Mathematical Modeling and Forecasting of COVID-19 in Saudi Arabia under Fractal-Fractional Derivative in Caputo Sense with Power-Law

: This manuscript is devoted to investigating a fractional-order mathematical model of COVID-19. The corresponding derivative is taken in Caputo sense with power-law of fractional order µ and fractal dimension χ . We give some detailed analysis on the existence and uniqueness of the solution to the proposed problem. Furthermore, some results regarding basic reproduction number and stability are given. For the proposed theoretical analysis, we use ﬁxed point theory while for numerical analysis fractional Adams–Bashforth iterative techniques are utilized. Using our numerical scheme is veriﬁed by using some real values of the parameters to plot the approximate solution to the considered model. Graphical presentations corresponding to different values of fractional order and fractal dimensions are given. Moreover, we provide some information regarding the real data of Saudi Arabia from 1 March 2020 till 22 April 2021, then calculated the fatality rates by utilizing the SPSS, Eviews and Expert Modeler procedure. We also built forecasts of infection for the period 23 April 2021 to 30 May 2021, with 95% conﬁdence.


Introduction
Recently the COVID-19 pandemic has greatly affected the whole world. The mentioned disease was originated in the end of 2019 in Wuhan city of China. Later on, the infection was transmitted throughout the whole globe in the next few months. WHO announced that it was a pandemic in the whole world. According to the reports published by WHO, nearly fifty million people have gotten infected around the globe in which more than three million people died. Many countries have implemented strict lockdown in their community advised the public to keep social distance. These necessary measures have produced some positive impact on the control of COVID-19 in various countries. The concerned infection has greatly destroyed the economical situation of various countries.
Here we remark that some countries have now succeeded in creating COVID-19 vaccines including USA, UK, Germany, China, etc.

Compartments and Parameters Description
S 1 (t) Susceptible class E 1 (t) Exposed class I 1 (t) Infected class A 1 (t) In-transmutable infected people showing no clinical symptoms R 1 (t) Recovered people M 1 (t)  in the first identified case Λ The birth rate υ The natural death ratē The rate of Transmutable infected people into M 1 The rate of in-transmutable infected people into M 1 ν The rate of virus leaving in M 1 for the M 1 class 1/ν The total life period of COVID-19 virus η The disease transmission coefficient ψ The transmittable multiple of A 1 to I 1 ( 0 ≤ψ ≤ 1) η ι Infected people due to an interactivity M 1 with S 1 η ω The transmission rate from M 1 to S 1 ι Transmission rate of the exposed persons the infection to I 1 after the incubation period ρ Transmission rate of the exposed persons the infection to A 1 after the incubation period Θ In-transmutable infection 1/χ The infectious period of transmutable I 1 persons 1/χ a The infectious period of in-transmutable A 1 persons The susceptible people who get the infection after an effective contact with the people in I 1 and A 1 at the rate ofη (I 1 +ψA 1 )S 1 N . Now, we shall reconsider the model (1) by including fractional order derivative 0 < µ ≤ 1 and fractal dimension 0 < χ ≤ 1 as follows where FFP D µ,χ 0,t is the fractal-fractional derivative of order 0 < µ ≤ 1 and fractal dimension 0 < χ ≤ 1 in Caputo sense with power law. We must construct a model (3) by means of the derivative of fractal fractional order in Caputo sense with power law as it provides an extremely realistic result and possesses a greater degree of freedom than integer-order. Precisely, we consider model (3) possesses fractional-order µ and fractal dimension χ describing the situation that lies between two integer values. The result will be accomplished by having the whole density of every compartment converging faster at a low order.

Foundations
Let ∆ = [0, T] (T < ∞), and U = C(∆, R 6 ) is a Banach space equipped with the norm given by Then, the fractalfractional derivative of Θ of order µ in the frame of Riemann-Liouville and Caputo with the power law are supplied by

Definition 2 ([21]
). Let u(t) is continuous in (a, b). Then the fractal-fractional integral of u with order γ in the definition Riemann-Liouville with power law is given by Lemma 1 ( [21]). If f is continuous on (a, b), then the following fractal FDE has a unique solution

Qualitative Analysis of the Proposed COVID-19 Model
In this section, we discuss the positivity and equilibrium analysis of the model (3). Then we investigate the uniqueness, existence, and Hyers-Ulam-Rassias stability results of the proposed model.

Positivity of the Model (3)
For the positivity of the model solution, let us structure the following set: Theorem 1. A solution z(t) of the given fractal fractional model (3) exists and belongs to R 6 + . Moreover, the solution will be non-negative.
Proof. Form the model (3), we conclude that Consequently, we infer that the solution will remain in R 6 + for all t ≥ 0. The total dynamics of the individuals can be acquired by the first five equations of the model (3) By replacing Rt D µ 0,t with C D µ 0,t and applying the Laplace transform, we get where E µ,ν is called the Mittag-Leffler function. Taking into account the fact that E µ,ν has asymptotic behavior [10]; therefore, we get lim t→∞N (t) ≤ Λ υ . The feasible region for model (3) is structured as:
In addition, by utilizing the next generation approach we get the following equation for the basic reproduction number: The following theorem provides us with the necessary part.
For the EE of the model (3), we denote it by D * , and D * = (S * 1 , E * 1 , which satisfies the following equation The coefficients in (7) are m 1 =νk 1 k 2 k 3 , and

Existence and Uniqueness Results
To begin with, we will express the differentiation in the model (3) as integrals, that is Due to the integrals in the model (8) being differentiable, we can formulate the model (8) as where By replacing RL D µ 0,t with C D µ 0,t then applying the initial conditions and fractional integral operator, we turn model (9) into the following integral equations: To prove the qualitative properties of the solution for model (3), we make use of the fixed point technique and the Picard-Lindel'f approach. First, we reformulate the model (3) which takes the form: where In view of Lemma 1, the system (11) gives Theorem 3 (Existence of unique solution). Assume that the assumption (12) holds. Then the system (11) has a unique solution if P := χB(µ,χ) Proof. Consider the Picard operator Π : U → U defined by and set sup σ∈∆ K(σ, 0) = K 0 . It should be noted that the solution of the system (11) is bounded, i.e., Now, using Picard operator (13) with given any Θ 1 , Θ 2 ∈ U , we obtain Thus Π is a contraction, and hence model (11) has a unique solution due to Banach contraction principle [23].

Stability Results
In this part, we discuss the stability of Ulam-Hyers and Ulam-Hyers-Rassias for the considered model (11). Furthermore, since stability is essential for approximate solution, we intend to use nonlinear functional analysis on these types of stability for the given model.
Proof. In view of Theorem 3, the system (11) has the unique solution Θ ∈ U , that is From (15) and keeping in mind (16), we have Hence where C K,ϕ := It is clear that when ϕ(t) = 1 in (17), the Ulam-Hyers stability result is obtained.

Numerical Scheme
In this section, we present a numerical approach for the solution of the model (3) by relying upon the procedure described in [21,24]. By using the systems (8)-(10) at the point t κ+1 , we obtain Then we approximate the integrals obtained in (18) to On [t , t +1 ], we approximate the expression σ χ−1 K i (σ, S 1 , E 1 , I 1 , A 1 , R 1 , M 1 ) where i = 1, 2, 3, 4, 5, 6 utilizing the Lagrangian piecewise interpolation as where i = 1, 2, . . . , 6. Thus, (19) and (20) give After simplifying the integrals in (21), we get the numerical solutions for the COVID-19 epidemic model (3) under the fractal fractional derivative in the Caputo sense with power law as follows: where and is the step size.

Statistical Analysis and Forecasts
This part is dedicated in providing statistical data for the COVID-19 pandemic in Saudi Arabia (see [25]), on it accordingly, we have computed the future predictions of the confirmed cases and deaths by applying the Expert Modeler procedure and SPSS software. A brief discussion of the redobtained outcomes is presented and supported by figures and statistical tables.

Results and Discussion
The most recent statistics about the COVID-19 pandemic and the total number of affirmed cases and deaths in the territory of Saudi Arabia for the period from 1 March 2020 to 22 April 2021 are displayed in Figures 1 and 2. The numbers likewise express that there is a fast and ceaseless expansion in the number of new cases, particularly in the months of May, June, July, and August 2020. The affirmed cases that came to notice during these four months were in excess of 293,419 cases, an average of 0.72 of the all affirmed cases till the end of our study, which added up to 408,078 cases, and there were 3713 deaths-rates of 0.54 from the entire death number. We have seen that it is on the ascent, as the affirmed cases expanded from 1563 in March with an average of 50 cases in a day, then, at that point expanded during April to reach 19,839 with an average of 661 cases. Nonetheless, the number of affirmed cases during May came to 61,982, with an average of 2000 cases. Then, at that point, the quantity of the affirmed cases expanded to reach 103,052 cases toward the end of June with an average of 3435 cases. We additionally saw that during June the cases expanded quickly, which is a much more noteworthy number than the cases during the period 1 March 2020 until 30 May 2020.
We likewise observed that the affirmed cases are persistently diminishing during the period of July to reach 87,783 with an average of 1243, then, at that point, it diminished in August to reach 40,602 with an average 1028 until the end of December 2020 with 5473 cases. In 2021, affirmed recorded cases reached 5212 during the period of January, then, at that point expanded altogether during the long stretches of February and March until reaching 18,656 cases for the 22 days of April.
On the other hand, there has been an increase in the number of deaths from this virus, as nine deaths were recorded for the month of March 2020, then it reached 1243 during the month of July 2020, and then decreased from the months August to December 2020. In 2021, deaths were recorded during January 155, then decreased during February, and then increased during March and April. Table 1 and Figures 3 and 4 illustrate that.
The Expert modeler is an ad-hoc procedure of time series models applied by SPSS for forecasting. It tries to construct a convenient predictive model for one or more series of dependent variables automatically. If there are independent variables regarding the dependent variable, the Expert Modeler Procedure automatically selects only those independent variables that are statistically significant. By default, the Expert Modeler Procedure considers both exponential smoothing and ARIMA models. However, one can limit the Expert Modeler to only search for ARIMA models or to the only search for exponential smoothing models. Furthermore, it is easy to perform and helps in quickly identifying the best models that achieve the required features, making it easier to obtain their forecasts in record time. For more details see [28].
Time series models play a critical role in predicting the actions of phenomena and variables over time and their impact climate science, economics, finance, epidemiology, health engineering, and other different sciences. Many researchers have lately used it to model and forecast future trends in the behavior of many diseases and epidemics. Most of these models include a procedure of four essential steps: initially, identifying the model, secondly, estimating anonymous parameters, thirdly, diagnosis and finally, forecast [29][30][31][32][33]. There are numerous kinds of models for time series that are fitting in predicting, for example, AR, MA, ARMA, ARIMA, ARCH, GARCH. In this manner, the nature and trial of the information for the two series under examination and every one of the hypotheses connected to them (and the consistency of the time series) were confirmed to be utilized in the forecast process. In this regard, there are many transformations that must be used to convert the original data of the non-stationary time series into a stationary time series to be used in the prediction process, such as natural log and differences of the first and second degree. Here, we processed the original data then took the first differences for two series to remove the effect of the general trend, and both series become stable, in order for the model to be valid for predicting deaths and confirmed cases, as in Figures 5 and 6. So they can be utilized in the expectation process. The statistical analysis software (SPSS) version 23 and the Expert Modeler Procedure were utilized to predict new everyday affirmed cases and deaths at a certainty span (95%) for COVID-19 in Saudi Arabia for the period from  22 April 2021 to 31 May 2021 as in Figures 7 and 8 and Table 5. Table 2 shows the value of the determination coefficient R-squared = 0.748 and 0.981 which are appropriate, which means the model quality used for prediction, and it signifies the data optimally. In addition, there is no problem in the model through the value of the Ljung-Box Q(18) Statistic. A test of the randomness of the residual errors in the model is necessary to be random and must be the level of statistical significance (Sig.) greater than (5%) in order for the data to be distributed randomly. It means that the data follows a random distribution. In this regard, we find the Ljung-Box statistics 6.951 and 31.240 and the statistically significant Sig. = 0.974 and 0.270, which is greater than 0.05; this indicates that the data follows a random distribution.
To examine the predictive capability of the model, the Eviews9 program was used and the least-squares method was applied to estimate a linear model by taking the estimated values of the model as an independent variable and the actual values as a dependent variable. So the closer the estimated parameter is to one, the more the estimated values are close to the actual values. The output of the analysis is observed in Tables 3 and 4 in which the estimated parameters 0.990706 and 0.983151 are close to one. This means the convergence of the estimated values from the actual values, in addition to the model quality in the estimate, and there is a statistically significant (prob. = 0.000) that is less than the approved level of significance, which is statistically significant (α = 0.05).

Simulations and Discussion
For the numerical simulations to study the behavior of the susceptible, infectious, treated, and recovered population related to this pandemic that occurred in Saudi Arabia during March 2020 to April 2021, we consider the parameters as in the table below.

Compartments and Parameters
Numerical Values Here some are estimated data and some are fitted with the help of available real data. In addition, the pandemic model is simulated under the fractal fractional order case using a numerical scheme as structured with total population N = 35.266155 Million.
In Figure 9, we see that at different fractal fractional order the population of susceptible class is decreasing with a different decay curve. The smaller the values of orders, the faster the decay process and vice versa. In addition, in Figures 10-13, the populations of various compartments are growing. The growth rate is faster at larger values of fractional-fractal order and vice versa. In Figure 14, we see that the population of numbers of corona virus is also increasing with high speed as infection has increased in the last two months. From these graphical presentations, we conclude that fractal fractional order derivatives explain population dynamics of model of infectious disease more frequently and are easier to understand.      In Figures 15 and 16, we compare our simulated results in case of infected reported and deaths of KSA as given in Table 5

Conclusions
Using fractal fractional-order derivative, we have successfully established theoretical and computational analysis for a COVID-19 mathematical model. Upon using fixed point approach and Adam-Bashforth method we have achieved the required results. We have presented the numerical results graphically by using various fractional order derivatives and different values of fractal dimension. Some statistical analysis has been provided by taking some real data about Saudi Arabia from 1 March 2020 till 22 April 2021; we have also calculated the fatality rates by using the SPSS, Eviews, and Expert Modeler Procedure, then built forecasts of infection for the period 23 April 2021 to 30 May 2021. The entire investigation of this article revealed that control of the dynamic transmission rate is vital for stopping the transmission of the spreading epidemic.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data were used to support this study.