On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative.

The major purpose of the presented study is to analyze and find the solution for the model of nonlinear fractional differential equations (FDEs) describing the deadly and most parlous virus so-called coronavirus (COVID-19). The mathematical model depending of fourteen nonlinear FDEs is presented and the corresponding numerical results are studied by applying the fractional Adams Bashforth (AB) method. Moreover, a recently introduced fractional nonlocal operator known as Atangana-Baleanu (AB) is applied in order to realize more effectively. For the current results, the fixed point theorems of Krasnoselskii and Banach are hired to present the existence, uniqueness as well as stability of the model. For numerical simulations, the behavior of the approximate solution is presented in terms of graphs through various fractional orders. Finally, a brief discussion on conclusion about the simulation is given to describe how the transmission dynamics of infection take place in society.

Introduction

One of the greatest missions given to humankind is to control the environment within which they live. However, the growth of the human population speedily is increasing in relation to different cultures and ways of life. Although Humanity has evolved many species of devices and equipment to obtain a beautiful life. But this excessive development sometimes leads to disasters in the environment. Especially, the use of nutrients, carriages, mobiles, cosmetics, electrified and petroleum equipment, etc, making the environment highly contaminated and infused with viruses.

Newly, the whole world hardship a new coronavirus comprehensive and it was named (COVID-19) which was claimed to flowed first in Wuhan city, China. It has been considered that the origin of COVID-19 is the transportation from animal to human as numerous infected status claimed that they had been to a local fish and wild animal market in Wuhan on 28 November [1]. Soon, some investigators assured that transportation also occurs person to other [2]. This virus dates back to 1965 when Tyrrell and Bynoe were identified when they passage a virus so-called B814 [3]. This virus is found in the cultures of the organs of the human embryonic trachea acquired by the respiratory system of an adult [4].

Because infectious diseases are a major threat to humans as well as the country’s economy. A proper understanding of the dynamics of the disease plays a significant role in reducing infection in society.

Application of an appropriate strategy against the disease transportation is another challenge. The mathematical modeling tactic is one of the main tools in order to handle these challenges. Many disease models have been evolved in the existing literature which authorizes us to explore and dominance the prevalence of infectious diseases in a better style. Most of these models rely on classical differential equations. However, in the past few years, it is paying attention that fractional differential equations can be applied to model global phenomena with a greater grade of precision and its applications can be found in various fields for instance dynamic, biology, engineering, control theory, economics, finance and in epidemiology.

The Fractional calculus transacts with differentiation and integration involving fractional order, which is further outstanding and beneficial than the ordinary integer order in the explanation of the real-world problems, also in the modeling of real phenomena due to a characterization of the memory and hereditary properties [5], [6], the integer-order derivative doesn’t familiarize the dynamics among two various points. Various types of fractional order or nonlocal derivatives were proposed in the present literature to transact the reduction of a traditional derivative. For instance, based on power-law, Riemann-Liouville introduced the idea of fractional derivative. After that Caputo-Fabrizio in [7] have proposed a new fractional derivative utilizing the exponential kernel. This derivative has a few troubles related to the locality of the kernel. Newly, to overcome Caputo-Fabrizio’s problem, Atangana and Baleanu (AB) in [8] have proposed a new modified version of a fractional derivative with the aid of popularized Mittag-Leffler function (MLF) as nonsingular kernel and nonlocal. Since the generalized MLF is used as the kernel and it’s guaranteed no singularity, the AB fractional derivative supply a stellar description of memory [9], [10], [11].

Because of mathematical models, we can know the rate of change in the COVID-19 how the disease can affect people at risk and quarantined. The area dedicated to the study of the biological model of infectious diseases is a warm area for recent research. Numerous studies on mathematical models were presented to the study of stability theory and the results of existence and improvement of biological models, see [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. For example, Chen, et al., in [12] presented a big mathematical model for simulating the phase-based transmissibility of a novel coronavirus. Hussain et al., in [13] showed the existence and uniqueness of results for the fractional model involving fractional derivative in Atangana-Baleanu sense using Schaefer’s and Banach’s fixed point theorems. Moreover, they applied the Shehu transform and Picard method to explore the iterative solutions and its stability for the proposed fractional model. A conceptual model for the COVID-19 disease, which effectively catches the time line of this virus outbreak was proposed by Lin et. al. in [14]. In [16] A mathematical modeling and dynamics of a novel (2019-nCoV) are formulated under a fractional model with considering the available infection cases within 7 days. A mathematical model considered by Shaikh et al. [15] to study estimate of the effectiveness of prophylactic measures, prophesying future outbreaks and possibility control strategies of COVID-19.

Due to the success of this operator in modeling infectious diseases, and motivated by the above useful applications of some fractional operators in epidemic mathematical models, in this paper we are studying the dynamics of the novel coronavirus model suggested by Chen et.al., [12] in the form of the system of the nonlinear differential equations involving the AB Caputo fractional derivative:with the initial conditions where denotes the Atangana-Baleanu-Caputo fractional derivative of order ∝. The model is based on the following facts:

The bats were split into four closets: is a susceptible class of bats, is an exposed class of bats, is an infected class of bats, and is a removed class of bats.

The hosts were split into four classes: is a susceptible class of hosts, is an exposed class of hosts, is an infected class of hosts, and is a removed class of hosts.

The people were split into six classes: is a susceptible class of people, is an exposed class of people, is an infected class of people, is asymptomatic infected people, is a population of removed people due to death, and is the population of the virus in reservoir various.

and are the initial values corresponding to the three categories.

The parameters of the model (1) are described as follows:

Λ denote the birth rate of bats, m is the death rate of bats, and β is the transmission rate from to . The parameters and are the incubation period of bats, the infectious period of bats, the latent period of people, and the infectious period of asymptomatic infection of people, respectively. Λ is Recruitment from total number of hosts and m The death rate of hosts. The symbols β and β represent the transmission rate from to and from to respectively. Λ and m are the birth and death rate parameter of people, respectively. In the proposed model β and κ are define the transmission rate from to and multiple of the transmissibility of to respectively. β and β means the transmission rate from to and from to respectively. δ is the proportion of asymptomatic infection rate of people, γ is the infectious period of symptomatic infection of people and a is the retail purchases rate of the hosts in the market. The shedding coefficients from to and the shedding coefficients from to are denoted by μ and respectively. Finally, is the lifetime of the virus in and is the total number of hosts.

The major aim of the paper is to demonstrate the existence, uniqueness and Ulam stability of solution for the model (1)-(2) by using some fixed point techniques. Moreover, the numerical simulations via the fractional version of Adams Bashfully technique to approximate the ABC fractional operator are performed, which it is shown that the model displays rich dynamical behaviors through graphical representation of numerical solutions.

This paper is coordinated as follows: Section 1 transacts with the introduction which contains a survey of the literature. Section 2 consists of some foundation preliminaries related the fractional calculus and nonlinear analysis. The existence and Ulam stability results on a proposed model are obtained in Sections 3 and 4. The numerical solution and numerical simulations of the model at hand are presented in Section 5. For the numerical simulation, we use a powerful two step numerical tool called fractional AB method. The concerned numerical method is more powerful than usual Euler method as well Taylor method. Because the mentioned method is faster convergent and stable as compared to Taylor and Euler method which are slowly convergent. For detail about this method, see [32], [33], [34], [35], [36].

Preliminaries

For short, we will use the following notations For the next analysis, we define Banach space for onward analysis. Let 0 ≤ t ≤ T < ∞, we define the Banach space by using as under the supremum norm given by where and θ ∈ {B, H, ρ}.

[8] Let ∝ ∈ (0, 1] and σ ∈ H 1(0, T). Then the left-sided ABC fractional derivative with the lower limit zero of order ∝ for a function σ is defined bywhere is the normalization function which is defined as and satisfies the result Further is called the Mittag-Leffler function defined by the serieshere Re(∝) > 0 and Γ(.) is a gamma function.

[8] Let ∝ ∈ (0, 1] and σ ∈ L 1(0, T). Then the left-sided ABC fractional integral with the lower limit zero of order ∝ for a function σ is defined by

The Laplace transform of ABC fractional derivative of a function σ(t) is given by

[8] The solution of the proposed problem for ∝ ∈ (0, 1]  is given by

[25] Let ϝ be a Banach space. The operator Π: ϝ → ϝ is a Lipschitzian if there exists a constant κ > 0 such thathere κ is the Lipschitz constant for Π. If κ < 1 we say that Π is a contraction.

Let ϝ be a Banach space If is a contraction mapping. Then there exists a unique fixed point of Π.

(Let be a closed, convex, non-empty subset of a Banach space ϝ. Suppose that and map into ϝand that

for all ;

is compact and continuous;

is a contraction mapping. Then, there exists such that .

Existence of solutions for the proposed model (1)-(2)

Now, we debate the existence and uniqueness results of the model (1)-(2) by utilizing the fixed-point technique. Let us reformulated model (1) in the subsequent appropriate formwhereWe take our system as by using (1) whereHere the symbol A denotes the transpose operation. Utilizing Lemma 2.3.1, the model (6) can be turned to the fractional integral equation in the sense of AB fractional integral as followsExpressing some growth condition and Lipschitzian assumption for existence uniqueness as:

(Hypothesis 1) There exists two constants such that

(Hypothesis 2) There exists constants such that

for each Ψ ∈ ϝ and

Let the hypotheses 1,2 hold. The Integral equation (8) which equivalent of the considered model (1) - (2) has at least one solution if where

Consider is closed convex set with where We define the operators Π1  and Π2 as where Now, we offer the proof in several steps as:

Step1: for .

Indeed, From (Hypothesis 1) and equations (9) , (10) , we obtain This confirms that .

Step1: we show that Π1 is contraction.

Let . By (Hypothesis 2), we have As Π1 is contraction mapping.

Step 3: We show that Π2 is relatively compact.

To prove that, we show that Π2 is continuous, uniform bounded, and equicontinuous.

Since Ψ(t) is continuous, then Π2Ψ(t) is continuous. Next, let we have Hence Π2 is uniformly bounded on Finally, we show that Π2 equicontinuous. Let and such that t 1 < t 2 . Then As t 1 → t 2 the right-hand side of the above inequality tends to zero. Consequently, by Arzelá–Ascoli theorem, Π2 is relatively compact and so completely continuous. Thus by Theorem 2.6 , the integral equation (8) has at least one solution and consequently the model under consideration has at least one solution. □

Under Hypothesis 2, the integral equation (8) has unique solution which yields that the model (1) - (2) has unique result if

Let the operator Π: ϝ → ϝ defined byLet Ψ and Ψ* in ϝ and ThenDue to (11), Π is contraction. Therefore the integral equation (8) has unique solution. Hence our model (1)-(2) has unique solution. □

Ulam-Hyers stability

The notion of Ulam stability was initiated by Ulam [26], [27]. Then aforesaid stability has been scrutinized for classical fractional derivatives in many of the research articles, we refer to some of them like [28], [29], [30], [31]. Additionally, since stability is a prerequisite in respect of approximate solution, so we endeavor on Ulam type stability for the model (1) via using nonlinear functional analysis.

System (1)-(2) is U-H stable if there exists λ > 0 with the following property: For some ϵ > 0, and each ifthen there exists Ψ ∈ ϝ satisfying the model (1) with the following initial conditionsuch thatwhere

Consider a small perturbation g ∈ C[0, T] such that comply with the following properties.

|g(t)| ≤ ϵ, for and ϵ1 > 0.

For we have the following modelwhere

The solution of the perturbed problem satisfies the given relation where is a solution of (15) , is satisfies ( 13 -a), and

Thanks to Remark 1 (2), and Lemma 2.3.1, the solution of (15) is given byAlso, we haveIt follows from Remark 1(1) that  □

Under the presumptions of Theorem 3.3 . Then the model (1) - (2) will be U-H stable in ϝ.

Let be the solution of the inequality (13-a) and the function Ψ ∈ ϝ is a unique solution of equation (1-a) with the conditionThat isDue to (16), the equation (17) becomes

Thus by Hypothesis 1 and Lemma 4.1.1, we obtainwhich impliesDue to Θ3 < 1. For we get

Hence the model (1)-(2) is U-H stable. □

Numerical approach

In this part, we give approximation solutions of the ABC fractional model (1)-(2). Then the numerical simulations are acquired via the suggested scheme. To this aim, we employ the modified fractional version for Adams Bashforth technique to approximate the fractional integral in the sense AB.

Using the initial conditions and fractional integral operator, we convert model (1) into the integral equationswhich impliesTo procure an iterative scheme, we go ahead with the first equation of the model (19) as follows:Set for it follows thatNow, we approximate the function on the interval through the interpolation polynomial as followswhich impliesNow, we compute the following integrals and I ℓ,∝ as follwsandPut we getandSubstituting (22) and (23) into (21), we getSimilarly

Numerical interpretation and discussion

Now to present the numerical simulations of the ABC fractional model (1)-(2), we apply the iterative solution contained in (24)-(37). Taking the time in (Days) . The numerical amounts of the parameters applied in the simulations are specified in Table 1 . The graphical representations of numerical solution for species at various fractional-order of the considered model (1) are given in Figs. 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 respectively. We simulate the results for first hundred days and hence our interval of study in numerical simulation is We take initial values in percentage of the total papulation as

Table 1

Numerical values and the physical interpretation of the parameters involve in the proposed model (1).

Parameters	physical description	Numerical value
ΛB	The birth rate parameter of bats	0.0300
mB	The death rate of bats	0.011
βB	The transmission rate from I_B to S_B	0.01166
1ω_B	The incubation period of bats	0.00073
1γ_B	The infectious period of bats	0.0345
ΛH	Recruitment from total number of hosts	1millions
mH	The death rate of hosts	0.011
βH	The transmission rate from I_H to S_H	0.0300
βBH	The transmission rate from I_B to S_H	0.0405
Λρ	The birth rate parameter of people	1.00
mρ	The death rate of people	0.0091
βρ	The transmission rate from I_p to S_p	0.004
κ	The multiple of the transmissibility of A_p to I_p	0.003
βω	The transmission rate from W to S_p	0.005
βρ	The transmission rate from I_p to S_p	0.001
δρ	The proportion of asymptomatic infection rate of people	0.002
1ωρ^′	The latent period of people	0.003
γρ	The infectious period of symptomatic infection of people	0.005
a	The retail purchases rate of the hosts in the market	0.0321
1γρ^′	The infectious period of asymptomatic infection of people	0.0121
μ	The shedding coefficients from I_ρ to W	0.0121
μρ^′	The shedding coefficients from A_ρ to W	0.0761
1ε	The lifetime of the virus in W	0.0261
N_H	The total number of hosts	10 millions

Fig. 1

Graphical representation of numerical solution for susceptible class of bats at various fractional of the considered model (1).

Fig. 2

Graphical representation of numerical solution for exposed bats at various fractional of the considered model (1).

Fig. 3

Graphical representation of numerical solution for infected bats at various fractional of the considered model (1).

Fig. 4

Graphical representation of numerical solution for removed class of bats at various fractional of the considered model (1).

Fig. 5

Graphical representation of numerical solution for susceptible host at various fractional of the considered model (1).

Fig. 6

Graphical representation of numerical solution for exposed hosts at various fractional of the considered model (1).

Fig. 7

Graphical representation of numerical solution for infected host at various fractional of the considered model (1).

Fig. 8

Graphical representation of numerical solution for removed host at various fractional of the considered model (1).

Fig. 9

Graphical representation of numerical solution for susceptible people at various fractional of the considered model (1).

Fig. 10

Graphical representation of numerical solution for exposed people at various fractional of the considered model (1).

Fig. 11

Graphical representation of numerical solution for symptomatic infected people at various fractional of the considered model (1).

Fig. 12

Graphical representation of numerical solution for asymptomatic infected people at various fractional of the considered model (1).

Fig. 13

Graphical representation of numerical solution for population of removed people due to death or recovered various fractional of the considered model (1).

Fig. 14

Graphical representation of numerical solution for population of virus in reservoir various fractional of the considered model (1).

In Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, we have given a global dynamics of each compartment in the considered model (1) by using the numerical values in Table 1 against different fractional order. The growth and decay of various compartment is different that some compartments are growing with different rate due to fractional order and similar behavior may be observed for decay of some compartments. Lower the order faster is the growing rate and vice versa and similarly the decay also is different at different fractional order. Therefore fractional calculus can help in understanding the transmission dynamics of novel coronavirus-19 disease. Also the concerned numerical scheme can be used as a powerful tools to perform numerical simulation of such complicated model. In Fig. 1 as susceptible papulation is decreasing as they are exposing so the density of expose class is going on growing as in Fig. 2. On the other hand the infected bats density is increasing as presented in Fig. 3, while the number of removed bats are also increasing in Fig. 4 until stable in the market. On the other hand as the density of susceptible host is decreasing in Fig. 5 because they are exposing to infection. Therefore the density of exposed host is growing as in Fig. 6 as a results the host catching more infecting and hence the density of host infected is also increasing. This process may be observed from Fig. 7. As a results of infection the removed (either death or get ride from infection) will increasing as in Fig. 8. In same way the density of susceptible people is decreasing in Fig. 9 because they are exposing to infection so the density of exposed people will decrease (see Fig. 10), because the people rapidly catching infection, hence the density of symptomatic infected people will grow as in Fig. 11. The similar behavior one can observed for density of asymptomatic and removed (either death or recovered from infection) in Figs. 12 and 13 respectively. Since as the density of infected bats is increasing, also the infection in host papulation is growing. As a result the papulation of virus in reservoir will grow further as presented in Fig. 14.

Conclusion

This manuscript has been devoted to comprehensively investigate a mathematical model for calculating the transmissibility of novel Coronavirus (COVID-19) disease by using nonsingular fractional order derivative. The existence and uniqueness of the considered model has been guaranteed by applying Krasnoselskii and Banach fixed point theorems. Also some stability results of Ulam type have been constructed. With the help of fractional Adams Bashforth method method we have simulated the results corresponding to various fractional orders. The obtained results play important role in developing the theory of fractional analytical dynamic for the current pandemic due to Coronavirus -19 which has badly affected the whole world. From the simulation we noted that the decrease in susceptibility is faster at lower fractional order of the derivative and in same line the increase in infections is also rapid with smaller order. The presented results may be helpful in understanding the present pandemic more comprehensively and can help in taking precautionary measure to reduce the infection to minimum.

CRediT authorship contribution statement

Mohammed S. Abdo: Writing - original draft. Kamal Shah: Methodology. Hanan A. Wahash: Formal analysis. Satish K. Panchal: Writing - review & editing.

Declaration of Competing Interest

We declare that none of the author has the competing or conflict of interest.