# A Study on the Fractal-Fractional Epidemic Probability-Based Model of SARS-CoV-2 Virus along with the Taylor Operational Matrix Method for Its Caputo Version

Authors: Shahram Rezapour; Sina Etemad; İbrahim Avcı; Hijaz Ahmad; Azhar Hussain
Journal: Journal of Function Spaces
 (2022)

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## Abstract

SARS-CoV-2 is a strain of the large coronavirus family that has led to COVID-19 disease. The virus has been one of the deadliest known viruses in the world to date. Rapid mutations and the creation of new strains cause researchers to focus on the dynamic behaviors of the virus and to analyze it accurately through clinical research and mathematical models. In this paper, from the point of view of mathematical modeling, we intend to focus on the dynamic behavior of the system and examine its analytical and numerical aspects in two different structures. In other words, by recalling newly formulated hybrid fractional-fractal operators, we present a fractal-fractional probability-based model of SARS-CoV-2 virus for the first time and extract its equivalent compact fractal-fractional IVP to investigate its existence and stability criteria. A type of special admissible contractions will help us in this regard. Moreover, based on the source data, we simulate our system according to algorithms derived by Adams-Bashforth method and explain the effects of variation of the dimension of fractal and fractional order on dynamics of solutions. Finally, we transform our fractal-fractional model into a Caputo probability-based model of SARS-CoV-2 to derive solutions via the operational matrix method under Taylor’s basis. The numerical simulations show close behaviors for both of models.


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## Body

## 1. Introduction

From birth to death, humans are always at risk for a variety of diseases; the source of these infectious diseases is mainly microorganisms such as parasites, fungi, viruses, and bacteria. Over the centuries, various epidemics have killed millions of people everywhere on the planet and caused great loss of life and property to families and governments. Recently, in late 2019, the international community contracted a new type of viral respiratory disease that was reported to have originated in Wuhan, China. For the sake of rapid spread of this unknown disease in Wuhan, scientists have used a variety of terms to describe the viral cause of the disease. According to the standard classifications in virology and considering its geographical location, it was first temporarily named Wuhan coronavirus and then the new coronavirus 2019 (2019-nCov). Finally, in 2020, an international committee from the World Health Organization (WHO), which works to classify viruses, used the official title SARS-CoV-2, which interprets the severe acute respiratory syndrome of coronavirus 2 [1]; and later, in order not to be confused with the SARS virus, the committee used the abbreviated title COVID-19 [2].Extensive medical research was conducted worldwide to identify the nature and spread of the virus, and on January 20, human-to-human transmission of the virus was proved [3]. The virus has also been shown to be transmitted via respiratory droplets such as coughing and sneezing and even talking indoors without ventilation [4, 5]. In addition, subsequent studies have shown that the best site for infection is the nasal cavity, through which it gradually and immediately enters the lungs and infects it [6]. However, other studies have shown that some wild animals, such as bats, mice, rabbits, and mink, can also transmit the SARS-CoV-2 virus to humans [7]. In these two years, no part of the human environments has been spared from the virus, even the most remote islands. As of March 10, 2022, more than 450 million people have been infected with COVID-19, of which more than six million have died, based on the approved reports of the World Health Organization [8]. In some cases, people with the SARS-CoV-2 virus have severe clinical symptoms and are hospitalized, but in most cases, patients with the SARS-CoV-2 virus do not need to be admitted to treatment centers and are treated with antiviral drugs such as remdesivir [9].Due to the high rate of transmission of the virus and the development of its various strains, there was a need for definitive treatment to control the epidemic. Therefore, knowledge of the pathobiology of the SARS-CoV-2 virus was essential. Because vaccines are always an important tool in the fight against all epidemics, we have also seen extensive efforts to produce safe and effective vaccines for the SARS-CoV-2 virus by large pharmaceutical companies. Of course, it should be noted that in addition to mass vaccination to eradicate the virus completely, it is necessary for human societies to continue to maintain social distance and use masks indoors. It is still unknown whether the vaccines are effective in killing the disease.In this regard, to accurately analyze the prevalence of the SARS-CoV-2 virus worldwide and predict its upward or downward trends, researchers turned to simulating the dynamics of the virus by mathematical models. Of course, it should be noted that in recent decades, mathematical models have always been helpful in studying the dynamics of various types of diseases and engineering processes, and through various modeling, scientists and researchers have been able to achieve their study goals. In this direction, fractional mathematical models are among the most widely used methods in the field of accurate analysis and evaluation of data. Known fractional operators such as Caputo, Atangana-Baleanu, and Caputo-Fabrizio fractional derivatives are efficient mathematical tools for defining and designing mathematical systems, so that their role can be clearly observed in newly published papers, for example, the modelling of anthrax in animals [10], genetic regulatory networks [11], mumps virus [12], Zika virus [13], mosaic disease [14], computer viruses [15], thermostat control [16], pantograph equation [17], Q-fever [18], hybrid equation of p-Laplacian operators [19], geographical models [20, 21], codynamics of COVID-19 and diabetes [22], chemical compounds such as methylpropane [23], and immunogenic tumor [24]. Also, due to difficulties of solving fractional differential equations analytically, developing efficient numerical methods with different fractional operators for such equations becomes an important focus for researchers; for example, in [25], fractional derivative generalized Atangana-Baleanu differentiability has been implemented to solve fuzzy fractional differential equations. Also, in 2021, Erturk et al. [26] used fractional calculus theory to investigate the motion of a beam on an internally bent nanowire. In [27], Jajarmi et al. presented a new and general fractional formulation to investigate the complex behaviors of a capacitor microphone dynamical system. Alqhtani et al. [28] presented that two important physical examples that are of current and recurring interests are considered, in which the classical time derivative was modeled with the Caputo fractional derivative leading the system of equations to subdiffusive fractional reaction-diffusion models of predator-prey type, together with some numerical experiments. In [29], Aljhani et al. discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. They also utilize the fractional homotopy analysis transformation method to obtain approximate solutions.Numerous articles about SARS-CoV-2 or COVID-19 have recently been published in scientific journals around the world, including a few examples: DarAssi et al. [30] presented a model of SARS-CoV-2 with hospitalization in the form of a variable-order fractional model of Caputo’s differential equations, in which they studied the asymptotic stability of the system. In the same direction, Gu et al. [31] also designed the comprehensive Caputo model of SARS-CoV-2 virus in the framework of the constant-order operator and analyzed the stable solutions of the system w.r.t. the index R0 (reproduction number). Under a five-compartmental SEIRD model, and using real data from Italian medical authorities, Rajagopal et al. [32] conducted a case study of the disease and analyzed system behavior in both classical and fractional modes. In another case research of the prevalence of SARS-CoV-2 in France and Colombia, Quintero and Gutiérrez-Carvajal [33] examined the evolution of the disease under the bound optimization method. In 2021, Zamir et al. [34] formulated a model of COVID-19 in nine subclasses and focused the elimination and control of the infection caused by COVID-19. Jain et al. [35] presented a prediction model of COVID-19 by using numerous machine learning models, such as SVM, Naïve Bayes, K-nearest neighbors, AdaBoost, gradient boosting, XGBoost, random forest, ensembles, and neural networks. Baleanu et al. [36] introduced a generalized version of fractional models for the COVID-19 pandemic, including the effects of isolation and quarantine. In [37], Ali et al. investigate the transmission dynamics of a fractional-order mathematical model of COVID-19 under five subclasses, susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered, using the Caputo fractional derivative. In 2022, Ozkose et al. [38] developed a new model of the Omicron strain of SARS-CoV-2 virus and, based on data collected across the United Kingdom, studied the relationship between this strain and heart attack. They also analyzed the sensitivity of the system and fitting of the parameters using the LCM method.In addition to these articles, many other researchers have published articles on COVID-19 dynamics and evaluated a variety of models under different conditions and assumptions. For instances, we can mention stochastic models of COVID-19 [39], or even various case studies of COVID-19 from all over the world like [40–44]. Most researchers simultaneously studied the models of COVID-19 analytically and numerically and evaluated the types of dynamic behaviors of the solutions under singular and nonsingular systems that can be mentioned like [45, 46]. In the theoretical study of all these mentioned models, the theoretical results are among the basic parts of the analysis of mathematical models, because the existence of a solution for a system allows us to continue to study other properties such as stable solutions, equilibrium solutions, numerical solutions, and their simulations. Usually, fixed point theory is effective in this field, and its role can be observed in boundary and initial value problems [47].By defining mathematical models and the refinement of numerical approaches, there is a need to use new mathematical operators with high computational capabilities to model processes. As a result, Atangana [48] used fractal derivatives to introduce a new type of hybrid operators and introduced fractional-fractal derivatives into the world of modeling in 2017. In fact, to define these advanced operators, he used two arguments to represent the order of the operator and the dimension of the operator, which he called the fractional order and the fractional dimension of the fractional-fractional derivatives, respectively [48]. Atangana then divided these derivatives into three different categories and, with the help of different integral kernels, extracted the numerical algorithms associated with them. Then, in the last year, these numerical techniques were used in some new studies in which researchers simulated the approximate solutions of fractional-fractal models of new infectious diseases. In 2021, Arfan et al. [49] designed a prey-predation structure for the four-compartmental fractal-fractional model of syn-ecosymbiosis and examined some conditions for species survival in an ecological system. Abdulwasaa et al. [50] conducted a case study with these fractal-fractional operators in which they examined the dynamics of new cases and the number of deaths from the COVID-19 epidemic over a specific period of time in India. Shah et al. [51] conducted the same study on a new model in Pakistan. Khan et al. [52] simulated and evaluated models of smoking at the incidence rate under the Caputo fractal-fractional derivative operator. Arif et al. [53] utilized the same fractal-fractional operators in engineering to analyze MHD stress fluid in a single channel. Alqhtani et al. [54] studied three models of fractal-fractional Michaelis-Menten enzymatic reaction (FFMMER) and presented these models based on three different kernels, namely, power law, exponential decay, and Mittag-Leffler kernels.In this work, considering the importance of symptomatic and asymptomatic populations in spreading of virus, we present the new fractal-fractional probability-based model of SARS-CoV-2 virus by dividing the total population into four subclasses such as susceptible, asymptomatic, symptomatic, and recovered individuals. In [55], the authors designed a five-compartmental Caputo fractional epidemic model for the novel coronavirus in which the impact of environmental transmission is considered in the final result. This model motivates us to study an extended model of transmission of SARS-CoV-2 virus via advanced hybrid operators. In this paper, we get help from these newly extended hybrid fractal-fractional operators and discuss a new hybrid model of transmission of SARS-CoV-2 virus analytically and numerically. If we want to focus on the novelty and contribution of this manuscript, it is notable that for the first time, our system is a fractal-fractional probability-based model of SARS-CoV-2 virus in which we apply new hybrid fractal-fractional derivatives for modeling of the power law type kernel. Also, in this model, a probability-based structure of transmission of virus is considered. In other words, if p is the probability that both categories susceptible and infected interact and this leads to the asymptomatic category, in that case, 1−p stands for the portion of the infected persons that may automatically belong to the symptomatic category. On the other hand, it should be kept in mind that when people become infected with the SARS-CoV-2 virus, they may not have any symptoms, but at the same time, some people may experience severe complications and show specific symptoms. Therefore, the feature of our model is that we have divided the group of people infected with the virus into two categories: symptomatic and asymptomatic. Also, from mathematical point of view, a specific approach of fixed point methods is applied via ϕ-admissible ϕ-ψ-contractions to discuss the existence criterion, in which it shows the applicability of new fixed point techniques in the applied problems. Also, another novelty of this study is that in addition to fractal-fractional analysis of the SARS-CoV-2 model, we extend its Caputo-type version to compare our previous results with solutions of the fractional model under the Taylor operational matrix method. Also, note that in this paper, we consider both fractional and fractal-fractional derivatives as the full memory. We can study similar models by using the short memory. In this direction, refer to [56, 57].In this study, from a numerical point of view, we present two numerical techniques for the approximate solution of the considered model of SARS-CoV-2 virus under two different fractional operator derivative. The first technique is the Adams-Bashforth technique which is applied to probability-based model of SARS-CoV-2 under fractal-fractional operator in this study. The ABM is a very stable technique and allows us explicitly to determine the numerical solution at an instant time from the solutions in the previous instants. Using the higher-order Adams-Bashforth method actually becomes more unstable as the timestep is reduced. So that, the corrector step need to be added to avoid much of this instability. This can be mentioned as a disadvantage of AB technique. The second method is a collocation type of the well-known spectral methods; fractional Taylor operational matrix method is applied to solve probability-based model of SARS-CoV-2 under Caputo operator first time in this paper. The main advantage of spectral methods is that they are easy to apply for both finite and infinite intervals and when the solution of a given problem is smooth, spectral methods have very good error properties, namely, the so-called “exponential convergence.” Thanks to these advantages, for solving many different types of integral and differential equations numerically, spectral methods received considerable interest in recent years. When the solution is not smooth enough, the stability and accuracy of these methods are decreasing, which is an important disadvantage because of limiting the applicability. In this work, we compare our results obtained from FTOMM with the Adams-Bashforth simulations.The structure of the manuscript is arranged as follows: some definitions ofϕ-admissible ϕ-ψ-contractions and fractal-fractional derivatives are presented in the next section. We describe our main fractal-fractional model in Section 3 along with the meanings of parameters. Under two different fixed point methods, we guarantee the existence property for solutions of the system in Section 4. Section 5 deals with the Lipschitz and uniqueness properties. Then, in Section 6, we discuss UHR-stable solutions for each four state functions separately. To predict the future of state functions and their analysis numerically, we simulate them via the Adams-Bashforth method in two subsections of Section 7. In the next step, in Section 8, we give the Caputo-type of transmission of the SARS-CoV-2 virus, and in several subsections, we describe our method via the Taylor operational matrix technique, and after some simulations, we compare our numerical results in both fractal-fractional and fractional systems in the context of some graphs and tables. The conclusions and further study suggestions are presented in Section 9.


## 2. Basic Concepts

Some basic notions on the fractal-fractional operators and fixed point theory are assembled.LetΨ display a subclass of nondecreasing operators like ψ:0,∞⟶0,∞ s.t.
(1)∑j=1∞ψjt<∞,ψt<t,∀t>0.Definition 1 (see [58]).
LetX be a normed space and F:X⟶X and ϕ:X2⟶ℝ≥0.
pF is ϕ-ψ-contraction if for ù1,ù2∈X,
(2)ϕͷ1,ͷ2dFͷ1,Fͷ2≤ψdͷ1,ͷ2.
(q) F is ϕ-admissible if ϕͷ1,ͷ2≥1 yields ϕFͷ1,Fͷ2≥1.Definition 2 (see [48]).
LetF be fractal differentiable on a,b of order ν. The fractional-fractal ωth-derivative of the function F via the power law type kernel in the Riemann-Liouville sense is defined by
(3)FFPDa,tω,νFt=1Γn−ωddtν∫att−mn−ω−1Fmdm,n−1<ω,ν≤n∈ℕ,where dFm/dmν=limt⟶mFt−Fm/tν−mν is the fractal derivative.It is known that ifν=1, then the fractal-fractional derivative FFPDa,tω,ν is reduced to the standard derivative RLDa,tω of order ω.Definition 3 (see [48]).
LetF be continuous on a,b. The fractional-fractal integral of the function F with fractional order ω and fractal order ν is
(4)FFPIa,tω,νFt=νΓω∫atmν−1t−mω−1Fmdm.


## 3. Description of the Model for SARS-CoV-2 Virus

Khan et al. [59, 60] modeled a mathematical structure of dynamics of SARS-CoV-2 virus in the form of four initial value problems equipped with four state functions S, P1, P2, and R, which are a part of total population. This model is
(5)dStdt=Θ−rP1tSt−rsP2tSt−b+b1St,dP1tdt=prP1tSt+rsP2tSt−b1+b2+r1P1t,dP2tdt=1−prP1tSt+rsP2tSt+qr1P1t−b1+b3+r2P2t,dRtdt=r11−qP1t+r2P2t+bSt−b1Rt,where St stands for the people belonging to the susceptible category, P1t is the people belonging to the asymptomatic category, P2t is the people belonging to the symptomatic category, and Rt stands for the people belonging to the recovered category at the time t∈J≔0,T,T>0. Based on these assumptions, the infected categories are taken to be symptomatic class and asymptomatic class, because asymptomatic persons are considered as the main factor of transmission of disease. It is to be noted that the variables, constants, and parameters are nonnegative.Inspired by the aforesaid standard epidemic model, we here consider the fractal-fractional epidemic probability-based model of the SARS-CoV-2 virus in the following structure:(6)FFPD0,tω,νSt=Θ−rP1tSt−rsP2tSt−b+b1St,FFPD0,tω,νP1t=prP1tSt+rsP2tSt−b1+b2+r1P1t,FFPD0,tω,νP2t=1−prP1tSt+rsP2tSt+qr1P1t−b1+b3+r2P2t,FFPD0,tω,νRt=r11−qP1t+r2P2t+bSt−b1Rt,subject to
(7)S0=S0>0,P10=P1,0≥0,P20=P2,0≥0,R0=R0≥0,where FFPD0,tω,ν is the fractional-fractal derivative of the fractional order ω∈0,1 and the fractal order ν∈0,1 via the power law type kernel. We have
(8)Nt=St+P1t+P2t+Rt,in which as we said above, Nt means the total population at the time t∈J≔0,T,T>0.About parameters, the total natural death rate along with the rate of disease-related death for both infected groups is specified by the symbolsb1, b2, and b3, respectively. We show the rate of transmission of disease by r, and its reduced rate is denoted by the symbol s. The vaccination rate is given by b, and Θ stands for the newborn rate. The probability of the asymptomatic persons is illustrated by p, and the probability of these persons that recover in the symptomatic step is specified by q. Moreover, r2 is the recovery rate in relation to asymptomatic persons and accordingly, and r1 is the recovery rate in relation to the symptomatic persons.


## 4. Existence of Solutions

In this position, we shall get help fixed point theory to the suggested fractal-fractional IVP (6). For the qualitative analysis, we define the Banach space X=Y4, where Y=CJ,ℝ, as
(9)XX=S,P1,P2,RX=maxSt+P1t+P2t+Rt:t∈J.We write the R.H.S. of model (6) by
(10)F1t,St,P1t,P2t,Rt=Θ−rP1tSt−rsP2tSt−b+b1St,F2t,St,P1t,P2t,Rt=prP1tSt+rsP2tSt−b1+b2+r1P1t,F3t,St,P1t,P2t,Rt=1−prP1tSt+rsP2tSt+qr1P1t−b1+b3+r2P2t,F4t,St,P1t,P2t,Rt=r11−qP1t+r2P2t+bSt−b1Rt.Hence,(11)RLD0,tωSt=νtν−1F1t,St,P1t,P2t,Rt,RLD0,tωP1t=νtν−1F2t,St,P1t,P2t,Rt,RLD0,tωP2t=νtν−1F3t,St,P1t,P2t,Rt,RLD0,tωRt=νtν−1F4t,St,P1t,P2t,Rt.By (11), we derive the following IVP:
(12)RLD0,tωXt=νtν−1Ft,Xt,ω,ν∈0,1,X0=X0,where
(13)Xt=St,P1t,P2t,RtT,X0=S0,P1,0,P2,0,R0T,Ft,Xt=F1t,St,P1t,P2t,Rt,F2t,St,P1t,P2t,Rt,F3t,St,P1t,P2t,Rt,F4t,St,P1t,P2t,Rt.Now, the fractional-fractal integral acts on (12), and it becomes
(14)Xt=X0+νΓω∫0tmν−1t−mω−1Fm,Xmdm.In other words, the extended form of the above fractal-fractional integral is represented as(15)St=S0+νΓω∫0tmν−1t−mω−1F1m,Sm,P1m,P2m,Rmdm,P1t=P1,0+νΓω∫0tmν−1t−mω−1F2m,Sm,P1m,P2m,Rmdm,P2t=P2,0+νΓω∫0tmν−1t−mω−1F3m,Sm,P1m,P2m,Rmdm,Rt=R0+νΓω∫0tmν−1t−mω−1F4m,Sm,P1m,P2m,Rmdm.Consider the operatorG:X⟶X as
(16)GXt=X0+νΓω∫0tmν−1t−mω−1Fm,Xmdm.In the preceding, we recall the required fixed point theorem in connection with our aim for proving the existence results.Theorem 4 (see [58]).
AssumeX,d as a Banach space, ϕ:X×X⟶ℝ, ψ∈Ψ, and F:X⟶X as an ϕ-ψ-contraction s.t.
(1)
F is ϕ-admissible(2)
∃ù0∈X,s.t.ϕͷ0,Fͷ0≥1(3)
for any sequenceͷn in X with ͷn⟶ͷ and ϕͷn,ͷn+1≥1 for all n≥1, we have ϕͷn,ͷ≥1,∀n∈ℕ
Then,∃ͷ∗ s.t. Fͷ∗=ͷ∗.Now, the first existence result is proved here under some special operators.Theorem 5.
Let∃T:ℝ×ℝ⟶ℝ, ∃ a continuous map F:J×X⟶X, and ∃ a nondecreasing map ψ∈Ψ. Let (B1)∀X1,X2∈X, and t∈J,
(17)Ft,X1t−Ft,X2t≤ℓ~ψX1t−X2t,with TX1t,X2t≥0, where ℓ~=Γν+ω/νTν+ω−1Γν.
(B2) X0∈X exists so that ∀t∈J,
(18)TX0t,GX0t≥0,and also, the inequality
(19)TX1t,X2t≥0,gives
(20)TGX1t,GX2t≥0,for any X1,X2∈X and t∈J.
(B3)∀Xnn≥1 belonging to X with Xn⟶X and
(21)TXnt,Xn+1t≥0,for each n and t∈J, we get
(22)TXnt,Xt≥0.
In such a case,∃ is a solution for the fractal-fractional problem (12), and so there exists a solution for the given fractal-fractional epidemic model of SARS-CoV-2 virus (6).Proof.
LetX1 and X2 be two members belonging to X with
(23)TX1t,X2t≥0,for each t∈J. Then, by definition of the Beta function, we may write
(24)GX1t−GX2t≤νΓω∫0tmν−1t−mω−1Fm,X1m−Fm,X2mdm≤νΓω∫0tmν−1t−mω−1ℓ~ψX1m−X2mdm≤νℓ~Tν+ω−1Bν,ωΓωψX1−X2X=νTν+ω−1ΓνΓν+ωℓ~ψX1−X2X.
Consequently, we have(25)GX1−GX2X≤νTν+ω−1ΓνΓν+ωℓ~ψX1−X2X=ψX1−X2X.
Now,ϕ:X×X⟶0,∞ is introduced by the this rule:
(26)ϕX1,X2=1ifTX1t,X2t≥0,0otherwise.
Then, for everyX1,X2∈X, we will get
(27)ϕX1,X2dGX1,GX2≤ψdX1,X2.
Thus,G is found as an ϕ-ψ-contraction. To verify that G is ϕ-admissible, let X1,X2∈X be arbitrary and ϕX1,X2≥1. By definition of ϕ, we have
(28)TX1t,X2t≥0.
Then, byP2, TGX1t,GX2t≥0 is satisfied. Again, the definition of ϕ gives ϕGX1,GX2≥1. Thus, G is ϕ-admissible.
On the other hand, the conditionP2 guarantees the existence of X0∈X. In this case, for each t∈J, TX0t,GX0t≥0 holds. Clearly, we get ϕX0,GX0≥1. These show that the conditions (1) and (2) of Theorem 4 are fulfilled.
Now, we assume thatXnn≥1 is a sequence in X s.t. Xn⟶X, and for all n, ϕXn,Xn+1≥1. By virtue of definition of ϕ,
(29)TXnt,Xn+1t≥0.
Therefore, in the light of hypothesisP3, we obtain
(30)TXnt,Xt≥0.
This indicates thatϕXn,X≥1 for every n. This guarantees the condition (3) of Theorem 4. Ultimately, by utilizing Theorem 4, we conclude that it found a fixed point for G like X∗∈X. This implies that X∗=S∗,P1∗,P2∗,R∗T is interpreted as a solution of the fractal-fractional model of SARS-CoV-2 (6) and the argument is finally completed.In the sequel, we use the Leray-Schauder criterion to prove the existence result.Theorem 6 (see [61]).
RegardX as a Banach space, E as a bounded closed set in X with the convexity property, and an open set O⊆E with 0∈O. The compact continuous map G:O¯⟶E, either
(i)
G possesses fixed point in O¯ or(ii)
∃∈∂O and μ∈0,1 s.t. ϰ=μGϰ.Theorem 7.
AssumeF∈CJ×X,X along with the following:
(C1):φ∈L1J,ℝ+ and an increasing map B∈C0,∞,0,∞ exist provided that
(31)Ft,Xt≤φtBXt
(C2): There existγ>0 with
(32)γΛ+Δφ0∗Bγ>1,in which φ0∗=supt∈Jφt and Λ,Δ are given in () and ()
Then, a solution exists for fractal-fractional problem (12), and so a solution exists for the given fractal-fractional model of SARS-CoV-2 virus (6) on J.Proof.
We define a mapG:X⟶X as in (15) and the ball
(33)Vε=X∈X:XX≤ε,for some ε>0. From the continuity of F, we yield the continuity of operator G. (C1) gives
(34)GXt≤X0+νΓω∫0tmν−1t−mωFm,Xmdm≤X0+νΓω∫0tmν−1t−mωφmBXmdm≤X0+νTν+ω−1Bν,ωΓωφ0∗BXX≤X0+νTν+ω−1ΓνΓν+ωφ0∗Bε,for X∈Vε. Consequently, we obtain
(35)GXt≤X0+νTν+ω−1ΓνΓν+ωφ0∗Bε<∞.
This gives the uniformally boundedness of the operatorG on X. We now verify the equicontinuity of operator G. For the purpose, arbitrarily, take t,t′∈0,T such that t<t′ and X∈Vε. Assuming
(36)supt,X∈J×VεFt,Xt=F∗<∞,estimate
(37)GXt′−GXt=νΓω∫0t′mν−1t′−mω−1Fm,Xmdm−νΓω∫0tmν−1t−mω−1Fm,Xmdm≤νF∗Γω∫0t′mν−1t′−mω−1dm−∫0tmν−1t−mωdm≤νF∗Bν,ωΓωt′ν+ω−1−tν+ω−1=νF∗ΓνΓν+ωt′ν+ω−1−tν+ω−1,which is independent of X, as t′⟶t, the R.H.S. of above, tends to 0. It implies that
(38)GXt′−GXtX⟶0.
This confirms the equicontinuity ofG. Arzelà-Ascoli’s theorem implies the compactness of operator G on Vε. The hypothesis of Theorem 6 on the operator G has now been verified. Utilizing (C2), we construct
(39)ℙ=X∈X:XX<γ,for some γ>0 via
(40)X0+νTν+ω−1ΓνΓν+ωφ0∗Bγ<γ.
Utilizing (C1) and by (35), we write
(41)GXX≤X0+νTν+ω−1ΓνΓν+ωφ0∗BX.
Now, we assume the existence ofX∈∂ℙ and α∈0,1 subject to X=αGX. For such α and X, by (41), one may write that
(42)γ=XX=αGXX<X0+νTν+ω−1ΓνΓν+ωφ0∗BXX<X0+νTν+ω−1ΓνΓν+ωφ0∗Bγ<γ,which is impossible. Therefore, (ii) is not valid, and by Theorem 6, G possesses a fixed point in ℙ¯. Therefore, the fractal-fractional model of SARS-CoV-2 virus (6) admits a solution and so proof is complete.


## 5. Uniqueness Result

Lemma 8.
AssumeS,P1,P2,R,S∗,P1∗,P2∗,R∗∈Y=CJ,ℝ. Let (H1) S≤λ1, P1≤λ2, P1≤λ3, and R≤λ4 for some λ1,λ2,λ3,λ4>0.
Then, the kernelsF1, F2, F3, and F4 given in (10) satisfied the Lipschitz property w.r.t. the corresponding components if ϖ1,ϖ2,ϖ3,ϖ4<1, where
(43)ϖ1=qλ1+r,ϖ2=r+s,ϖ3=r+b,ϖ4=r.Proof.
Starting from the kernelF1, for each S,S∗∈Y, we estimate
(44)F1t,St,P1t,P2t,Rt−F1t,S∗t,P1t,P2t,Rt=p−qStP2t−rSt−p−qS∗tP2t−rS∗t≤qP2t+rSt−S∗t≤qλ3+rSt−S∗t=ϖ1St−S∗t.
This shows that the kernelF1 is Lipschitz w.r.t. S with constant ϖ1<1. Regarding the kernel function F2, for each P1,P1∗∈Y≔CJ,ℝ, we estimate
(45)F2t,St,P1t,P2t,Rt−F2t,St,P1∗t,P2t,Rt=qStP2t−r+sP1t−qS∗tP2t−r+sP1∗t≤r+sP1t−P1∗t=ϖ2P1t−P1∗t.
This leads thatF2 is Lipschitz w.r.t. P1 with constant ϖ2<1. Now for each P2,P2∗∈Y, we have
(46)F3t,St,P1t,P2t,Rt−F3t,St,P1t,P2∗t,Rt=sP1t−r+bP2t−sP1t−r+bP2∗t≤r+bP2t−P2∗t=ϖ3P2t−P2∗t.
This shows thatF3 is Lipschitz w.r.t. P2 with constant ϖ3<1. Now for each R,R∗∈Y, we have
(47)F4t,St,P1t,P2t,Rt−F4t,St,P1t,P2t,R∗t=bP2t−rRt−bP2t−rR∗t≤rRt−R∗t=ϖ4Rt−R∗t.
This shows thatF4 is Lipschitz w.r.t. R with constant ϖ4<1. From the above, we conclude that the kernels Fi,i=1,2,3,4, are Lipschitzian w.r.t. the corresponding component with constants ϖi,i=1,2,3,4, respectively.We study the uniqueness result for solution to the presumed fractal-fractional model (6) based on the conclusions gained in Lemma 8.Theorem 9.
Assume (H1), then the given fractal-fractional model of SARS-CoV-2 virus (6) has a unique solution if
(48)νTν+ω−1ΓνΓν+ωϖi<1,i∈1,2,3,4.Proof.
The outcome of the theorem is assumed to be invalid. That is to say, there is another solution for the given fractional-fractal model of SARS-CoV-2 virus (6). Assume that S∗t,P1∗t,P2∗t,R∗t is another solution with S0,,P1,0,P2,0,R such that by (16), we have
(49)S∗t=S0+νΓω∫0tmν−1t−mω−1F1m,S∗m,P1∗m,P2∗m,R∗mdm,P1∗t=P1,0+νΓω∫0tmν−1t−mω−1F2m,S∗m,P1∗m,P2∗m,R∗mdm,P2∗t=P2,0+νΓω∫0tmν−1t−mω−1F3m,S∗m,P1∗m,P2∗m,R∗mdm,R∗t=R0+νΓω∫0tmν−1t−mω−1F4m,S∗m,P1∗m,P2∗m,R∗mdm.
Now, we can estimate(50)St−S∗t≤νΓω∫0tmν−1t−mω−1×F1m,Sm,P1m,P2m,Rm−F1m,S∗m,P1∗m,P2∗m,R∗mdm≤νΓω∫0tmν−1t−mω−1ϖ1S−S∗dm≤νTν+ω−1ΓνΓν+ωϖ1S−S∗,and so
(51)1−νTν+ω−1ΓνΓν+ωϖ1S−S∗≤0.
It is true ifS−S∗=0, and accordingly, S=S∗. Next, from
(52)P1−P1∗≤1−νTν+ω−1ΓνΓν+ωϖ2P1−P1∗,we get
(53)1−νTν+ω−1ΓνΓν+ωϖ2P1−P1∗≤0.
This implies thatP1−P1∗=0 and so P1=P1∗. Also, we have
(54)P2−P2∗≤1−νTν+ω−1ΓνΓν+ωϖ3P2−P2∗.
This gives(55)1−νTν+ω−1ΓνΓν+ωϖ3P2−P2∗≤0.
This implies thatP2−P2∗=0 and so P2=P2∗. Finally, from
(56)R−R∗≤1−νTν+ω−1ΓνΓν+ωϖ4R−R∗,we get
(57)1−νTν+ω−1ΓνΓν+ωϖ4R−R∗≤0.
This implies thatR−R∗=0 and so R=R∗. Consequently, we get
(58)St,P1t,P2t,Rt=S∗t,P1∗t,P2∗t,R∗t.
This shows that the fractal-fractional model of SARS-CoV-2 virus (6) has exactly one solution.


## 6. UH and UHR Stability Criterion

We now proceed to review stable solutions in the context of the Ulam-Hyers (UH) and Ulam-Hyers-Rassias (UHR) to the given fractal-fractional model of SARS-CoV-2 virus (6).Definition 10.
The fractal-fractional model of SARS-CoV-2 virus (6) is UH-stable if ∃0<MFi∈ℝ,i=1,2,3,4 s.t. ∀εi>0 and ∀,S∗,P1∗,P2∗,R∗∈X fulfilling
(59)DFFP0,tω,νS∗t−F1t,S∗t,P1∗t,P2∗t,R∗t<ε1,DFFP0,tω,νP1∗t−F2t,S∗t,P1∗t,P2∗t,R∗t<ε2,DFFP0,tω,νP2∗t−F3t,S∗t,P1∗t,P2∗t,R∗t<ε3,DFFP0,tω,νR∗t−F4t,S∗t,P1∗t,P2∗t,R∗t<ε4.There exist S,P1,P2,R∈X satisfying the given fractal-fractional model of SARS-CoV-2 virus (6) with
(60)S∗t−St≤MF1ε1,∀t∈J,P1∗t−P1t≤MF2ε2,∀t∈J,P2∗t−P2t≤MF3ε3,∀t∈J,R∗t−Rt≤MF4ε4,∀t∈J.Definition 11.
The given fractal-fractional model of SARS-CoV-2 virus (6) is generalized UH-stable if ∃MFi∈Cℝ+,ℝ+,i=1,2,3,4 with MFi0=0 s.t. ∀εi>0 and ∀S∗,P1∗,P2∗,R∗∈X fulfilling
(61)DFFP0,tω,νS∗t−F1t,S∗t,P1∗t,P2∗t,R∗t<ε1,DFFP0,tω,νP1∗t−F2t,S∗t,P1∗t,P2∗t,R∗t<ε2,DFFP0,tω,νP2∗t−F3t,S∗t,P1∗t,P2∗t,R∗t<ε3,DFFP0,tω,νR∗t−F4t,S∗t,P1∗t,P2∗t,R∗t<ε4.There exist a solution S,P1,P2,R∈X of the given fractal-fractional model of SARS-CoV-2 virus (6) with
(62)S∗t−St≤MF1ε1,∀t∈J,P1∗t−P1t≤MF2ε2,∀t∈J,P2∗t−P2t≤MF3ε3,∀t∈J,R∗t−Rt≤MF4ε4,∀t∈J.Remark 12.
Note thatS∗,P1∗,P2∗,R∗∈X is a solution of (59) iff ∃η1,η2,η3,η4∈C0,T,ℝ (depending upon S∗,P1∗,P2∗,R∗, respectively) so that for all t∈J,
(i)
ηit<εi,i=1,2,3,4(ii)
We have(63)FFPD0,tω,νS∗t−F1t,S∗t,P1∗t,P2∗t,R∗t+η1t,FFPD0,tω,νP1∗t−F2t,S∗t,P1∗t,P2∗t,R∗t+η2t,FFPD0,tω,νP2∗t−F3t,S∗t,P1∗t,P2∗t,R∗tη3t,FFPD0,tω,νR∗t−F4t,S∗t,P1∗t,P2∗t,R∗tη4tDefinition 13.
The fractal-fractional model of SARS-CoV-2 virus (6) is UHR-stable w.r.t. functions Ψi,i=1,2,3,4, if ∃0<MFi,Ψi∈ℝ,i=1,2,3,4 s.t. ∀εi>0 and ∀S∗,P1∗,P2∗,R∗∈X fulfilling
(64)DFFP0,tω,νS∗t−F1t,S∗t,P1∗t,P2∗t,R∗t<ε2Ψ2t.DFFP0,tω,νP1∗t−F2t,S∗t,P1∗t,P2∗t,R∗t<ε2Ψ2t,DFFP0,tω,νP2∗t−F3t,S∗t,P1∗t,P2∗t,R∗t<ε3Ψ3t,DFFP0,tω,νR∗t−F4t,S∗t,P1∗t,P2∗t,R∗t<ε4Ψ4t.There exist S,P1,P2,R∈X satisfying the given fractal-fractional model of SARS-CoV-2 virus (6) with
(65)S∗t−St≤MF1,Ψ1ε1Ψ1t,∀t∈J,P1∗t−P1t≤MF2,Ψ1ε2Ψ2t,∀t∈J,P2∗t−P2t≤MF3,Ψ3ε3Ψ3t,∀t∈J,R∗t−Rt≤MF4,Ψ1ε4Ψ4t,∀t∈J.Definition 14.
The given fractal-fractional model of SARS-CoV-2 virus (6) is generalized UHR-stable w.r.t. Ψi,i=1,2,3,4, if ∃MFi,Ψi∈ℝ,i=1,2,3,4 with MFi0=0 s.t. ∀εi>0 and ∀S∗,P1∗,P2∗,R∗∈X fulfilling
(66)DFFP0,tω,νS∗t−F1t,S∗t,P1∗t,P2∗t,R∗t<Ψ1t,DFFP0,tω,νP1∗t−F2t,S∗t,P1∗t,P2∗t,R∗t<Ψ2t,DFFP0,tω,νP2∗t−F3t,S∗t,P1∗t,P2∗t,R∗t<Ψ3t,DFFP0,tω,νR∗t−F4t,S∗t,P1∗t,P2∗t,R∗t<Ψ4t.There exist a solution S,P1,P2,R∈X of the given fractal-fractional model of SARS-CoV-2 virus (6) with
(67)S∗t−St≤MF1,Ψ1Ψ1t,∀t∈J,P1∗t−P1t≤MF2,Ψ2Ψ2t,∀t∈J,P2∗t−P2t≤MF3,Ψ3Ψ3t,∀t∈J,R∗t−Rt≤MF4,Ψ4Ψ4t,∀t∈J.Remark 15.
Note thatS∗,P1∗,P2∗,R∗∈X is a solution of (64) iff there exists η1,η2,η3,η4∈C0,T,ℝ (depending upon S∗,P1∗,P2∗,R∗, respectively) so that for all t∈J,
(i)
ηit<ΨiTεi,i=1,2,3,4(ii)
We have(68)FFPD0,tω,νS∗t=F1t,S∗t,P1∗t,P2∗t,R∗t+η1t,FFPD0,tω,νP1∗t=F2t,S∗t,P1∗t,P2∗t,R∗t+η2t.FFPD0,tω,νP2∗t=F3t,S∗t,P1∗t,P2∗t,R∗t+η3t,FFPD0,tω,νR∗t=F4t,S∗t,P1∗t,P2∗t,R∗t+η4tTheorem 16.
The given fractal-fractional model of SARS-CoV-2 virus (6) is UH-stable on J≔0,T, and it is generalized UH-stable such that
(69)νTν+ω−1ΓνΓν+ωϖi<1,i∈1,2,3,4,where ϖi are given by () provided that the assumption (H1) is valid.Proof.
Letε1>0 and S∗∈Y be arbitrary so that
(70)FFPD0,tω,νS∗t−F1t,S∗t,P1∗t,P2∗t,R∗t<ε1.
Then, in view of Remark12, we can find a function η1t satisfying
(71)FFPD0,tω,νSastt=F1t,Sastt,P1astt,P2astt,R∗t+η1t,with η1t≤ε1. So
(72)S∗t=S0+νΓω∫0tmν−1t−mω−1F1m,S∗m,P1∗m,P2∗m,R∗mdm+νΓω∫0tmν−1t−mω−1η1mdm.
By Theorem9, let S∈Y be the unique solution of the given fractal-fractional model of NOV-COV-2 virus (). Then, St is given by
(73)St=S0+νΓω∫0tmν−1t−mω−1F1m,Sm,P1m,P2m,Rmdm.
Then,(74)S∗t−St≤νΓω∫0tmν−1t−mω−1η1mdm+νΓω∫0tmν−1t−mω−1×F1m,S∗m,P1∗m,P2∗m,R∗m−F1m,Sm,P1m,P2m,Rmdm,≤νTν+ω−1ΓνΓν+ωε1+νTν+ω−1ΓνΓν+ωϖ1S∗−S.
Hence, we get(75)S∗−S≤νTν+ω−1Γν/Γν+ωε11−νTν+ω−1Γν/Γν+ωϖ1.
If we letMF1=νTν+ω−1Γν/Γν+ω/1−νTν+ω−1Γν/Γν+ωϖ1, then S∗−S≤MF1ε1. Similarly, we have
(76)P1∗−P1≤MF2ε2,P2∗−P2≤MF3ε3,R∗−R≤MF4ε4,where
(77)MFi=νTν+ω−1Γν/Γν+ω1−νTν+ω−1Γν/Γν+ωϖi,i∈2,3,4.
Hence, the UH stability of the given fractal-fractional model (6) is fulfilled. Next, by assuming
(78)MFiεi=νTν+ω−1Γν/Γν+ωεi1−νTν+ω−1Γν/Γν+ωϖi,i∈1,2,3,4,with MFi0=0, the generalized UH stability of the given fractional-fractal model (6) is fulfilled.In the next result, UHR stability for the given fractal-fractional model of SARS-CoV-2 (6) is studied:Theorem 17.
The conditionH1 is assumed to be held:
(H′): ∃ increasing mappings Ψi∈C0,T,ℝ+,i∈1,2,3,4 and ∃ΛΨi>0 such that ∀t∈J,(79)FFPJ0,tω,νΨit<ΛΨiΨit,i∈1,2,3,4.
Then, the given fractal-fractional model of SARS-CoV-2 virus (6) is UHR and generalized UHR-stable.Proof.
For everyε1>0 and ∀S∗∈Y satisfying
(80)FFPD0,tω,νS∗t−F1t,S∗t,P1∗t,P2∗t,R∗t<ε1Ψ1t,∃ηts.t.FFPD0,tω,νS∗t=F1t,S∗t,P1∗t,P2∗t,R∗t+η1t,with η1t≤ε1Ψ1t. It follows that
(81)S∗t=S0+νΓω∫0ttν−1t−tω−1F1t,S∗t,P1∗t,P2∗t,R∗tdt+νΓω∫0ttν−1t−tω−1η1tdt.
By Theorem9, let S∈Y be the unique solution of the given fractal-fractional model of SARS-CoV-2 virus (6). Then, St is given by
(82)St=S0+νΓω∫0ttν−1t−tω−1F1t,St,P1t,P2t,Rtdt.
Then, by (61),
(83)S∗t−St≤νΓω∫0tmν−1t−mω−1h1mdm+νΓω∫0tmν−1t−mω−1×F1m,S∗m,P1∗m,P2∗m,R∗m−F1m,Sm,P1m,P2m,Rmdm≤ε1νΓω∫0tmν−1t−mω−1Ψ1mdm+νTν+ω−1ΓνΓν+ωϖ1S∗−S≤ε1ΛΨ1Ψ1+νTν+ω−1ΓνΓν+ωϖ1S∗−S.
Accordingly, it gives(84)S∗−S≤ε1ΛΨ1Ψ11−νTν+ω−1Γν/Γν+ωϖ1.
If we let(85)MF1,Ψ1=ΛΨ11−νTν+ω−1Γν/Γν+ωϖ1,then S∗−S≤ε1MF1,Ψ1Ψ1. Similarly, we have
(86)P1∗−P1≤ε2MF2,Ψ2Ψ2,P2∗−P2≤ε3MF3,Ψ3Ψ3,R∗−R≤ε4MF4,Ψ4Ψ4,where
(87)MFi,Ψi=ΛΨi1−νTν+ω−1Γν/Γν+ωϖi,i∈1,2,3,4.
Hence, the given fractal-fractional model of SARS-CoV-2 virus (6) is stable in the sense of UHR. Along with this, by setting εi=1;i∈1,2,3,4, the mentioned fractal-fractional model of SARS-CoV-2 virus (6) is generalized UHR-stable.


## 7. Numerical Algorithms and Simulations

### 7.1. Numerical Adams-Bashforth Method

In this section, we describe the numerical scheme in relation to the fractal-fractional model of SARS-CoV-2 virus (6). For this, we have taken help from the technique regarding two-step Lagrange polynomials called fractional Adams-Bashforth method (ABM). To begin this process, we follow the numerical method of fractal-fractional integral equations (15) using a new approach at tn+1. In other words, we discretize the mentioned equation (15) for t=tn+1, and we have
(88)Stn+1=S0+νΓω∫0tn+1tn+1−mω−1H1mdm,P1tn+1=P1,0+νΓω∫0tn+1tn+1−mω−1H2mdm,P2tn+1=P2,0+νΓω∫0tn+1tn+1−mω−1H3mdm,Rtn+1=R0+νΓω∫0tn+1tn+1−mω−1H4mdm,where
(89)H1m=mν−1F1m,Sm,P1m,P2m,Rm,H2m=mν−1F2m,Sm,P1m,P2m,Rm,H3m=mν−1F3m,Sm,P1m,P2m,Rm,H4m=mν−1F4m,Sm,P1m,P2m,Rm.By approximating above integrals, we get(90)Stn+1=S0+νΓω∑l=0n∫tltl+1tn+1−mω−1H1mdm,P1tn+1=P1,0+νΓω∑l=0n∫tltl+1tn+1−mω−1H2mdm,P2tn+1=P2,0+νΓω∑l=0n∫tltl+1tn+1−mω−1H3mdm,Rtn+1=R0+νΓω∑l=0n∫tltl+1tn+1−mω−1H4mdm.In the sequel, we approximate the functionsH1, H2, H3, and H4, introduced by (89), on the interval tl,tl+1 via two-step Lagrange interpolation polynomials with the step size h=tl−tl−1 as
(91)H1,l∗m≃m−tl−1htlν−1F1ml,Sl,P1,l,P2,l,Rl−m−tlhtl−1ν−1F1ml−1,Sl−1,P1,l−1,P2,l−1,Rl−1,H2,l∗m≃m−tl−1htlν−1F2ml,Sl,P1,l,P2,l,Rl−m−tlhtl−1ν−1F2ml−1,Sl−1,P1,l−1,P2,l−1,Rl−1,H3,l∗m≃m−tl−1htlν−1F3ml,Sl,P1,l,P2,l,Rl−m−tlhtl−1ν−1F3ml−1,Sl−1,P1,l−1,P2,l−1,Rl−1,H4,l∗m≃m−tl−1htlν−1F4ml,Sl,P1,l,P2,l,Rl−m−tlhtl−1ν−1F4ml−1,Sl−1,P1,l−1,P2,l−1,Rl−1.Then, we have(92)Stn+1=S0+νΓω∑l=0n∫tltl+1tn+1−mω−1H1,l∗mdm,P1tn+1=P1,0+νΓω∑l=0n∫tltl+1tn+1−mω−1H2,l∗mdm,P2tn+1=P2,0+νΓω∑l=0n∫tltl+1tn+1−mω−1H3,l∗mdm,Rtn+1=R0+νΓω∑l=0n∫tltl+1tn+1−mω−1H4,l∗mdm.By evaluating above integrals directly, the approximate solutions of the given fractional-fractal model of SARS-CoV-2 virus (6) are given by
(93)Sn+1=S0+νhωΓω+2∑l=0ntlν−1F1tl,Sl,P1,l,P2,l,RlYn,l−tl−1ν−1F1tl−1,Sl−1,P1,l−1,P2,l−1,Rl−1Y^n,l,P1,n+1=P1,0+νhωΓω+2∑l=0ntlν−1F2tl,Sl,P1,l,P2,l,RlYn,l−tl−1ν−1F2tl−1,Sl−1,P1,l−1,P2,l−1,Rl−1Y^n,l,P2,n+1=P2,0+νhωΓω+2∑l=0ntlν−1F3tl,Sl,P1,l,P2,l,RlYn,l−tl−1ν−1F3tl−1,Sl−1,P1,l−1,P2,l−1,Rl−1Y^n,l,Rn+1=R0+νhωΓω+2∑l=0ntlν−1F4tl,Sl,P1,l,P2,l,RlYn,l−tl−1ν−1F4tl−1,Sl−1,P1,l−1,P2,l−1,Rl−1Y^n,l,where
(94)Yn,l=n+1−lωn−l+2+ω−n−lωn−l+2+2ω,Y^n,l=n+1−lω+1−n−lωn−l+1+ω,where ω is the fractional order of the given fractal-fractional system (6).


### 7.2. Simulations Based on Adams-Bashforth Method

In this section, using the AB method for fractal-fractional, we present approximate solutions for the fractal-fractional probability-based model of SARS-CoV-2 virus (6). We demonstrate simulations to observe the behavior of four subclasses of SARS-CoV-2, which are S, P1, P2, and R under the different set of parameters.To provide a numerical simulation, we start by determining the value of the parameters by using reported cases in Turkey from 01 January 2021 to 03 July 2021. The birth rate for the Turkey in 2021 is 15.408 births per 1000 people, and the death rate isb1=5.5 per 1000 people. The Turkey’s population on 1st of January was N=84339067. Since we use the day as time limit, we can calculate the newborn rate as Θ=84339067×15.408/1000×365. To estimate the remaining parameters, we use the curve fitting technique with the data reported for SARS-CoV-2. Using this method, we determine the parameters as follows: p=0.4, r=0.003,s=0.05, b=0.05, r1=0.05, and r2=0.6, and we assume q=0.2, b2=0.04, and b3=0.6. Also, the stepsize for the time interval is choosen as h=10−3. As a first visualization, in Figure 1, we demonstrate the real data versus present model simulation. Then, behaviors of four subclasses are presented in Figures 2(a)–2(d) with the chosen initial values S=100, P1=90, P2=80, and R=70, respectively, and under various fractal-fractional orders ω and v.Figure 1
Simulated data vs. Real data for the SARS-CoV-2 cases in Turkey from 01 January 2021 to 03 July 2021.Figure 2
Behaviors of each compartments: (a)St, (b) P1t, (c) P2t, and (d) Rt under various fractal-fractional orders ω=v=1,0.95,0.90,0.85,0.80 with the estimated parameters from the reported data.
(a)(b)(c)(d)Now, we simulate and discuss the dynamics of the model based on the parameters provided by [60]. Based on this source, we assume Θ=30, r=0.003, s=0.05, b=0.05, b1=0.05, p=0.4, b2=0.04, r1=0.05, q=0.2, b3=0.6, and r2=0.6. Finally, the initial values for state functions are S0=0.5, P10=0.3, P20=0.2, and R0=0.1. In different figures, we will show the behaviors of four state functions S, P1, P2, and R by assuming different values for fractal and fractional orders ω=ν=1.00,0.99,0.98,0.97,0.96,0.95.In Figures3(a) and 3(b), we illustrate the obtained dynamics of all four state functions S, P1, P2, and R by the use of ABM with the vaccination rate (a) b=0.05 and (b) b=0.1, respectively. The great impact of the vaccine can be clearly observed from these illustrations as increasing the vaccination rate decreases the infected population and increases the recovered population.Figure 3
Behaviors of four trajectories for (a) vaccination rateb=0.05 and (b) vaccination rate b=0.1, under fractal-fractional order ω=v=1.
(a)(b)In Figure4, the susceptible subclass St is demonstrated with the initial value S0=0.5. From this illustration, we observed that the graphs of this category of people converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, we can see that by increasing the fractal-fractional orders, the density of St also increases.Figure 4
Behavior ofSt by changing fractal-fractional order ω and v.In Figure5, the asymptomatic subclass P1t is demonstrated with the initial value P1t=0.3. From this illustration, we observed that the graphs of this category of people converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, we can see that by increasing the fractal-fractional orders, the density of asymptomatic category P1t also increases.Figure 5
Behavior ofP1t by changing fractal-fractional order ω and v.In Figure6, the symptomatic subclass P2t is presented with the initial value P2t=0.2. From this illustration, we can see that the graphs of this category of people converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, we can see that by increasing the fractal-fractional orders, the density of symptomatic category P2t also increases.Figure 6
Behavior ofP2t by changing fractal-fractional order ω and v.In Figure7, the recovered category Rt is demonstrated with the initial value Rt=0.1. From this illustration, we observed that the graphs of this category of people converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, we can see that by increasing the fractal-fractional orders, the density of recovered population Rt also increases.Figure 7
Behavior ofRt by changing fractal-fractional order ω and v.It is seen that the graphs of all four category of people have the similar behaviors regarding to different values of fractal-fractional orders, and they converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, the densities of all four group of population are increasing as the fractal-fractional order increases.


## 7.1. Numerical Adams-Bashforth Method

In this section, we describe the numerical scheme in relation to the fractal-fractional model of SARS-CoV-2 virus (6). For this, we have taken help from the technique regarding two-step Lagrange polynomials called fractional Adams-Bashforth method (ABM). To begin this process, we follow the numerical method of fractal-fractional integral equations (15) using a new approach at tn+1. In other words, we discretize the mentioned equation (15) for t=tn+1, and we have
(88)Stn+1=S0+νΓω∫0tn+1tn+1−mω−1H1mdm,P1tn+1=P1,0+νΓω∫0tn+1tn+1−mω−1H2mdm,P2tn+1=P2,0+νΓω∫0tn+1tn+1−mω−1H3mdm,Rtn+1=R0+νΓω∫0tn+1tn+1−mω−1H4mdm,where
(89)H1m=mν−1F1m,Sm,P1m,P2m,Rm,H2m=mν−1F2m,Sm,P1m,P2m,Rm,H3m=mν−1F3m,Sm,P1m,P2m,Rm,H4m=mν−1F4m,Sm,P1m,P2m,Rm.By approximating above integrals, we get(90)Stn+1=S0+νΓω∑l=0n∫tltl+1tn+1−mω−1H1mdm,P1tn+1=P1,0+νΓω∑l=0n∫tltl+1tn+1−mω−1H2mdm,P2tn+1=P2,0+νΓω∑l=0n∫tltl+1tn+1−mω−1H3mdm,Rtn+1=R0+νΓω∑l=0n∫tltl+1tn+1−mω−1H4mdm.In the sequel, we approximate the functionsH1, H2, H3, and H4, introduced by (89), on the interval tl,tl+1 via two-step Lagrange interpolation polynomials with the step size h=tl−tl−1 as
(91)H1,l∗m≃m−tl−1htlν−1F1ml,Sl,P1,l,P2,l,Rl−m−tlhtl−1ν−1F1ml−1,Sl−1,P1,l−1,P2,l−1,Rl−1,H2,l∗m≃m−tl−1htlν−1F2ml,Sl,P1,l,P2,l,Rl−m−tlhtl−1ν−1F2ml−1,Sl−1,P1,l−1,P2,l−1,Rl−1,H3,l∗m≃m−tl−1htlν−1F3ml,Sl,P1,l,P2,l,Rl−m−tlhtl−1ν−1F3ml−1,Sl−1,P1,l−1,P2,l−1,Rl−1,H4,l∗m≃m−tl−1htlν−1F4ml,Sl,P1,l,P2,l,Rl−m−tlhtl−1ν−1F4ml−1,Sl−1,P1,l−1,P2,l−1,Rl−1.Then, we have(92)Stn+1=S0+νΓω∑l=0n∫tltl+1tn+1−mω−1H1,l∗mdm,P1tn+1=P1,0+νΓω∑l=0n∫tltl+1tn+1−mω−1H2,l∗mdm,P2tn+1=P2,0+νΓω∑l=0n∫tltl+1tn+1−mω−1H3,l∗mdm,Rtn+1=R0+νΓω∑l=0n∫tltl+1tn+1−mω−1H4,l∗mdm.By evaluating above integrals directly, the approximate solutions of the given fractional-fractal model of SARS-CoV-2 virus (6) are given by
(93)Sn+1=S0+νhωΓω+2∑l=0ntlν−1F1tl,Sl,P1,l,P2,l,RlYn,l−tl−1ν−1F1tl−1,Sl−1,P1,l−1,P2,l−1,Rl−1Y^n,l,P1,n+1=P1,0+νhωΓω+2∑l=0ntlν−1F2tl,Sl,P1,l,P2,l,RlYn,l−tl−1ν−1F2tl−1,Sl−1,P1,l−1,P2,l−1,Rl−1Y^n,l,P2,n+1=P2,0+νhωΓω+2∑l=0ntlν−1F3tl,Sl,P1,l,P2,l,RlYn,l−tl−1ν−1F3tl−1,Sl−1,P1,l−1,P2,l−1,Rl−1Y^n,l,Rn+1=R0+νhωΓω+2∑l=0ntlν−1F4tl,Sl,P1,l,P2,l,RlYn,l−tl−1ν−1F4tl−1,Sl−1,P1,l−1,P2,l−1,Rl−1Y^n,l,where
(94)Yn,l=n+1−lωn−l+2+ω−n−lωn−l+2+2ω,Y^n,l=n+1−lω+1−n−lωn−l+1+ω,where ω is the fractional order of the given fractal-fractional system (6).


## 7.2. Simulations Based on Adams-Bashforth Method

In this section, using the AB method for fractal-fractional, we present approximate solutions for the fractal-fractional probability-based model of SARS-CoV-2 virus (6). We demonstrate simulations to observe the behavior of four subclasses of SARS-CoV-2, which are S, P1, P2, and R under the different set of parameters.To provide a numerical simulation, we start by determining the value of the parameters by using reported cases in Turkey from 01 January 2021 to 03 July 2021. The birth rate for the Turkey in 2021 is 15.408 births per 1000 people, and the death rate isb1=5.5 per 1000 people. The Turkey’s population on 1st of January was N=84339067. Since we use the day as time limit, we can calculate the newborn rate as Θ=84339067×15.408/1000×365. To estimate the remaining parameters, we use the curve fitting technique with the data reported for SARS-CoV-2. Using this method, we determine the parameters as follows: p=0.4, r=0.003,s=0.05, b=0.05, r1=0.05, and r2=0.6, and we assume q=0.2, b2=0.04, and b3=0.6. Also, the stepsize for the time interval is choosen as h=10−3. As a first visualization, in Figure 1, we demonstrate the real data versus present model simulation. Then, behaviors of four subclasses are presented in Figures 2(a)–2(d) with the chosen initial values S=100, P1=90, P2=80, and R=70, respectively, and under various fractal-fractional orders ω and v.Figure 1
Simulated data vs. Real data for the SARS-CoV-2 cases in Turkey from 01 January 2021 to 03 July 2021.Figure 2
Behaviors of each compartments: (a)St, (b) P1t, (c) P2t, and (d) Rt under various fractal-fractional orders ω=v=1,0.95,0.90,0.85,0.80 with the estimated parameters from the reported data.
(a)(b)(c)(d)Now, we simulate and discuss the dynamics of the model based on the parameters provided by [60]. Based on this source, we assume Θ=30, r=0.003, s=0.05, b=0.05, b1=0.05, p=0.4, b2=0.04, r1=0.05, q=0.2, b3=0.6, and r2=0.6. Finally, the initial values for state functions are S0=0.5, P10=0.3, P20=0.2, and R0=0.1. In different figures, we will show the behaviors of four state functions S, P1, P2, and R by assuming different values for fractal and fractional orders ω=ν=1.00,0.99,0.98,0.97,0.96,0.95.In Figures3(a) and 3(b), we illustrate the obtained dynamics of all four state functions S, P1, P2, and R by the use of ABM with the vaccination rate (a) b=0.05 and (b) b=0.1, respectively. The great impact of the vaccine can be clearly observed from these illustrations as increasing the vaccination rate decreases the infected population and increases the recovered population.Figure 3
Behaviors of four trajectories for (a) vaccination rateb=0.05 and (b) vaccination rate b=0.1, under fractal-fractional order ω=v=1.
(a)(b)In Figure4, the susceptible subclass St is demonstrated with the initial value S0=0.5. From this illustration, we observed that the graphs of this category of people converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, we can see that by increasing the fractal-fractional orders, the density of St also increases.Figure 4
Behavior ofSt by changing fractal-fractional order ω and v.In Figure5, the asymptomatic subclass P1t is demonstrated with the initial value P1t=0.3. From this illustration, we observed that the graphs of this category of people converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, we can see that by increasing the fractal-fractional orders, the density of asymptomatic category P1t also increases.Figure 5
Behavior ofP1t by changing fractal-fractional order ω and v.In Figure6, the symptomatic subclass P2t is presented with the initial value P2t=0.2. From this illustration, we can see that the graphs of this category of people converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, we can see that by increasing the fractal-fractional orders, the density of symptomatic category P2t also increases.Figure 6
Behavior ofP2t by changing fractal-fractional order ω and v.In Figure7, the recovered category Rt is demonstrated with the initial value Rt=0.1. From this illustration, we observed that the graphs of this category of people converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, we can see that by increasing the fractal-fractional orders, the density of recovered population Rt also increases.Figure 7
Behavior ofRt by changing fractal-fractional order ω and v.It is seen that the graphs of all four category of people have the similar behaviors regarding to different values of fractal-fractional orders, and they converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, the densities of all four group of population are increasing as the fractal-fractional order increases.


## 8. Model Dynamics in the Caputo Sense

In this section, we convert the presented fractal-fractional epidemic probability-based model of SARS-CoV-2 virus (6) into a Caputo-type model. The main motivation of this replacement is to compare the proposed model in two different type and capture the memory effects on the given model by using different fractional-order dynamics. The new formulation of the proposed model is as follows:
(95)CD0,tωSt=Θ−rP1tSt−rsP2tSt−b+b1St,CD0,tωP1t=prP1tSt+rsP2tSt−b1+b2+r1P1t,CD0,tωP2t=1−prP1tSt+rsP2tSt+qr1P1t−b1+b3+r2P2t,CD0,tωRt=r11−qP1t+r2P2t+bSt−b1Rt,where
(96)CD0,tωut=RLI0,tη−ωdηdtηut,η−1<ω≤η,η∈ℕ.The Caputo fractional derivative satisfies the Newton–Leibniz formula for every0<ω<1, that is,
(97)RLI0,tωCD0,tωut=ut−∑j=0ω−1uj0tjj!.In recent years, many researchers have developed a number of numerical methods to solve different types of fractional-order models. In this section, our aim is to use a new method called the FTOMM (see ref. [62, 63]) method (fractional Taylor operational matrix method), to solve the probability-based model of the SARS-CoV-2 virus (95) in the Caputo settings.


### 8.1. Function Approximation and Operational Matrix

The Taylor vector of the fractional order is given as [64]
(98)Tnκ=1,tκ,t2κ,⋯,tnκ,where n∈ℕ and κ>0. Let Tnκ⊂S where S∈L20,1. For any φ∈M, since M=span1,tκ,t2κ,⋯,tnκ is a vector space of finite dimension in S, thus φ possesses a unique best apporoximation φ∗, that is,
(99)∀φ^∈M,φ−φ∗≤φ−φ^.Then, the functionφ is approximated by the fractional-order Taylor vector by
(100)φ≃φ∗=∑j=0ngjtjκ=GTTnκ,where GT=g0,g1,⋯,gm are the unique coefficients.ConsiderFt,w as an operational matrix of ωth-integration with m+12 dimension. Then, the ωth-R-L-integration of the Taylor vector defined in equation (98) is
(101)RLI0,tωTnκt=Ft,ωTnkt.By applying theωth-R-L integral for Tnκ, it becomes
(102)RLI0,tωTnκt=1Γω+1tω,Γκ+1Γκ+ω+1tκ+ω,⋯,Γnκ+1Γnκ+ω+1tnκ+ω.Thus, (102) can be reformulated as
(103)RLI0,tωTnκt=tωSωTnκt,where
(104)Sω=diag1Γω+1,Γκ+1Γκ+ω+1,Γ2κ+1Γ2κ+ω+1,⋯,Γnκ+1Γnκ+ω+1.SetΦt,ω=tωSω. In this case, the fractional Taylor operational matrix of integration is reformulated by
(105)Ft,ω=diagΦt,ω,Φt,ω,⋯,Φt,ω.The product of two Taylor basis vectors is(106)RLI0,tωTnκtTnκTt=tωPω∗TnκtTnκTt,where
(107)Pω=1Γω+1Γκ+1Γκ+ω+1⋯Γnκ+1Γnκ+ω+1Γκ+1Γκ+ω+1Γ2κ+1Γ2κ+ω+1⋯Γn+1κ+1Γn+1κ+ω+1⋮⋮⋮⋮Γnκ+1Γκ+ω+1Γn+1κ+1Γn+1κ+ω+1⋯Γ2nκ+1Γ2nκ+ω+1,(108)TnκtTnκTt=1tωt2ω⋯tnωtωt2ωt3ω⋯tn+1ωt2ωt3ωt4ω⋯tn+2ω⋮⋮⋮⋮⋮tnωtn+1ωtn+2ω⋯t2nω.Again, by utilizingRLI0,tω on the matrix (108), we get
(109)RLI0,tωTnκtTnκTt=1Γω+1Γκ+1Γκ+ω+1tω⋯Γnκ+1Γnκ+ω+1tnωΓκ+1Γκ+ω+1tωΓ2κ+1Γ2κ+ω+1t2ω⋯Γn+1κ+1Γn+1κ+ω+1tn+1ω⋮⋮⋮⋮Γnκ+1Γκ+ω+1tnωΓn+1κ+1Γn+1κ+ω+1tn+1ω⋯Γ2nκ+1Γ2nκ+ω+1t2nω.


### 8.2. Application of FTOMM on the SARS-CoV-2 Model

In this part, the suggested FTOMM method is utilized to the model of SARS-CoV-2 virus given in (95).We start by expandingCD0,tωSt,CD0,tωP1t,CD0,tωP2t, and CD0,tωRt with the help of a fractional Taylor basis vector as following:
(110)CD0,tωSt≈CTTnκt,CD0,tωP1t≈KTTnκt,CD0,tωP2t≈LTTnκt,CD0,tωRt≈NTTnκt.Next, operating theωth-R-L integral on above equations and using initial values S0,P10,P20, and R0, we get
(111)St≈CTFt,ωTnκt+S0,P1t≈KTFt,ωTnκt+P10,P2t≈LTFt,ωTnκt+P20,Rt≈NTFt,ωTnκt+R0.Substituting (110) and (111) into SARS-CoV-2 model (95), we get
(112)CTTnκt=Θ−rKTFt,ωTnκt+S10CTFt,ωTnκt+S0−rsKTFt,ωTnκt+P10CTFt,ωTnκt+S0−b+b1CTFt,ωTnκt+S0,KTTnκt=prKTFt,ωTnκt+P10CTFt,ωTnκt+S0+prsKTFt,ωTnκt+P10CTFt,ωTnκt+S0−b1+b2+r1KTFt,ωTnκt+P10,LTTnκt=1−prKTFt,ωTnκt+P10CTFt,ωTnκt+S0+1−prsKTFt,ωTnκt+P10CTFt,ωTnκt+S0+qr1KTFt,ωTnκt+P10−b1+b3+r2LTFt,ωTnκt+P20,NTTnκt=r11−qKTFt,ωTnκt+P10+r2LTFt,ωTnκt+P20+bCTFt,ωTnκt+S0−b1NTFt,ωTnκt+R0.Now, by using above equations and collocation pointstj=j/n, where j=0,1,⋯n, we derive a system of 4n+4 algebraic nonlinear equations with 4n+4 unknown coefficients. This system is solved efficiently for the unknown coefficient vectors CT, KT, LT, and NT by using the Newton method in MATLAB software.As a final step, substituting the vectors of coefficientsCT, KT, LT, and NT into (111), we obtain for St, P1t, P2t, and Rt approximately.


### 8.3. Simulations Based on FTOMM Method and Comparison with Adams-Bashforth Method

In this section, all graphical results of the fractional SARS-CoV-2 model (95) by using FTOMM and their comparison between ABM are illustrated through Figures 8–14. To see the correctness and having a comparison, we illustrate the graphical representation of the presented model at several values of ω.Figure 8
Comparisons between the ABM and FTOMM for the parametric values.
(a)(b)(c)(d)Figure 9
Absolute error comparisons between the ABM and FTOMM for the parametric values.
(a)(b)(c)(d)Figure 10
St by changing ω where m=7.Figure 11
P1t by changing ω where m=7.Figure 12
P2t by changing ω where m=7.Figure 13
Rt by changing ω where m=7.Figure 14
Comparisons between the ABM and FTOMM for the parametric values ofS2.
(a)(b)(c)(d)In Figure8, we present a comparison of obtained solutions by use of the ABM and FTOMM for the parametric values assumed in subsection 7.2. From Figures 8(a)–8(d), we can clearly conclude that the both acquired numerical solutions of four state functions St, P1t, P2t, and Rt by use of ABM and FTOMM are identical.In Table1, we present the solutions of four subclasses St, P1t, P2t, and Rt obtained by use of the Adams-Bashforth and fractional Taylor operational matrix methods.Table 1
Compared approximate results of four state functions obtained by ABM and FTOMM. (a)S, (b) P1, (c) P2, and (d) R).
(a)tSABMSFTOMM00.500.5025266.9033269.791750129.1402130.481975114.4625115.8542100115.6538117.1644125115.6506115.2917150115.6489115.1839(b)tP1ABMP1FTOMM00.300.30252.68591.97535054.181552.99187553.131653.321410052.660152.744312552.671353.586315052.671753.9160(c)tP2ABMP2FTOMM00.200.20251.97530.87325010.754210.1061759.26299.19541009.26829.25011259.27029.37291509.27029.1423(d)tRABMRFTOMM00.100.1025150.9550158.087550270.5701270.572175271.7889271.7783100269.6428269.7108125269.2052269.9315150269.0798266.3295In Figure9, we give the graphical illustration of the absolute errors of four subclasses St, P1t, P2t, and Rt obtained by the Adams-Bashforth and fractional Taylor operational matrix methods.In this part, by using FTOMM, we simulate and discuss the behavior of the model based on the parametric values of the setS2 provided by [60]. From this source, we assume the new parametric values to be Θ=20, r=0.079, s=0.0001, b=0.9, b1=0.16, p=0.29, b2=0.11, r1=0.45, q=0.2, b3=0.8, and r2=0.9. Finally, the initial values for state functions are the following:
(113)S0=0.5,P10=0.3,P20=0.2,R0=0.1.In Figures10–13, we present the behaviors of solutions of four state functions St, P1t, P2t, and Rt, respectively, which are obtained by using FTOMM for some values of ω=1.00,0.90,0.80,0.70,0.60,0.50 where t∈0,150.From Figure10, we can see the illustration of St with initial value S0=0.5 for several values of ω. It can be observed from this graph that the order of fractional derivative has an effect on convergence of people of susceptible category to stable case. Namely, at higher fractional orders, it converges slowly to a stable case, while at lower fractional order, this process is more quickly. About the density of St, we can observe that by increasing the fractional order, the density also increases. Also, we can clearly see that the fractional orders are highly consistent with integer order when using FTOMM.From Figures11–13, we can see the illustration of P1t, P2t, and Rt with P10=0.3, P20=0.2, and R0=0.1, respectively, for some values of ω. It can be observed from these graphs that at higher fractional orders, people of asymptomatic, symptomatic, and recovered categories converge slowly to a stable case, while at lower fractional order, it is more quickly. Also, we observe that by increasing the fractional orders, the densities of St, P1t, P2t, and Rt increases too.In Figure14, we present the comparison of the obtained solutions by use of the ABM and FTOMM for the parametric values of set S2. From Figures 14(a)–14(d), we can clearly see that the both obtained approximate solutions of four state functions St, P1t, P2t, and Rt by use of ABM and FTOMM are behaving identical.It is clear from all figures that both obtained solutions by fractional Taylor operational matrix method and Adams-Bashforth method are identical. We can conclude that fractional Taylor operational matrix method gives almost the same results as the results acquired by Adams-Bashforth technique. Also, more accurate results can be obtained by enhancing the value ofm and κ. Due to the simplicity of FTOMM, it is effective and has advantages for mathematical modelling of dynamics of SARS-CoV-2 virus.


## 8.1. Function Approximation and Operational Matrix

The Taylor vector of the fractional order is given as [64]
(98)Tnκ=1,tκ,t2κ,⋯,tnκ,where n∈ℕ and κ>0. Let Tnκ⊂S where S∈L20,1. For any φ∈M, since M=span1,tκ,t2κ,⋯,tnκ is a vector space of finite dimension in S, thus φ possesses a unique best apporoximation φ∗, that is,
(99)∀φ^∈M,φ−φ∗≤φ−φ^.Then, the functionφ is approximated by the fractional-order Taylor vector by
(100)φ≃φ∗=∑j=0ngjtjκ=GTTnκ,where GT=g0,g1,⋯,gm are the unique coefficients.ConsiderFt,w as an operational matrix of ωth-integration with m+12 dimension. Then, the ωth-R-L-integration of the Taylor vector defined in equation (98) is
(101)RLI0,tωTnκt=Ft,ωTnkt.By applying theωth-R-L integral for Tnκ, it becomes
(102)RLI0,tωTnκt=1Γω+1tω,Γκ+1Γκ+ω+1tκ+ω,⋯,Γnκ+1Γnκ+ω+1tnκ+ω.Thus, (102) can be reformulated as
(103)RLI0,tωTnκt=tωSωTnκt,where
(104)Sω=diag1Γω+1,Γκ+1Γκ+ω+1,Γ2κ+1Γ2κ+ω+1,⋯,Γnκ+1Γnκ+ω+1.SetΦt,ω=tωSω. In this case, the fractional Taylor operational matrix of integration is reformulated by
(105)Ft,ω=diagΦt,ω,Φt,ω,⋯,Φt,ω.The product of two Taylor basis vectors is(106)RLI0,tωTnκtTnκTt=tωPω∗TnκtTnκTt,where
(107)Pω=1Γω+1Γκ+1Γκ+ω+1⋯Γnκ+1Γnκ+ω+1Γκ+1Γκ+ω+1Γ2κ+1Γ2κ+ω+1⋯Γn+1κ+1Γn+1κ+ω+1⋮⋮⋮⋮Γnκ+1Γκ+ω+1Γn+1κ+1Γn+1κ+ω+1⋯Γ2nκ+1Γ2nκ+ω+1,(108)TnκtTnκTt=1tωt2ω⋯tnωtωt2ωt3ω⋯tn+1ωt2ωt3ωt4ω⋯tn+2ω⋮⋮⋮⋮⋮tnωtn+1ωtn+2ω⋯t2nω.Again, by utilizingRLI0,tω on the matrix (108), we get
(109)RLI0,tωTnκtTnκTt=1Γω+1Γκ+1Γκ+ω+1tω⋯Γnκ+1Γnκ+ω+1tnωΓκ+1Γκ+ω+1tωΓ2κ+1Γ2κ+ω+1t2ω⋯Γn+1κ+1Γn+1κ+ω+1tn+1ω⋮⋮⋮⋮Γnκ+1Γκ+ω+1tnωΓn+1κ+1Γn+1κ+ω+1tn+1ω⋯Γ2nκ+1Γ2nκ+ω+1t2nω.


## 8.2. Application of FTOMM on the SARS-CoV-2 Model

In this part, the suggested FTOMM method is utilized to the model of SARS-CoV-2 virus given in (95).We start by expandingCD0,tωSt,CD0,tωP1t,CD0,tωP2t, and CD0,tωRt with the help of a fractional Taylor basis vector as following:
(110)CD0,tωSt≈CTTnκt,CD0,tωP1t≈KTTnκt,CD0,tωP2t≈LTTnκt,CD0,tωRt≈NTTnκt.Next, operating theωth-R-L integral on above equations and using initial values S0,P10,P20, and R0, we get
(111)St≈CTFt,ωTnκt+S0,P1t≈KTFt,ωTnκt+P10,P2t≈LTFt,ωTnκt+P20,Rt≈NTFt,ωTnκt+R0.Substituting (110) and (111) into SARS-CoV-2 model (95), we get
(112)CTTnκt=Θ−rKTFt,ωTnκt+S10CTFt,ωTnκt+S0−rsKTFt,ωTnκt+P10CTFt,ωTnκt+S0−b+b1CTFt,ωTnκt+S0,KTTnκt=prKTFt,ωTnκt+P10CTFt,ωTnκt+S0+prsKTFt,ωTnκt+P10CTFt,ωTnκt+S0−b1+b2+r1KTFt,ωTnκt+P10,LTTnκt=1−prKTFt,ωTnκt+P10CTFt,ωTnκt+S0+1−prsKTFt,ωTnκt+P10CTFt,ωTnκt+S0+qr1KTFt,ωTnκt+P10−b1+b3+r2LTFt,ωTnκt+P20,NTTnκt=r11−qKTFt,ωTnκt+P10+r2LTFt,ωTnκt+P20+bCTFt,ωTnκt+S0−b1NTFt,ωTnκt+R0.Now, by using above equations and collocation pointstj=j/n, where j=0,1,⋯n, we derive a system of 4n+4 algebraic nonlinear equations with 4n+4 unknown coefficients. This system is solved efficiently for the unknown coefficient vectors CT, KT, LT, and NT by using the Newton method in MATLAB software.As a final step, substituting the vectors of coefficientsCT, KT, LT, and NT into (111), we obtain for St, P1t, P2t, and Rt approximately.


## 8.3. Simulations Based on FTOMM Method and Comparison with Adams-Bashforth Method

In this section, all graphical results of the fractional SARS-CoV-2 model (95) by using FTOMM and their comparison between ABM are illustrated through Figures 8–14. To see the correctness and having a comparison, we illustrate the graphical representation of the presented model at several values of ω.Figure 8
Comparisons between the ABM and FTOMM for the parametric values.
(a)(b)(c)(d)Figure 9
Absolute error comparisons between the ABM and FTOMM for the parametric values.
(a)(b)(c)(d)Figure 10
St by changing ω where m=7.Figure 11
P1t by changing ω where m=7.Figure 12
P2t by changing ω where m=7.Figure 13
Rt by changing ω where m=7.Figure 14
Comparisons between the ABM and FTOMM for the parametric values ofS2.
(a)(b)(c)(d)In Figure8, we present a comparison of obtained solutions by use of the ABM and FTOMM for the parametric values assumed in subsection 7.2. From Figures 8(a)–8(d), we can clearly conclude that the both acquired numerical solutions of four state functions St, P1t, P2t, and Rt by use of ABM and FTOMM are identical.In Table1, we present the solutions of four subclasses St, P1t, P2t, and Rt obtained by use of the Adams-Bashforth and fractional Taylor operational matrix methods.Table 1
Compared approximate results of four state functions obtained by ABM and FTOMM. (a)S, (b) P1, (c) P2, and (d) R).
(a)tSABMSFTOMM00.500.5025266.9033269.791750129.1402130.481975114.4625115.8542100115.6538117.1644125115.6506115.2917150115.6489115.1839(b)tP1ABMP1FTOMM00.300.30252.68591.97535054.181552.99187553.131653.321410052.660152.744312552.671353.586315052.671753.9160(c)tP2ABMP2FTOMM00.200.20251.97530.87325010.754210.1061759.26299.19541009.26829.25011259.27029.37291509.27029.1423(d)tRABMRFTOMM00.100.1025150.9550158.087550270.5701270.572175271.7889271.7783100269.6428269.7108125269.2052269.9315150269.0798266.3295In Figure9, we give the graphical illustration of the absolute errors of four subclasses St, P1t, P2t, and Rt obtained by the Adams-Bashforth and fractional Taylor operational matrix methods.In this part, by using FTOMM, we simulate and discuss the behavior of the model based on the parametric values of the setS2 provided by [60]. From this source, we assume the new parametric values to be Θ=20, r=0.079, s=0.0001, b=0.9, b1=0.16, p=0.29, b2=0.11, r1=0.45, q=0.2, b3=0.8, and r2=0.9. Finally, the initial values for state functions are the following:
(113)S0=0.5,P10=0.3,P20=0.2,R0=0.1.In Figures10–13, we present the behaviors of solutions of four state functions St, P1t, P2t, and Rt, respectively, which are obtained by using FTOMM for some values of ω=1.00,0.90,0.80,0.70,0.60,0.50 where t∈0,150.From Figure10, we can see the illustration of St with initial value S0=0.5 for several values of ω. It can be observed from this graph that the order of fractional derivative has an effect on convergence of people of susceptible category to stable case. Namely, at higher fractional orders, it converges slowly to a stable case, while at lower fractional order, this process is more quickly. About the density of St, we can observe that by increasing the fractional order, the density also increases. Also, we can clearly see that the fractional orders are highly consistent with integer order when using FTOMM.From Figures11–13, we can see the illustration of P1t, P2t, and Rt with P10=0.3, P20=0.2, and R0=0.1, respectively, for some values of ω. It can be observed from these graphs that at higher fractional orders, people of asymptomatic, symptomatic, and recovered categories converge slowly to a stable case, while at lower fractional order, it is more quickly. Also, we observe that by increasing the fractional orders, the densities of St, P1t, P2t, and Rt increases too.In Figure14, we present the comparison of the obtained solutions by use of the ABM and FTOMM for the parametric values of set S2. From Figures 14(a)–14(d), we can clearly see that the both obtained approximate solutions of four state functions St, P1t, P2t, and Rt by use of ABM and FTOMM are behaving identical.It is clear from all figures that both obtained solutions by fractional Taylor operational matrix method and Adams-Bashforth method are identical. We can conclude that fractional Taylor operational matrix method gives almost the same results as the results acquired by Adams-Bashforth technique. Also, more accurate results can be obtained by enhancing the value ofm and κ. Due to the simplicity of FTOMM, it is effective and has advantages for mathematical modelling of dynamics of SARS-CoV-2 virus.


## 9. Conclusions

In this manuscript, a fractal-fractional epidemic probability-based model of the SARS-CoV-2 virus with four compartments including susceptible, asymptomatic, symptomatic, and recovered was designed. By recalling a special group of contractions, namedϕ-admissible ϕ-ψ-contractionas, we proved the existence property for fixed points of a fractal-fractional operator which is the same soultion of the mentioned system. Furthermore, other theoretical properties like stable solutions and their uniqueness for each compartments of the fractal-fractional model were established. We derived numerical solutions via the Adams-Bashforth and simulated them from several aspects such as variations of fractal-fractional dimension orders. Further, we formulated a Caputo type of the fractional model and compared its solutions obtained by the FTOMM method, with the previous ones of the fractal-fractional model. All simulations showed similar and close outcomes. From all illustrations presented in this work, we observed that the population of infected people converge quickly to a stable case at higher fractal-fractional orders and slowly to such a stable case at lower fractal-fractional orders. Also, we can see that by increasing the fractal-fractional orders, the density of susceptible population also increases. Also, from Figure 3, we can see that the probability of disease extinction increases with vaccination rate. All the numerical results and calculations are obtained with the help of MATLAB version R2019A. In the future, we aim to compare the results of our methods in the framework of other types of nonsingular kernels. Also, as a future study, the techniques introduced in this study can be modified to apply to other diseases and new variants of SARS-CoV-2 for different compartments.



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*Source: 2388557-2022-09-01.xml*