Modeling and Dynamics of the Fractional Order SARS-CoV-2 Epidemiological Model

Department of Mathematics and Statistics, Woman University Swabi, Khyber Pakhtunkhwa, Pakistan Department of Computing, Muscat College Muscat, Muscat, Oman Department of Mathematics, Kuwait College of Science and Technoloy, Safat 13133, P. O. Box 27235, Kuwait Department of Mathematics, University of Malakand, Chakdara Dir Lower, Khyber Pakhtunkhwa, Pakistan Department of Mathematical Sciences, UAE University, P. O. Box 15551, Al-Ain, UAE Department of Clinical Laboratory Sciences, Central Research Laboratory, College of Applied Medical Sciences, King Khalid University, Abha, Saudi Arabia


Introduction
e coronavirus family causes infections in humans, beginning with a common cold and progressing to SARS. Two coronavirus epidemics have been recorded in the preceding twenty years [1][2][3]. SARS was one of them, and it created a large-scale outbreak in several nations. Approximately, 8000 people were affected by this outbreak, with 800 of them dying. In December 2019, a serious respiratory sickness outbreak began in Wuhan, China [4]. In early January 2020, the causal agent, a new coronavirus, was identified and isolated from a single patient . According to scientific evidence, animals were the earliest source of virus transmission, although the majority of cases are caused by infected humans contacting susceptible humans. e spread of this virus is a hot topic that has touched practically every corner of the globe and has been reported in over 200 countries. According to current records, there have been over 401,288,380 confirmed cases, with 5,783,182 deaths occurring till February 9th 2022. According to the WHO, this is a public health emergency.
e World Health Organization (WHO) has designated it a public health emergency of worldwide concern due to the severity of the condition. is virus appears to be highly contagious, spreading rapidly to nearly every country on the planet, prompting the declaration of a global pandemic. It signifies that it is a highly major public health threat, with symptoms such as cough, fever, lethargy, and breathing problems after infection.
Fractional computing is an emerging area of mathematics and attracted the attention of researchers. Because of the wide applications to express the axioms of heritage and recall different physical situations that occur in various fields of applied science. Many classical models have been shown with less accuracy in prediction about the temporal dynamics of the disease, while on the other hand, models with noninteger order provide better information in allocating and preserving data for large-scale analysis [5][6][7]. Moreover, the derivative of integer order does not find the dynamics between two various points [8,9]. Furthermore, the comparison of integer and noninteger order epidemic models reveals that models with noninteger order are the generalization of integer order and provide more and accurate dynamics rather than the classical order, (for detail see [9][10][11]). A model with noninteger order demonstrating the complex dynamics of a biological system has been proposed by Asma et al. [12]. A fractional-order epidemic model has been investigated to explore the dynamics of toxoplasmosis in feline and human populations [13]. Another study has been reported and studied the stability analysis of pests in tea with fractional order [14]. Similarly, many authors studied dynamics of infectious diseases with fractional-order derivatives, for e.g., Hadamard and Caputo and Rieman and Liouville [15][16][17][18][19]. For the solution of Caputofractional order, epidemiological models of many iterative and numerical methods have been developed; however, the complications of singular kernel arise. So, Caputo and Fabrizio presented an idea based on the nonsingular kernel to overcome the limitation that arises in the above fractional-order derivatives [20].
Coronavirus disease 2019 (COVID-19) is one of the top infectious diseases among other ones and therefore has been recognized as a global threat by World Health Organization (WHO). Due to novel characteristics of the coronavirus disease various researchers have taken a keen interest. Several researchers formulated various epidemiological models to study the dynamics of communicable diseases (see for instance [21][22][23][24][25]). e current pandemic of novel coronavirus disease is also a burning issue and many biologists and mathematicians reported different studies. For example, Wu et al. introduced a model to describe the transmission of the disease based on reported data from 31.12.2019 to 28.01.2020 [26]. Imai et al. studied the transmission of the disease with the help of computational modeling to estimate the disease outbreak in Wuhan, whose main focus was on the human-to-human transmission [27]. Another study has been investigated by Zhu et al. [28] to analyze the infectivity of the novel coronavirus. e dynamical analysis of the novel disease of COVID-19 under the effect of the carrier with environmental contamination has been performed by Hattaf et al. [29]. All the reported studies indicate that bats and minks may be two animal hosts of the novel coronavirus. Similarly, many more studies have been reported on the dynamics of a novel coronavirus, for instance, see [30,31]. Nevertheless, the literature reveals that the work proposed is an excellent contribution, however, it could be possible to improve further by incorporating some interesting and important factors related to the novel coronavirus disease. e spreading of coronavirus disease globally rises from the human-to-human transmission, while the initial source of the disease was an animal/reservoir. e characteristic of SARS-CoV-2 confirms that various infection phases are significant and affect the transmission. e role of asymptomatic is notable because with no symptoms it becomes the major source of transmission of the infection. So, a small number of this population leads to a big disaster. We develop a mathematical model according to the novel disease of coronavirus and keeping in view the aesthetic of the virus. To do this, first, we formulate the model and then fractionalize it to perform the fractional type analysis of the proposed model. e fractional derivative used in this study is a particular case of the new generalized Hattaf fractional (GHF) derivative [32,33]. We show that the proposed fractional-order epidemiological model is bounded and possesses positive solutions. We also find the steady states of the epidemic problem and discuss asymptotic stabilities. For this, we use the dynamical systems theory. Particularly, we utilize the linearization, mean value theorem, and Barbalat's Lemma. Moreover, the sensitivity analysis will be performed for the threshold parameter to find the impact of each epidemic parameter involved in the model mechanism. We use the sensitivity index formula for this purpose. We perform the numerical visualization of the analytical results to verify the theocratical part and show the effectiveness of the control strategy. We also show the difference between integer and noninteger order epidemiological cases.

Formulation of the Model with
Fractional Analysis e proposed problem is formulated by taking into account the characteristics of the novel coronavirus illness. We divide the total human population N h (t) into four various compartments and assume that M(t) represents the reservoir. In the proposed study, we also consider several transmission routes, such as from human-to-human and from a reservoir-to-human. Before we show the model, we make the following assumption: (i) e parameters and variables involved in the model are positive or non-negative values (ii) e inflow of newborn are susceptible (iii) e novel disease is transmitted by several routes, such as from latent and symptomatic individuals as well as from reservoirs and so accordingly incorporated. (iv) ose who have a strong immune system got natural recovery (v) Two types of recoveries i.e., from latent and symptomatic populations are taken (vi) e death rate due to disease is taken in the symptomatic infected compartment 2 Complexity As a result of combining all the above assumptions, the following system of nonlinear differential equations emerges: and the initial population sizes are assumed to be as follows: In the proposed epidemiological model, the parameters described as Λ is the new birth rate, and the disease transmission rates are symbolized by, β 1 , β 2 , and β 3 , which represent the transmission from latent, symptomatic, and reservoir, respectively. We also denote the reduced transmission coefficient by c and ψ, while c 1 is the moving ratio of latent to infected and c 2 denotes the recovery rate. We also denote the recovery rate under treatment by c 3 , while d is the natural mortality. e disease-induced rate is d 1 . Furthermore, η 1 and η 2 are the two ratios that contribute production of the virus in the seafood market. We denote the removing rate of the virus with α.

Fractional-Order Epidemiological Model.
Let σ be the fractional-order parameter 0 < σ < 1. We will extend the model to its associate fractional order. First, we give some fundamental concepts that will be used in getting our findings.
Definition 1. (see [9]). Let T > 0 and assume that ϕ ∈ H 1 (0, T), if n − 1 < σ < n and σ > 0 such that n ∈ N, then the derivative in the sense of Caputo as well as the Caputo-Fabrizio with σ order are given as follows: where CF and C are used for the representation of Caputo-Fabrizio and Caputo, respectively, while t > 0 and K(σ) are the normalization function, and K(0) � 0 � K(1).
Definition 2 see [9]). (If 0 < σ < 1 and φ(t) varies with time t, then the integral is described as follows: e above integral is known as the Riemann-Liouville integral.
e integral defined by Equation (5) is said to be the Caputo-Fabrizio-Caputo (CF) integral.
Since σ is the fractional order, and using the notion for the shake of simplicity, therefore the fractional order model looks like the following equation:

Complexity
We show that the proposed fractional-order epidemic model as reported by the above system is both biologically and mathematically feasible. For this, we discuss the positivity and boundedness of the model (6), which proves that the underconsidered problem is well-possed. We also investigate that the dynamics of the proposed model are confined to a certain region invariant positively. e following Lemmas is established for this purpose. (6) solutions and let us consider that it possessing non-negative initial sizes of population, then Proof. Since, σ is the fractional order and assuming that G represents the fractional operator with order σ, then system (1) leads to is implies that and ξ ∈ S h , L h , I h , R h , M , respectively. Following the methodology proposed in [34] and consequently used by Qureshi et al. [35], we reach to the conclusion that the solutions are non-negative for all non-negative t.

Lemma 2. Let us assume that the Ω is the feasible region of the model (6), then within it, the model that is under consideration is invariant and the feasible region is given by
Proof. Let N h (t) represent the total human population, then the use of the proposed fractional model leads to the assertion is given by the following equation: Solving equation (10), we get the following equation: It could be also noted that L h , I h ≤ N h , so the last equation of the fractional model (6) looks like the following equation: e solution of (12) leads to the following equation: In (11) and (13), E(.) denotes the Mittag-Leffler function and E σ (Z) � ∞ n�0 Γ(σi + 1)/Z n . Furthermore, it is obvious that when times grows without bound then (11) and (13) , then N h and M contained in Ω and will never leave. So, the dynamics of the fractional epidemic model can be investigated in feasible region Ω. □ □

Stability Analysis
In this section, we will examine the stability of fractional epidemiological model (6). We find the steady states first and threshold parameter (basic reproductive number) of the fractional model to investigate the stability conditions. We use the notion X 1 for disease-free equilibrium calculating at We now use the disease-free state and find the threshold parameter. is quantity represents the maximum epidemic potential of a pathogen, which describes what would happen if an infectious agent were to enter a susceptible community, and therefore is an estimate based on an idealized scenario. e effective threshold quantity depends on the nature of the population's current susceptibility.
is measure the potential transmission, which is likely lower than the basic reproduction number, depends on various factors e.g., whether some individuals have immunity due to prior exposure to the pathogen or whether some individuals are vaccinated against the disease. erefore, this quantity is effective and changes over time and is an estimate based on a more realistic situation within the population. We calculate this quantity i.e., the threshold quantity (R 0 ) of the proposed model by following [36], therefore following the next generation matrix approach, we calculate the associated matrices i.e., F and V as given by the following equation: e associated threshold quantity of model (6) is the spectral radius of the matrix It could be noted that the threshold quantity consists of three parts that describe various transmission routes. One may observe, that there is a transmission from an infected human, while the other from reservoirs.
Similarly, we use the above quantity (R 0 ) and assume that X 2 is the endemic equilibrium of the fractional order model, then the components are calculated by solving system (6) simultaneously at steady state. We also set and M � M * for the sake of convenience, then the corresponding endemic equilibrium leads to , whoso components are defined by the following equation: e endemic equilibrium reveals that X 2 exists only if R 0 > 1. For this, we sate the following result.
for the proposed problem (6) exists only whenever, the threshold quantity (R 0 ) is greater than unity.
We use the linear stability analysis to discuss the temporal dynamics of the fractional model (6) around X 1 and X 2 . So we have the following results. Theorem 1. If the threshold quantity (R 0 ) is less than unity, then the local, as well as global dynamics of the problem, is asymptotically stable around X 1 � (S h0 , 0, 0, 0, 0).
Proof. Following eorem 3 reported in [37] to obtain the required results. Since it is clear that all other compartments of the proposed model do not depend explicitly on the recovered class, so we study the dynamics of the model for only the three-compartment, which will be enough for the whole model. Let A(X 1 ) be the Jacobian matrix of the proposed model (6) around X 1 ; then, e calculation shows that A(X 1 ), obviously has two negative eigenvalue i.e., λ 1 � − d σ and λ 2 � − α σ . To find the nature of the remaining, we take the matrix given by the following equation: It is sufficient for the Routh-Hurwitz criteria that H 1 : trace(A(x 1 )) < 0, and det(A(X 1 )) > 0 holds. We calculate the trace(A(x 1 )) and det(A(X 1 )), such that It can be noted from the above equations (19)-(20) that trace(A(X1))<0 and det(A(X1))>0, if R1+R2<1.. So the Routh-Hurwitz criteria are satisfied if R 0 < 1. It proves the conclusion that the local dynamics of the model (6) is asymptotically stable, if R 0 < 1. □ e application of linear stability analysis is utilized to find the dynamics of the proposed model (6) around its Complexity 5 associated endemic equilibrium (16). For this, we describe the result as follows.
□ Theorem 2. If the threshold quantity (R 0 ) is greater than unity i.e., R 0 > 1, then the local as well as the global dynamics of the endemic equilibrium, Proof. Using the theory of a dynamical system, we discuss the local dynamics of the proposed system around the endemic equilibrium. Let A(X 2 ) be the Jacobian matrix of system (6) around X 2 ; then, We find the characteristic polynomial of the matrix (21), such that   Complexity Eigenvalues (roots) of (22) are negative or having negative real parts, if (H 0 ): k i > 0, i � 1, 2, . . . , 4 and δ > 0 holds, where It could be noted, from the above equations, that obviously H 0 holds, if R 0 > 1. us, we conclude that the local dynamics of the proposed model (6) at endemic equilibrium (X 2 ) is asymptotically stable, if the threshold quantity is greater than unity. □ Theorem 3. If R 0 ≤ 1, then the point X 1 of the model (6) is globally asymptotically stable, while the same holds for X 2 whenever R 0 > 1.
Proof. To perform the global analysis of the proposed fractional order epidemiological model, first we assume that M(t)) and if t ⟶ ∞, then it has finite limit, therefore by following the result 3.1 in [38], then from first equation of the model (7), we may write the following equation: Since for every ϕ ≤ ϕe t , so by following the result eorem 1 in [38] with the application of mean value theorem, the above (25) may leads to the following equation:

Complexity 7
where a � ‖S h (0)‖e − T + KT σ e − T /σΓ(σ) + Λ σ , and U > 0. Consequently, we may derive the following equation: Similarly, lim of L h (t), I h (t), R h (t), and M(t) can be proved. We also assume that us, in the light of mean value theorem, there exists constants C 1 > 0, C 2 > 0, such that So, eorem 2.1 and 1 in [39] implies that is uniformly continuous. us, the application of Barbalat's Lemma (see for detail, [40]) gives the following equation: lim CF t⟶∞ D α 0,t (X(t)) � (0, 0, 0, 0, 0, 0). Consequently, is the equilibrium of model (6) and by the similar procedure as adopted in [41], we reach to the following equation: So, it could be concluded that the disease endemic state X 2 does not exists if R 0 < 1, and so limX(t) � X 1 whenever t approaches ∞ and if R 0 � 1 then X 2 � X 1 , and limX(t) � X 1 as t approaches ∞, while on the other hand whenever R 0 > 1, then X 2 exists and thus limX(t) � X 2 as t tend to ∞.

Numerical Simulation
In this section, we present the numerical simulation of the proposed epidemic problem. We divided the section into two subsections in which we discuss the sensitivity analysis of every epidemic parameter and its relative impact on disease transmission. We also discuss the temporal dynamics for the long run and present the significance of fractional parameters.

Sensitivity Analysis.
We discuss the local sensitivity analysis of the model parameters to define the relation between threshold quantity and the epidemic parameters.
is allows us to measure the relative impact of every epidemic parameter on disease transmission. We follow the work presented in [42] to perform the sensitivity analysis. Using the sensitivity index formula, we get the sensitivity indices as given in Table 1. It may be noted from the sensitivity indices that the set of parameters S 1 � β 1 , β 2 , η 1 , η 2 , c has a direct relation with threshold quantity, which means that increase in the value of these parameters causes an increase in the value of the threshold quantity of the model. On the other hand, there is an inverse relationship between the set of parameters c 1 , c 2 , c 3 and so an increase in these parameters will cause a decrease in the value of threshold quantity. e highest sensitivity index parameter is β 2 having the sensitivity index 0.999999, which means that an increase in the value of this parameter say by 10% would increase the value of the threshold quantity by 9.99999%, as shown in Figure 1(a). Similarly the collectively impact of other parameters of S 1 is approximately 9.89%, if their values increase or decrease by 10%, as shown in Figures 1(a) and 1(b). Moreover, the parameters of S 2 have a negative relation and therefore its collectively impact is 9.94% whenever the value of the parameters given in S2 increases or decreases by 10 % (see Figures 1(c) and Figures  1(d)). More precisely, if the value of the parameters of S 2 are  increased by 10%, the basic reproductive number will be decreased by 9.94%, while if one decreases the value of the parameters, the threshold quantity will be increased by 9.94%.

Verification of Stability
Results. We find out the numerical simulation to verify the theocratical work carried out for the fractional-order SARS-CoV-2 transmission epidemiological model (6). To show the validity of the analytical findings we present the large-scale simulation. ere are not many choices like the traditional numerical methods to choose various schemes for the numerical simulation of fractional-order models [43], therefore extensive attention is required to formulate new and convenient techniques for the simulation of fractional models. We follow a numerical scheme formulated in [18,44]. We assume the time step h � 10 − 3 for integration with the simulation interval [0, t], n � T/h and n ∈ N. We also assume that u � 0, 1, 2, . . . , n, therefore the discretization for the proposed model looks like the following equation: Furthermore, we chose the value of epidemic parameters biologically while the initial population sizes are assumed to be non-negative values (100, 90, 80, 70, 60). We use the MATLAB software package to execute the model for numerical simulations. We justify the stabilities results to show the dynamics of the disease-free and endemic states as given in Figures 2 and 3.
is investigates the graphical verification of the dynamics of the considered problem around disease-free state X 1 . Besides from a mathematical point of view, the biological interpretation reveals that whenever the value of the threshold parameter is less than unity, each solution curve of S h will tend to its equilibrium position as shown in Figure 2(a). is shows that there will be always a susceptible population. Moreover, the dynamics of the other compartments around the disease-free state are depicted in Figures 2(b)-2(d), and which describe that the solution curves will tend to the associated equilibrium position and remain stable. So it could be noted that the elimination of the contagious disease of the novel coronavirus from the community depends on the value of R 0 , and the disease could be easily eliminated if R 0 < 1. Furthermore, the dynamics of the fractional-order model around endemic equilibrium are shown in Figures 3(a)-3(e), which respectively show the temporal dynamics of susceptible, latent, infected, recovered, and reservoirs. From these results, we observed that if proper control measures are not adopted the disease will attain the endemic position. It is clear that the susceptible population decreases from the beginning and then has no effect after some time and so becomes stable as shown in Figure 3(a). e dynamics of the latent population state that there will be a sudden increase in the initial period of infection, while then decreases after some unit of time and become stable, as shown in Figure 3(b), which verifies that there will be always latent population. Similarly, the dynamics of the infected population are shown in Figure 3(c).
is reveals that the infected ratio increases day by day and     reaches its endemic position in a few units of time. Furthermore, the simulation of the model for recovered population and reservoir is given in Figures 3(d)-3(e). All these results suggest that if no proper control measure is implemented, the disease will attain its endemic position whenever the value of the threshold quantity (R 0 ) is greater than unity.
We also show the significance of the fractional-order via disease transmission as shown in Figures 4-8, which respectively visualizes the temporal dynamics of S h , A h , I h , R h , and M. Particularly, the temporal dynamics of the susceptible are shown in Figure 4. We noted a significant impact of the fractional order on the transmission dynamics of susceptible individuals that if the fractional parameter σ increases than the number of susceptible individuals decreases as shown in Figure 4. e long run of the latent, infected, and the recovered population for various orders of fractional order are presented in Figures 5-7. We noted that there is a strong influence of (σ) on disease transmission. e temporal dynamics of the reservoir are presented in Figure 8. us, we investigate that the CF model gives more accurate dynamics of the disease and provides valuable outputs instead of classical models.

Conclusion
We investigated the dynamics of SARS-CoV-2 with asymptomatic, symptomatic, and quarantined individuals using an epidemic model. First, the formulation of the model is proposed, and then consequently fractionalized due to the increasing development in fractional calculus. Particularly, we used the well-known Caputo-Fabrizio operator for the said purposes. Both the biological and mathematical feasibilities are discussed in detail for the proposed model and proved that the problem is well-passed. We also calculated the threshold parameter and performed stabilities of the fractional model. e detailed sensitivity is also discussed and quantified the role of every epidemic parameter and its relative impact on the disease transmission. We showed that the proposed model is stable in both local and global sense. Finally, we gave some graphical representations and showed the validations of the obtained results. We also presented the relative impact of the fractional parameter on the various groups of the compartmental populations graphically and proved that the major outcome of the reported work is that the fractional-order CF epidemic models are more appropriate and the best choice rather than the classical order. We believe that the findings of this work will be helpful for the audience working in the field of mathematical epidemiology.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.