key: cord-1007248-y75wdipl
authors: Zhou, Jian-Cun; Salahshour, Soheil; Ahmadian, Ali; Senu, Norazak
title: Modeling the dynamics of COVID-19 using fractal-fractional operator with a case study
date: 2021-12-30
journal: Results Phys
DOI: 10.1016/j.rinp.2021.105103
sha: bcf68971cda2ef68c60d4bd8f639d3f5267951d1
doc_id: 1007248
cord_uid: y75wdipl

This research study consists of a newly proposed Atangana–Baleanu derivative for transmission dynamics of the coronavirus (COVID-19) epidemic. Taking the advantage of non-local Atangana–Baleanu fractional-derivative approach, the dynamics of the well-known COVID-19 have been examined and analyzed with the induction of various infection phases and multiple routes of transmissions. For this purpose, an attempt is made to present a novel approach that initially formulates the proposed model using classical integer-order differential equations, followed by application of the fractal fractional derivative for obtaining the fractional COVID-19 model having arbitrary order [Formula: see text] and the fractal dimension [Formula: see text]. With this motive, some basic properties of the model that include equilibria and reproduction number are presented as well. Then, the stability of the equilibrium points is examined. Furthermore, a novel numerical method is introduced based on Adams–Bashforth fractal-fractional approach for the derivation of an iterative scheme of the fractal-fractional ABC model. This in turns, has helped us to obtained detailed graphical representation for several values of fractional and fractal orders [Formula: see text] and [Formula: see text] , respectively. In the end, graphical results and numerical simulation are presented for comprehending the impacts of the different model parameters and fractional order on the disease dynamics and the control. The outcomes of this research would provide strong theoretical insights for understanding mechanism of the infectious diseases and help the worldwide practitioners in adopting controlling strategies.

J o u r n a l P r e -p r o o f Journal Pre-proof regards, the researchers use the tools of mathematical modeling to study the transmission and make further plan to prevent the mankind form the effects of mentioned infectious disease. In this regards, many researchers developed different mathematical models for the current COVID-19", for detail see [11] [12] [13] [14] .

A large portion of the mathematical models of COVID-19 are formulated in terms of the integer order derivatives which have a few restrictions to portray the realistic aspects of a phenomena under consideration. To manage those constraints, non-integer order derivatives provide a practical mean to the sickness dynamic and beneficial results that need to comprehend the models. non-integer order models have memory appropriateness and give a superior situation to depict an epidemic model. Many mathematical models on the elements various illnesses in term of noninteger order derivatives were proposed see for occasion [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] and the literature referenced therein. Fractal fractional calculus is the generalization of classical calculus [25] [26] [27] [28] [29] . To get a better insight into a mathematical model and to deeply understand phenomena, non-integer order operators can be used.

For the ease of understanding this research is organized as follows: the mathematical model with fractal fractional-order derivative is formulated in Sect. 2. In Sect. 3, the equilibrium points and basic reductive number R 0 are presented. The local stability of the disease-free and endemic equilibria for the determinist version model are presented in Sect. 4. Furthermore, the parameters estimation is shown in Sect. 5 . The existence and qualitative analysis with Hyers-Ulam Stability of fractional-order model in the sense of ABC presented in Sect. 6 . The numerical schemes and discussions are presented in Sect. 7, and in the last section we presented the concluding remarks.

A compartmental approach is used to develop the mathematical model for COVID-19 transmission dynamics. "The total population N is divided into six compartments named S, E, I, A, H, G, and R represent susceptible, exposed, symptomatically infected, asymptomatically infected, isolated, or hospitalized, and Recovered/immune cases respectively. In the mathematical model developed in this study, humans get into the suspected group S at the rate of G and infected with Coronavirus as a result of contact with individuals in the group of A or I.. The exposed group E gains population from infection induced by the Coronavirus. A proportion G 3 , (0 < G 3 < 1) of the members of the group E advance to the asymptomatic group A and the remaining proportion 1 − G 3 progresses to the symptomatic group I. People in the group I and A progress either to the Hospitalization group H or recovery group R at the rates indicated in Table 1 . In the construction of the mathematical model, the exposed compartment E is included because people who are contracted with the virus don't get infectious immediately; there is an incubation period for the virus to get infectious. The groups I and A are included in the model, as people infected with Coronavirus are either symptomatic or asymptomatic. COVID-19 induced death rate G 11 is also considered in the model. As a result, the authors are convinced that the model considered in this study named SEIAHR model incorporates all essential components of COVID-19 to study its transmission dynamics, in agreement with the definition of a mathematical model in" [30] . The 

A(t) = G 3 G 4 E(t) − (G 10 + G 7 + G 8 ) A(t), H(t) = G 5 I(t) + G 10 A(t) − (G 11 + G 9 + G 8 ) H(t), R(t) = G 6 I(t) + G 7 A(t) + (G 11

with initial condition

The detail of the used unknown variables and parameters are given below in Table 1 and 2 respectively: The class of exposed individuals I

The class of symptomatic infected individuals A

The class of asymptomatic infected individuals H

The class of Hospitalized individuals R

The class of Recovered individuals 11 Coronavirus induced death rate 0.0018 [30] Recently, it has been studied that the theory of fractional-calculus is rich for applications and researchers obtained more accurate results through fractional system rather than ordinary systems. Hence, we structured the above model (1) of COVID-19 infection in the framework of new fractal fractional derivative with a generalized Mittag-Leffler kernel as follows:

With initial condition

Where the symbol FFM D Ψ,Ξ 0,t represents the fractal fractional order derivative with fractional order 0 < Ψ ≤ 1 and the fractal dimension Ξ > 0. Now, applying the AB fractional integral to both sides of (3), we obtained the following system 

In order to proceeds the dynamical behaviour analysis, we firstly present some basic theoretical properties of the proposed model (1), including basic reproductive number, disease free and endemic equilibria. Additionally, an analytical expression for the important biological parameter termed as the basic reproductive number is provided. We obtained the following two equilibrium points for the proposed model (1):

J o u r n a l P r e -p r o o f Journal Pre-proof

The proposed epidemiological model (1) of the COVID-19 is examined for the disease free equilibrium, for this purpose let N 0 is the disease free equilibrium of the proposed model (1), then for analyzing this point the population under consideration is assumed to be infection free. Thus the system reported by N 0 = G G 8 , 0, 0, 0, 0, 0 . where S 0 = G G 8 .

The endemic equilibrium (EE) of the COVID-19 vaccine model (1) denoted by To track down R 0 for the above model (1), we utilize the next generation matrix technique [1, 31] . The Jacobian matrix around the DFE point N 0 is given by:

and

The dominant eigenvalue of ρ TV −1 is called the basic reproductive number, and is given by

R 0 can be written as

. . 6 are as defined above.

J o u r n a l P r e -p r o o f Journal Pre-proof

Let N * is the endemic equilibrium of model (1) , then the solution of the resultant algebraic equations will leads to the endemic equilibrium which define , N * = (S * , E * , I * , A * , H * , R * ), where

and

Thus, for R 0 > 1 a positive EE exists, with the assumption that G = G 8 N.

The previous section presented the basic reproductive number, disease free and endemic equilibria of the proposed model (1) . This analysis provides a clue for suggesting a better analysis of the dynamical behaviour of the model. Thus regarding the local as well as global analysis of the proposed model we have the following stability results. 

Prof. The jacobian matrix of system (3) around the DFE is given above (6) , has the following characteristic equation:

J o u r n a l P r e -p r o o f

where Y i , i = 1 . . . 6 are as defined above in Eq. 12. From (14), we have a 2 > 0 and a 1 =

Further more, the eigenvalues of the equation

have a negative real part, if the Routh-Hurwitz stability condition a 0 − a 2 a 1 < 0 and a 0 , a 2 , a 1 > 0 are satisfied. That is,

For

where k = 0, 1, 2, 3, ..., (m − 1). Similarly, we can find the arguments of the roots of the equation

are all greater than π 2M if R 0 < 1, having an argument less than π 2M for R 0 > 1. Thus, for R 0 < 1 the DFE N 0 is LAS.

Proof: Since, we know that for R 0 > 1 the EEP exists. Further the Jacobian matrix J at EEP is given by:

J o u r n a l P r e -p r o o f

The two eigenvalues λ 1 = −G 8 and λ 2 = Y 5 = − (G 11 + G 9 + G 8 ) of the matrix 17 are negative. Further more, for the remaining eigenvalues we can utilize the following equation

where

The coefficient B 3 can easily be shown to be positive and B 2 , B 1 , B 0 are also positive as shown below:

Since it isn't hard to show that B 0 B 2 3 B 2 1 − B 1 B 2 B 3 < 0, the Routh-Hurwitz stability conditions for Eq. (18) are satisfied. Thus all the eigenvalues of the Eq. (18) have a negative real part. Accordingly, the EEP N * is LAS for R 0 > 1.

The parameters used in the system (1) are estimated depend on the total number of conformed incidents, and deaths data in Khyber Pukhtunkhwa Pakistan. The ordinary Least Square Solution (OLS) is utilized to reduce the error terms for the daily reports, and the related relative error is used in the goodness of fit.

where I i is the reported total number of infected, andÎ i is the simulated total number of infected. The simulated cumulative number of infected are calculated by summing the individuals transit from the infected compartment to the recovered compartment for each day. The Fig. 1 shows the fit of model to the data. Estimated values of parameters are shown in the In the present section, we are going prove the uniqueness, existence, Ulam-Hyers stability of the solution for the proposed model with help of fixed point approaches. Before that, we rewrite the model (3) as

Where

We will communicate (23) with the below identical system

Where 

The system (22) can be turned to the following formula,

Next, for the analysis, the below assumptions H 1 and H 2 should be fulfilled:

• H 1 : : J × F −→ R is continuous and there exists two constants τ , η > 0 such that | (t, W(t))| ≤ τ + |W(t)|η for t ∈ J and W ∈ F.

• H 2 : there should be exists constant L > 0 such that 

and

Proof. We turn the given system (22) into a fixed point problems, i.e W = ΦW, W ∈ F Where the operator Φ :

Let

is close, convex, bounded subset with

where

Now we split the proof in the following steps as:

Step (1):

This proves that

J o u r n a l P r e -p r o o f

Step (3) : Φ 1 is contraction. Let W 1 , W 2 ∈ Φ ζ . Then via (H 2 ),we get

Step (3) : Φ 2 is relatively compact. case 1 : Φ 2 is continuous . Due to W(t) is continuous, then Φ 2 W(t) is continuous too. case 2 : Φ 2 is uniformly bounded on Φ ζ . Let W(t) ∈ Π ζ . Then, we have

Hence Φ 2 is uniformly bounded on Π ζ . case 3 : Φ 2 is eqicontinuous. Let W ∈ Π ζ and 0 < t 1 < t 2 < T. Then

It follows that

Thus, by Arzela-Ascoli theorem, we deduce that Φ 2 is completely continuous. The Eq. (26) has at least one solution, so the proposed model has unique solution.

then the integral equation (24) has a unique solution which implies that the model (3) has a unique solution.

Proof. Taking the operator Φ : F −→ F defined by (26) . Let W 1 , W 2 ∈ F and t ∈ J. Then

due to , Φ is contraction. Thus (26) has a unique solution, which yield that the model (3) has a unique solution.

Definition 1. [32] The fractal fractional integral system given by Equations (5) is said to be Hyers-Ulam stable if exist constants ∆ i > 0, i ∈ N 6 satisfying: For everyγ i > 0, i ∈ N 8 , for

there exist Ṡ ,Ė,İ,Ȧ,Ḣ,Ṙ which are satisfyinġ Proof. The fractal fractional model (5) has at least one solution (S, E, I, A, H, R) satisfying equations of system (5) . Then, we have

Similarly, we have the followings

Hence the proof is accomplished.

With the help of the numerical scheme as presented in the above sections, the models are simulated under various fractional orders for model (5) . This is very important to show the feasibility of the reported work and investigate the validity of the analytical work using large-scale numerical simulation. It is important to point out that, unlike traditional numerical analysis, there are not as many options to choose schemes for the numerical analysis of the fractional order epidemiological models simulations. For the numerical solution of the fractal fractional model (5) 

The numerical scheme for Newton polynomial we can see in Fig. 2 the initial days the susceptible class is decreasing at different fractional order. Consequently, the exposed class first increases which also increase the infected class for initial few days. In same line the recovered class is raising which indicate that the infection is considerably reducing or they are going to die due to infection. Also, the virus class first increase and after the due to reduction in infection this class is also went on reducing. First, we can express model (1) as follows:

J o u r n a l P r e -p r o o f

Now, we can rewrite the above system as: (52)

Applying the fractal fractional integral and plugging Newton polynomial into these equations, we can get;

Journal Pre-proof 

The numerical scheme for Adams Bashforth method we can see in Fig. 3-? ? the initial days the susceptible class is decreasing at different fractional order. Consequently, the exposed class first increases which also increase the infected class for initial few days. In same line the recovered class is raising which indicate that the infection is considerably reducing or they are going to die due to infection. Also, the virus class first increase and after the due to reduction in infection this class is also went on reducing. First, we can express model (1) as follows: 

Now, applying the fractal fractional integral to both sides of (53), we obtained the following system Set t = t m+1 for m = 0, 1, 2, ..., it follows that s, S, E, I, A, H, R) ds 

E(t) − E(0) = Ξ(1 − Ψ)t Ξ−1 M(Ψ) F 2 (t, S, E, I, A, H, R) + ΞΨ M(Ψ) t 0 s Ξ−1 (t − s) Ψ−1 F 2 (

I(t) − I(0) = Ξ(1 − Ψ)t Ξ−1 M(Ψ) F 3 (t, S, E, I, A, H, R) + ΞΨ M(Ψ) t 0 s Ξ−1 (t − s) Ψ−1 F 3 (

A(t) − A(0) = Ξ(1 − Ψ)t Ξ−1 M(Ψ)                                                                                         S (t m+1 ) − S(0) = Ξ(1 − Ψ)t Ξ−1 m M(Ψ) F 1 (t m , S,

+ ΞΨ M(Ψ) m ∑ n=1 t n+1 t n s Ξ−1 (t m+1 − s) Ψ−1 F 1 (E (t m+1 ) − E(0) = Ξ(1 − Ψ)t Ξ−1 m M(Ψ) F 2 (t m ,+ ΞΨ M(Ψ) m ∑ n=1 t n+1 t n s Ξ−1 (t m+1 − s) Ψ−1 F 2 (s, S,

A (t m+1 ) − A(0) = Ξ(1 − Ψ)t Ξ−1 m M(Ψ) F 3 (t m ,+ ΞΨ M(Ψ) m ∑ n=1 t n+1 t n s Ξ−1 (t m+1 − s) Ψ−1 F 3 (s,+ ΞΨ M(Ψ) m ∑ n=1 t n+1 t n s Ξ−1 (t m+1 − s) Ψ−1 F 4 (

R (t m+1 ) − R(0) = Ξ(1 − Ψ)t Ξ−1 m M(Ψ) F 5 (t m ,       (t − t n−1 ) h t Ξ−1 n F i (t n , S(t n ), E(t n ), A(t n ), H(t n ), R(t n )) − (t − t n ) h t Ξ−1 n−1 F i (t n                                                         S(t m+1 ) − S(0) = Ξ(1−Ψ)t Ξ−1 m M(Ψ) F 1 (t m , S,

By the approximate the functions x i (s), (56) becomes 

By simple calculations, we get

and

put t n = nh, we get

and 

J o u r n a l P r e -p r o o f Journal Pre-proof 

+ ΞΨ M(Ψ) ∑ m n=1 t Ξ−1 n F 5 (t n , S(t n ), E(t n ), A(t n ), H(t n ), M(t n )) h Ψ Ψ(Ψ+1) (m + 1 − n) Ψ (m − n + 2 + Ψ) − (m − n) Ψ (m − n + 2 + 2Ψ) −t Ξ−1 n−1 F 5 (t n−1 , S(t n−1 ), E(t n−1 ), A(t n−1 ), H(t n−1 ), R(t n−1 )) h Ψ Ψ(Ψ+1) (m + 1 − n) Ψ+1 − (m − n) Ψ (m − n + 1 + Ψ)

This study presents a novel approach for understanding the dynamics of the mathematical modeling approach that provides strong conclusions on the transmission mechanism of the newly but deeply investigated COVID-19 pandemic driven infections. To define the proposed model the COVID-19, the infected people are divided into two classes, namely, detected and undetected classes. The Fractal fractional order derivative with fractal dimension Ξ and fractional order Ψ in ABC sense is used to more readily investigate the infection dynamics. After the model definition, at first, we introduced the fundamental and essential numerical provisions of the fractal fractional COVID-19 pandemic model. We make use of the fractional order stability approach for the local stability of both endemic as well as the disease-free equilibrium points. The fractal and fractional order mathematical model in the ABC sense are solved numerically via Newton polynomial and J o u r n a l P r e -p r o o f

Adams-Bashforth techniques. We believe that the attempt made in this work will provide fruitful insights for adopting strategies in reducing the continuous COVID-19 pandemic.

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No conflict of interest exists regarding the publications of this paper.

Please check the following as appropriate: All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript