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.

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 (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 is the probability that both categories susceptible and infected interact and this leads to the asymptomatic category, in that case, 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 -admissible --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 of -admissible --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.

Let display a subclass of nondecreasing operators like s.t.

Definition 1 (see [58]). Let be a normed space and and .
is --contraction if for , (q) is -admissible if yields .

Definition 2 (see [48]). Let be fractal differentiable on of order . The fractional-fractal -derivative of the function via the power law type kernel in the Riemann-Liouville sense is defined by where is the fractal derivative.

It is known that if , then the fractal-fractional derivative is reduced to the standard derivative of order .

Definition 3 (see [48]). Let be continuous on . The fractional-fractal integral of the function with fractional order and fractal order is

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 , , , and , which are a part of total population. This model is where stands for the people belonging to the susceptible category, is the people belonging to the asymptomatic category, is the people belonging to the symptomatic category, and stands for the people belonging to the recovered category at the time . 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: subject to where is the fractional-fractal derivative of the fractional order and the fractal order via the power law type kernel. We have in which as we said above, means the total population at the time .

About parameters, the total natural death rate along with the rate of disease-related death for both infected groups is specified by the symbols , , and , respectively. We show the rate of transmission of disease by , and its reduced rate is denoted by the symbol . The vaccination rate is given by , and stands for the newborn rate. The probability of the asymptomatic persons is illustrated by , and the probability of these persons that recover in the symptomatic step is specified by . Moreover, is the recovery rate in relation to asymptomatic persons and accordingly, and 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 where , as

We write the R.H.S. of model (6) by

Hence,

By (11), we derive the following IVP: where

Now, the fractional-fractal integral acts on (12), and it becomes

In other words, the extended form of the above fractal-fractional integral is represented as

Consider the operator as

In the preceding, we recall the required fixed point theorem in connection with our aim for proving the existence results.

Theorem 4 (see [58]). Assume as a Banach space, , , and as an --contraction s.t. (1) is -admissible(2)(3)for any sequence in with and for all , we have Then, s.t. .

Now, the first existence result is proved here under some special operators.

Theorem 5. Let , a continuous map , and a nondecreasing map . Let (), and , with , where .
() exists so that , and also, the inequality gives for any and .
() belonging to with and for each and , we get 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. Let and be two members belonging to with for each . Then, by definition of the Beta function, we may write Consequently, we have Now, is introduced by the this rule: Then, for every , we will get Thus, is found as an --contraction. To verify that is -admissible, let be arbitrary and . By definition of , we have Then, by , is satisfied. Again, the definition of gives . Thus, is -admissible.
On the other hand, the condition guarantees the existence of . In this case, for each , holds. Clearly, we get . These show that the conditions (1) and (2) of Theorem 4 are fulfilled.
Now, we assume that is a sequence in s.t. , and for all , . By virtue of definition of , Therefore, in the light of hypothesis , we obtain This indicates that for every . This guarantees the condition (3) of Theorem 4. Ultimately, by utilizing Theorem 4, we conclude that it found a fixed point for like This implies that 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]). Regard as a Banach space, as a bounded closed set in with the convexity property, and an open set with . The compact continuous map , either (i) possesses fixed point in or(ii) and s.t. .

Theorem 7. Assume along with the following:
(C1): and an increasing map exist provided that (C2): There exist with in which and 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 .