Dynamical Behavior of a Fractional Order Model for Within-Host SARS-CoV-2

: The prime objective of the current study is to propose a novel mathematical framework under the fractional-order derivative, which describes the complex within-host behavior of SARS-CoV-2 by taking into account the effects of memory and carrier. To do this, we formulate a mathematical model of SARS-CoV-2 under the Caputo fractional-order derivative. We derived the conditions for the existence of equilibria of the model and computed the basic reproduction number R 0 . We used mathematical analysis to establish the proposed model’s local and global stability results. Some numerical resolutions of our theoretical results are presented. The main result of this study is that as the fractional derivative order increases, the approach of the solution to the equilibrium points becomes faster. It is also observed that the value of R 0 increases as the value of β and π v increases. between individuals, quarantine, and wearing masks. This paper presents an epidemiological model of within-host SARS-CoV-2 transmission dynamics under the fractional-order derivative effect. Firstly, the existence, non-negativity, and bounds of the solution of the model are discussed. It is obtained that the model has two equilibrium points, namely the non-infected point and the virus-infected point. The explicit expression for the effective reproduction number R 0 , which gives the actual number of infections to infectious cells at any time, is found. The dynamical behavior of the system concerning R 0 is investigated. It is observed that the non-infected equilibrium is globally stable if R 0 < 1. The system per-sistence near the virus-infected equilibrium when R 0 < 1 is discussed. From the obtained numerical simulation results, it is observed that as the fractional order derivative increases, the approach of the solution to the equilibrium points becomes faster. This paper uses the Caputo fractional-order derivative operator to form a fractional-order mathematical model of the spread of within-host SARS-CoV-2. The transmission rate effect on the dynamical behavior with a ﬁxed value of the fractional order is studied. It is obtained that the system solution approaches the virus-infected equilibrium point. Hence, the value of R 0 increases as β increases (see Figure 6). The effect of increasing the virus production rate on the dynamical behavior is studied, in addition, by using different values for the ﬁxed value of the fractional order. It is observed that the value of R 0 increases as π v increases (see Figure 7).


Introduction
In December 2019, coronavirus disease 2019 (COVID- 19), which began in Wuhan, China, was classified as a global pandemic. Since then, it has rapidly spread throughout the world, becoming the most pressing public health crisis and affecting millions of people [1]. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for this disease. SARS-CoV-2 is a single-stranded RNA virus with a positive-sense single-stranded RNA (+ssRNA) genome. SARS-CoV-2 is one of the viruses that belongs to the Coronaviridae family and the Nidovirales order [2]. In addition, SARS-CoV-2 belongs to the Beta coronavirus family, which includes two additional extremely deadly viruses, SARS-CoV and MERS-CoV [3]. COVID-19's infection dynamics are complicated by several factors, including uncertainty about the source of the infection, a long period of incubation during which an infected individual may not develop symptoms or be aware of their infection, and the inefficiency and lack of availability of drugs or vaccines. All of these factors contribute to the rapid spread of COVID-19, complicating disease control [4].
Mathematical modeling of diseases is a new research area in mathematical biology. Researchers have used mathematical modeling to observe the dynamics of disease spread in populations and the within-host dynamics in the body to control and manage disease effects [5][6][7][8]. Several modeling studies on the spread, transmission, and infection dynamics of COVID-19 disease have been conducted in various regions. By fitting the data of four Chinese provinces, Hubei, Guangdong, Zhejiang, and Henan, Liang [3] proposed a mathematical framework to detect the rules of the spread of the following pneumonia illnesses: COVID-19, SARS, and MERS. According to their result, COVID-19 has a much faster growth rate than SARS and MERS. Yang and Wang et al. [4] formulated a mathematical model that incorporates an environmental reservoir along with non-constant transmission rates to investigate the outbreak of COVID-19 disease in Wuhan, China, and demonstrated their findings with publicly available data. The authors of [9] investigated a mathematical model for the COVID-19 outbreak in Wuhan, taking into account some key elements such as individual behavioral responses, governmental actions, zoonotic transmission, and mass emigration in a short period. Chen et al. [10] developed a mathematical framework to simulate SARS-CoV-2 phase-based transmissibility and reported that in Middle Eastern countries, the transmissibility of SARS-CoV-2 is higher than MERS, whereas it is lower than MERS in the Republic of Korea.
Fractional calculus has recently gained more attention in mathematical modeling due to it's salient features: fractional derivatives can capture memory effects due to their non-local nature and give a more realistic scenario than the integer-order differential model. In [11], the authors used Lagrange's interpolation method to analyze a fractal-fractionalorder COVID-19 model and interpret their results numerically for various fractional orders. The optimal effects of physical distance on SARS-CoV-2 virus transmission were studied by Bushnaq et al. [12] using a fractional-order mathematical model. The researchers have considered control inputs such as social distancing and quarantine to minimize the number of susceptible and infected people while maximizing the number of people who can recover. Farman et al. [13] investigated a COVID-19 model using the Atangana-Baleanu fractional derivative and fractal-fractional Atangana-Baleanu with Mittag-Leffler kernel and derived a comparison analysis of the models with the considered fractional derivatives. Zhang et al. [14] simulated a new fractional-order mathematical model for the COVID-19 epidemic by employing the adaptive predictor-corrector algorithm and the fourth-order Runge-Kutta (RK4) method. To measure the spread of COVID-19, a SEIRD model was proposed by the authors in [15]. They estimated the parameters using both classical and fractional-order models fitted with data reported by WHO for Italy. It was observed that the root-mean-square error for the fractional-order model is lower than that of the classical one, and the fractional model provides a more accurate forecast of the data. In [16], a fractional-order model has been proposed to explore the effects of isolation, quarantine, and environmental viral load in the COVID-19 outbreak. Using data from Pakistan as input, they built a model and simulated the model using the Adams-Moulton scheme. Ahmad et al. [17] used the fractional-order derivative in Atangana-Baleanu sense to study a COVID-19 outbreak model for five different classes and fitted the model to data from Wuhan. A modified Adam-Bashforth method was used to arrive at an approximation of the numerical solution for this model. Moreover, one can find other works related to transmission dynamics of COVID-19 (see [18,19]).
A few modeling study works have been performed to understand the within-host behavior of SARS-CoV-2 dynamics. A mathematical model was proposed to demonstrate the within-host viral dynamics of SARS-CoV-2 such that the values of the parameters were estimated from score data of chest radiography, as shown in [20]. They also employed their mathematical model to analytically indicate the effect of drugs on virus growth and the immunity effect on patients. A few mathematical models were reviewed in [21] that are capable of explaining the dynamical behavior of SARS-CoV-2 and estimating the standard parameters of viral infections. Almocera et al. [22] explored the response of effector T cells to SARS-CoV-2 and gave a profound insight into how SARS-CoV-2 causes the disease by analyzing a within-host model. In [23], a comparative analysis of SARS-CoV-2 dynamics with that of MERS-CoV and SARS-CoV has been performed by investigating a quantitative model, which suggests that the infection of SARS-CoV-2 can be controlled if a combination therapy is used that triggers cytotoxicity and blocks de novo infection or virus production synergistically. Vaidya et al. [24] characterized the dynamical infection of SARS-CoV-2 within the host in ferrets by investigating a mathematical model fitted with experimental data. Yin et al. [25] proposed a viral kinetic delay-differential model to assess the effects of T cells and antibodies on SARS-CoV-2 ineffectiveness in patients with various symptoms. Afonyushkin et al. [26] investigated the distribution of the SARS-CoV-2 virus in human tissue and organs using a multi-compartmental model. Recently, Nath et al. [27] discussed the dynamical behavior of the model that was introduced by Li et al. [20]. This study aims to modify the model considered by Nath et al. [27] to a fractional-order one for understanding the complex behavior of SARS-CoV-2 in the host.
The remaining parts of the paper are assembled as follows: A fractional-order model that describes the dynamics of SARS-CoV-2 has been formulated in Section 2. Section 3 deals with a few preliminary concepts related to fractional calculus. The model has been analyzed by presenting the equilibrium points, stability analysis, and global stability analysis in Sections 4-6. A numerical simulation of the model has been performed in Section 7. Finally, concluding remarks of the paper have been presented in Section 8.

The Original COVID-19 Model
The original model [20,27] consists of three classes that categorize individuals with respect to the within-host coronavirus disease based on their status, as indicate in Figure 1. The class M p (t) represents virus-free pulmonary epithelial cells, the virus-infected pulmonary cells are represented by the class M * p (t), and the class v(t) represents the free virus. In addition, d M , d M * , and d v are the death rates of classes M p (t), M * p (t), and v(t), respectively. The transmission rate from virus-free cells to virus-infected cells is denoted by β. The free virus production rate is π v . In the following model, the constant regeneration rate of v(t) will be described by the term d M M p (0). (1)

The Fractional COVID-19 Model
For the model's modification by the general (M p , M * p , v) within-host coronavirus model (1), we can express the following fractional within-host coronavirus model: with the initial conditions Here, D α is the fractional Caputo derivative with α order as α ∈ (0, 1] that indicates the memory effects in the proposed M p M * p v within-host coronavirus model. We consider that the functions M p (t), M * p (t), v(t) with their Caputo fractional derivative order as α ∈ (0, 1] are continuous functions. All the parameters in the fractional-order model (2) are within the interval (0, 1] and the values are taken from [20,27] that were used in analysis of model (1).

Preliminaries of Fractional-Order Derivative
This section reviews the fundamental definitions, theorems, concepts, and results that we will use throughout the remainder of this paper. For more details, we refer the reader to [28][29][30][31]. Definition 1. The fractional-order derivative in Caputo sense for order x > 0 is defined below: Definition 2. The Atangana-Baleanu Caputo fractional derivative for a given function for order x is defined below: is a normalization function and E α (.) is the Mittag-Leffler function.

Definition 3. An integral order x under the Atangana-Baleanu fractional derivative is defined by
and the integral will be zero for a constant function f (t).

Lemma 2.
Let a nonlinear fractional-order system be D θ t x = φ(x), where x(t 0 ) = x 0 and 0 < θ < 1. If a point x * satisfies φ(x * ) = 0, then x * is called an equilibrium point of the system. The equilibrium x * is said to be locally asymptotically stable if all eigenvalues λ i of the Jacobian matrix J = ∂φ ∂x evaluated at x * satisfy the conditions |arg(λ i )| > θπ 2 .
x) is a continuous function in x and locally Lipschitz in x. Thus, ∀t > 0, ∃x(t) ∈ Ω such that v ∈ C 0,v with θ < v < 0 be a locally Hölder continuous function such that Lemma 4 ( [32,33]). Suppose that Ω is a bounded closed set. Every solution of D θ t x(t) = f (x) starts from a point in Ω and remains in Ω for all time. If ∃ φ(x) : Ω → R with continuous first partial derivatives satisfies the following condition, and M is the largest invariant set of E, then every solution x(t) originating in Ω tends to M as t → ∞. Specifically, when M = {0}, then x → 0, as t → ∞.

Analysis of the Model
The evaluation of the equilibrium of the model (2) is done by finding all the possible roots of the model (2) when all fractional derivatives in the model (2) are equal to zero: which gives The uninfected equilibrium (UIE) of the model (2) can be computed by solving the system (5) and given by the form e 1 = (M p (0), 0, 0). Furthermore, the model (2) can be written as where and then we have where the matrix (11) is called the next-generation matrix for the model and R 0 =  (2) is exists, if the following equations are satisfying:

Proof. A virus-infected equilibrium of the model
It is straightforward to calculate that there exists a unique solution to (12) that satisfies π v β > 0 ∀t, but M * p > 0 and v > 0 are hold if and only if R 0 > 1. Hence, the virus-infected equilibrium point of the model (2) can be computed as ) and it exists if and only if R 0 > 1.

Local Stability
Herein, the local stability analysis will be discussed near all equilibrium points of model (2) through the following theorems. Theorem 1. Suppose that g : Ω ⊂ 3 + → 3 + is a differentiable function and e 1 ∈ Ω; then, uninfected equilibrium point e 1 is locally asymptotically stable if the following condition holds: Otherwise, it becomes unstable.
Proof. The Jacobian matrix at uninfected equilibrium point e 1 can be written as Clearly, the characteristic equation of the above matrix J(e 1 ) is and here, (14) is satisfied, we find that the uninfected equilibrium point e 1 is locally asymptotically stable. Theorem 2. The virus-infected equilibrium point e 2 , is locally asymptotically stable whenever Proof. The Jacobian matrix at the virus-infected equilibrium point e 2 can be written as The characteristic equation of the above matrix J(e 2 ) is where Thus, under both conditions R 0 > 1 and (17), we have A i , which are positive, where i = 1, 2, 3 and A 1 A 2 − A 3 > 0, and hence, according to the Routh-Hurwitz criterion, the virus-infected equilibrium point e 2 is locally asymptotically stable.

Global Stability
Herein, the global stability of the system (2) will be discussed through the following theorems.
Theorem 3. The uninfected equilibrium point e 1 is globally asymptotically stable on the domain if and only if R 0 < 1.

Lemma 6.
Assume that there is a continuous and differentiable function x(t) ∈ + such that ∀ t ≥ t 0 and 0 < α < 1; we have  In Figure 4, we use the same initial condition and data but fix α = 0.9 with different values of β. We find that the number of virus-free cells decreases and all numbers of virusinfected cells and virus-free cells increases, while the solution of the model (2) approaches e 2 , which means that the e 1 becomes unstable.  In Figure 5, we use the same initial condition and data but fix α = 0.4 with different values of π v . We find that the number of virus-free cells decreases and all numbers of the virus-infected cells and virus-free cells increases, while the solution of the model (2) approaches e 2 , which means that the e 1 becomes unstable.

Conclusions
Epidemiological models play a basic role in constructing strategies for understanding, preventing, and controlling infectious diseases through vaccination, social distancing between individuals, quarantine, and wearing masks. This paper presents an epidemiological model of within-host SARS-CoV-2 transmission dynamics under the fractional-order derivative effect. Firstly, the existence, non-negativity, and bounds of the solution of the model are discussed. It is obtained that the model has two equilibrium points, namely the non-infected point and the virus-infected point. The explicit expression for the effective reproduction number R 0 , which gives the actual number of infections to infectious cells at any time, is found. The dynamical behavior of the system concerning R 0 is investigated. It is observed that the non-infected equilibrium is globally stable if R 0 < 1. The system persistence near the virus-infected equilibrium when R 0 < 1 is discussed. From the obtained numerical simulation results, it is observed that as the fractional order derivative increases, the approach of the solution to the equilibrium points becomes faster. This paper uses the Caputo fractional-order derivative operator to form a fractional-order mathematical model of the spread of within-host SARS-CoV-2. The transmission rate effect on the dynamical behavior with a fixed value of the fractional order is studied. It is obtained that the system solution approaches the virus-infected equilibrium point. Hence, the value of R 0 increases as β increases (see Figure 6). The effect of increasing the virus production rate on the dynamical behavior is studied, in addition, by using different values for the fixed value of the fractional order. It is observed that the value of R 0 increases as π v increases (see Figure 7). It will be interesting to consider the impact of the vaccine with and without time delay for further work. Vaccination is one of the most effective methods of mitigation and elimination during epidemics, as evidenced in such epidemics as hepatitis, poliomyelitis, and, most recently, COVID-19. Moreover, the model can be further extended to an optimal one by choosing control parameters with which the disease can be controlled. In this regard, the authors would like to suggest a few works [34][35][36] that are related to optimal control concepts. Moreover, in the present study, we only discussed the local and global stability of the considered model. While analysis of a fractional-order system, the Ulam-Hyers stability (which guarantees a close exact solution to the system) is also an essential topic [37][38][39], and it will be considered in our future work.