It follows for the whole population that
After applying the Laplace transform, we have
In this part, we will find the equilibrium points of the COVID19 model. By equating each equation of model
For
Hence, the disease free equilibrium (DFE) and endemic equilibrium (EE) are given by the following theorem.
By using the model
For details, see [
Stability analysis of the equilibrium points is clarified in [
Here we comment that
In this section, we apply the PicardLindel method and the Laplace transform to investigate the existence and uniqueness of solution for preventive and curative to fractional COVID19 disease model.
It follows from Theorem 3 in [
Applying the inverse Laplace transform give us
Now, we have
Let
Hence
By using Theorem 5.1, our model is equivalent to
The iterative scheme of the model
Taking the limit as
In this portion, we apply the PicardLindel method to investigate the existence of solution for the fractional COVID19 disease model, which its mathematical represente are presented by:
For sake of simpilicity, we define the functions
The model described in
By applying the operator
The kernels in
Repeating the same procedure as in
Now, we denote the difference between successive components by
Taking into account that
Then, we take the norm on both sides of
Next, we shall prove the main theorem based on the above results.
Consequently, the sequences in
Now, we show that the functions in
Then we show that the terms in
Repeating the procedure above, we get
As
To prove the uniqueness result, we assume that the model
It is clear that
This is satisfied by the hypothesis
Similarly, we obtain
This section gives the numerical solution of the COVID19 model
By applying the operator
For
Over
Set
Finally,
In an analogous manner for the remainder of the equations of the model
The biological parameters estimated from the actual data reported in KSA for the period June 01, 2021 to February 15, 2022, and classified in
We see that corresponding to different fractional order, the susceptible class is decreasing with various scenario in
In this research work, we have updated a COVID19 model to a fractional order derivative of generalized type. On the statistical aspect, we have used some statistical analysis to collect data on vaccination in KSA for 300 days, and then the concerned statistical analysis has been shown. Consequently, the forecast about the evolution of the COVID19 vaccination in 60 days has been presented. We have found, through the ARIMA(1, 1, 2) model, a decrease in the number of people who have been given full doses with (927164), and they constitute 2.6% of the total population in KSA. Data analysis showed that 67.81% of the population had been fully vaccinated during the study period. On the analytical aspect, we have established some adequate results for the existence and uniqueness of the solution through fixed point techniques. The respective results are important because a mathematical formulation should be preferentially checked for their existence. Finally, in terms of numerical aspects, we have extended the AdamBashforth method for the considered model to derive a scheme for numerical analysis. Moreover, we have then used the real data for parameters and some initial data of KSA to see the transmission dynamics of COVID19 with the vaccinated class. Finally, the concerned numerical simulations have been compared with the exact real available date given in
The authors would like to thank Imam Mohammad Ibn Saud Islamic University (IMSIU), for funding this research work.