key: cord-288894-2iaq3ayv
authors: Kumar, Sachin; Cao, Jinde; Abdel-Aty, Mahmoud
title: A novel mathematical approach of COVID-19 with non-singular fractional derivative
date: 2020-07-01
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110048
sha: 
doc_id: 288894
cord_uid: 2iaq3ayv

We analyze a proposition which considers new mathematical model of COVID-19 based on fractional ordinary differential equation. A non-singular fractional derivative with Mittag-Leffler kernel has been used and the numerical approximation formula of fractional derivative of function [Formula: see text] is obtained. A new operational matrix of fractional differentiation on domain [0, a], a ≥ 1, a ∈ N by using the extended Legendre polynomial on larger domain has been developed. It is shown that the new mathematical model of COVID-19 can be solved using Legendre collocation method. Also, the accuracy and validity of our developed operational matrix have been tested. Finally, we provide numerical evidence and theoretical arguments that our new model can estimate the output of the exposed, infected and asymptotic carrier with higher fidelity than the previous models, thereby motivating the use of the presented model as a standard tool for examining the effect of contact rate and transmissibility multiple on number of infected cases are depicted with graphs.

The fractional calculus is a classical branch has been developed recently to deal with a new discovered problems (see i.e. J. Liouville and N. H. Abel) [1] . Also, more information and detail description have been discussed in Kilbas et al. [2] , Podlubny [3] , Machado et al. [4] . Fractional calculus generalize the differentiation and integration of integer order to real or fractional order. Nowadays, this generalization is extended to the variable order and the differential equations and integral equation have real and variable orders have been discovered with a beautiful physical interpretation. Control theory and stochastic process have many applications of fractional order differential equation [5] . The researchers have found many type of fractional derivative such as Riemann-Liouville, Caputo, Riesz, Hadamard and Grunwald-Letnikov derivatives. The fractional differential equations (FDEs) are obtained from ordinary ones by replacing the integer order to real order. FDEs have many applications in science, engineering, biology, medical, finance, economics and groundwater flow [6, 7] .

The applications of FDEs are increasing day by day which leads to an urgent need to find the general solution of these FDEs. To find out the analytical solution of these complicated * Corresponding author.

E-mail address: sachinraghav522@gmail.com (S. Kumar) .

FDEs, one needs to invoke a numerical solution treatment. Alarge number of methods have been discovered to deal with FDEs and FPDEs. Some of them are eigen-vector expansion, homotopy perturbation method [8] , Adomain decomposition method [9] , predictor-corrector method [10] , fractional differential transform method [11] and generalized block pulse operational matrix method [12] etc. The spectral method known as operational matrix method is very efficient method and easy to apply. It has a very desirable accuracy. Some operational matrix based upon Legendre wavelets [13] , Chebyshev wavelets [14] , sine wavelets, Haar wavelets [15] are available in literature. Operational matrix based upon orthogonal and nonorthogonal polynomial are given in Legendre polynomial [16] , Laguerre polynomial [17] , Chebyshev polynomial and semi-orthogonal polynomial as Genocchi polynomial [18] .

The novel corona virus was emerged first time in December, 2019, in Wuhan city of China. The virus is a new type in its family. Later world health organization (WHO) named it COVID-19.

Due to this virus, any infected person faces many symptoms like as respiratory illness, cough, fever and difficulty in breathing [19, 20] . This spreads when a healthy person comes in a contact with the virus carried out by a infected person specially contact with the drops of cough and sneeze of infected person. Some approximate solution of the time-fractional equations involving fractional integrals without singular kernel can be used to heed some light on the expected time development [21] [22] [23] . WHO has declared it as a pandemic due to widely spread of this virus. Yet there is no medicine or vaccine to cure this virus infected people. Only precautions can be adopted to keep ourselves safe. Till the date 4, April, 2020, the number of confirmed COVID-19 infected cases is 1,118,045 and 59,201 are dead due to this. The effect of this virus is more on the people of age greater than 40. The only cure is our precautions, we should have to quarantine ourselves in our homes to decreases contact rate, transmissible multiple. The human kind has the power to change the environment around us. There are some boundaries that should not be violated. In this present era, the intention of competition between humans, countries has developed so many powerful instruments to control on sea, air and ground. The human has created so many weapons like as guns, atom bomb, dangerous chemicals and nuclear bomb. So they violated the fundamental law of nature and led to so many natural disasters. We have forgotten that without nature we can not exist and we are just passenger. In this paper, we introduce a new and novel mathematical approach to study the behavior and dynamics of COVID-19 with a new non-singular fractional derivative called Mittag-Leffler kernel's derivative. To solve the presented model, we use of a newly derived matrix with Legendre collocation method. We will present some numerical treatments based on the number of infected people increases with increment in contact rate.

The organization of thus article is as follows. In Section 2 , some preliminary definition of fractional derivative and ABC derivative are briefly discussed. The derivation of operational matrix of fractional differentiation based on orthogonal Legendre polynomial on interval [0, a ] is derived in Section 3 . The description of COVID-19 model, its related data and procedure of numerical solution are given in Section 4 . The results and discussion are presented in Section 5 and the conclusion of all over article is given in Section 6 .

The definition of fractional integration and differentiation are available in literature [23] . There are mainly two types i.e., Riemann-Liouville and Caputo [24, 25] . In starting, fractional derivatives with power law kernel are introduced. In recent years, many fractional derivative definitions with non-singular kernel are introduced as exponential kernel and Mittag-Leffler kernel.

Definition 1. We define the fractional integration of ( x ) of order

The definition of Riemann-Liouville integration is given as follows

Definition 2. The definition of fractional differentiation with power law kernel is given in literature as follows

With z ∈ [0, ∞ [ and n is an integer. The Caputo definition has a similarity with integer derivative that is

with M is a constant.

where γ is floor function. All fractional operator are linear in nature as they follow the linearity property

with M 1 and M 2 are constants. The Caputo and Riemann-Liouville operator can be relate by the following expression [26] [27] [28] Let a function ( x, t ) belongs to the Sobolev space H 1 (0, 1). Then this fractional derivative with Mittag-Leffler kernel which is also known as ABC fractional derivative can be defined as

.

In this section we derive the Legendre operational matrix of fractional differentiation on domain [0, a ], a ≥ 1, a ∈ N of Mittag-Leffler kernel derivative which is known as ABC derivative.

Proof. From the definition (9) D n y l = 0 , l = 0 , 1 , . . . , n − 1 and for

Now, we use the series expansion formula of Mittag-Leffler function and evaluate the above integral as follows

This is the desired approximation expression for ABC derivative of

Here, we give the brief definition and property of extended Legendre polynomial. We know that the Legendre polynomial are orthogonal polynomial defined on interval [ −1 , 1] . By using the transformation z = 2 y −a a we transform the Legendre polynomial from the interval [ −1 , 1] to the interval [0, a ]. The series form of this polynomial is given in the following expression

The Legendre polynomial are orthogonal on interval [ −1 , 1] with respect to the weight function 1. As we have extended the these polynomial to a larger interval [0, a ] so the orthogonality condition is changed according the transformation as follows

With the help of these extended Legendre polynomial a function χ( y ) belonging to L 2 [0, a ] can be written as a finite linear combination as

The coefficient r j are determined as follows with the help of orthogonal condition

where

Now in next theorem we will develop the Legendre operational matrix of fractional differentiation on the domain [0, a ] with the help of Eq. (13) .

If we denote the column vector of extended Legendre polynomial by N ( y ) then fractional differentiation of order n − 1 < γ < n is given by the formula,

Here R γ represents the operational matrix of fractional differentiation of order N × N. We can obtain this as the following

where ξ i,j,l is defined by the following expression 

Taking the help from the series expression of extended Legendre polynomial and the definition of ABC derivative

To find out the ( i, j ) th element ϖ i,j of operational matrix R ρ , we perform the inner product as follows

By using the orthogonal property of Legendre polynomial we determined the above inner products value as follows

where

We use a numerical integration scheme known as Simpson 1 3 rule and the interval of integration [0, b ] is divided into m equal sub parts with length of segment width h . Putting the value of both of inner product in Eq. (16) we obtained the following expression of

Assuming i, j = i l= ρ ξ i, j,l we get the final desired result

We have derived the operational matrix of differentiation for fractional order on domain [0, b ]. But for the integer order elements of operational matrix is obtained as follows The function ζ j is defined as

To study the measurement-induced by COVID-19 transition due to the time development with different parameters and data, we performed the time-evolution using exact diagonalization. In this section we focus on cases dynamics interspersed with fractional order measurements in the two-dimension basis, and demonstrate the fact that there is a qualitative difference once initial values are introduced differently. We suppose that N p denotes the total population of people. This is divided into 5 categories as (i) S p -Susceptible people.

(ii) E p -Exposed people (iii) I p -Infected people (iv) A p -Asymptotically infected people (v) R p -Recovered people and N p = S p + E p + I p + A p + R p . The parameter Π p denotes the birth rate of people and the parameter μ p represent the death rate of people in each case. The term η p S p I p represents the susceptible people will infected with a sufficient contact with infected people I p . Here, η p is disease transmissibility coefficient. And the term ψη p A p S p denotes that susceptible people will be infected from asymptotically infected people with ψ transmissibility multiple of A p to I p and values of ψ belong to the closed interval [0,1]. The ψ = 0 states no transmissibility and ψ = 1 then this contact with asymptotically infected people will be treated as contact with infected people. The rate of susceptible people, from which they join the class of infected and asymptotatic class are denoted by ω p and ϱ p respectively. The removal and recovery rate from the class I p and A p to the class R p are depicts by the parameters τ p and τ ap respectively. The unknown function M is related to the seafood market or reservoir. The parameter η w is disease transmission coefficient from M to S p with term η w S p M . The contribution of virus from asymptomatically infected and symptomatic infected to the reservoir is denoted by ϱ p and ω p . The parameter Λ denotes the removing rate of virus from reservoir. We present the model as follows [29] .

The prescribed initial conditions for the above model are

The value of used parameters are taken from the literature [29] ( Table 1 ) We have derived the Legendre operational matrix of fractional differentiation on domain [0, a ]. Now we will use this newly operational matrix to find the solution of COVID-19 model. So with the help of Eq. (12) we will approximate the unknown functions present in our model as follows Table 1 Numerical value of used parameters for model (21 ψ 1 (x ) , . . . , ψ N−1 (x )) T is a column vector. Now for approximating the left hand parts of all equations presented in model (19) we are operating the fractional operator of derivative on these sides and using Eq.

To find the solution we approximate the initial conditions by taking help of Eq. (12)

Now using the Eqs. (24) and (25) in our model we get the residual functions as

Now collocating Eq. (25) and (24) at suitable points between the interval [0; a ], one gets a nonlinear system of algebraic equations. Solving this system of equations and finding its dynamics, we discussed the numerical solution of our proposed model.

We study the dynamics of susceptible, exposed, infected and asymptotically infected people using different fractional order. Fig. 1 (a) shows the behavior between susceptible people versus time and Fig. 1 (b) is indicates the relations between exposed people versus time. We see in Fig. 1 (b) that number of exposed people increases with time. We observe that this growth increases as we increases the fractional order γ 2 from 0.7 to 1 while Fig. 1 (a) shows that number of susceptible people decreases with time because they are getting into exposed or infected class. Fig. 2 (a) and (b) are plotted between infected people I p ( t ) versus time and asymptotically infected people versus time respectfully. Fig. 2 (a) predicts that number of infected people will increase with time. Similar nature is also seen for the asymptotically infected people. The effect of fractional exponent on I p ( t ) and A p ( t ) is that it increases with increment in γ 3 and γ 4 from 0.7 to 1. As COVID-19 spread with social contacting, touch with infected people and infected surfaces. We see that number of infected people increase exponentially with time. This behavior can be seen by taking data of Italy and USA till today 4 April, 2020 as there infected people are increasing followed by this exponentially behavior.

To study the behavior of this virus with contacting to infected or asymptotically infected people, we plotted the graph between the infected people I p ( t ) versus time and asymptotically infected people A p ( t ) versus time. We see in Fig. 3 (a) that number of asymptotically infected people increases with time. And an important fact can be seen that it increases as contact rate η p increases. Fig. 3 (b) shows that number of infected people increases rapidly like exponentially behavior and this number increases as contact rate increases. Fig. 4 (a) and (b) show that the behavior of I p ( t ) and A p ( t ) with transmissibility condition ψ. We can observe that number of infected people and asymptotically infected people increases with increment in ψ. To study the behavior of model with different initial condition, we plotted two graphs Fig. 5 (a) and (b). Fig. 5 (a) is plotted between infected people versus time with different A p (0). We see as initial number of A p ( t ) increases the number of infected people also increases. Similarly, from Fig. 5 (b) , if at initially stage number of infected people is more than number of asymptotically infected people increases with this initially increment of infected people.

Measurement-induced COVID-19 transitions represent an interesting new class of phase transition which shine light on the resilience of the present kind of viruses against a known one. They were initially explored for systems at ordinary differential equations dynamics and integrable models. In this work we have demonstrated that the nature of the measurement induced COVID-19 transition can be well described by newly fractional calculus systems. The measurements have been made in a basis which is scrambled by different parameters and new controllers of the dynamics, then the transition from infected case to recovered case occurs at a nonzero measurement probability and can be controlled by changing the significant parameter, generalizing the previously studied chaotic systems. It is worth noting that one key difference between the model considered in this paper and previous models is that here there are more variables (spatial) disorder in the unitary part of the dynamics. This is noteworthy because we could derive an approximation formula for the fractional derivative of ABC type of function ( t ≥ a ) on domain [0; a ]; a ≥ 1; a ∈ N : for the first time (as far as we know) and have developed the operational matrix of fractional differentiation with Mittag-Leffler kernel. The use of this newly derived matrix with Legendre collocation method is applied to solve a system of fractional ordinary differential equation. We find out the dynamics of susceptible, exposed, infected and asymptotically infected people, that how is behave with different fractional fractional order. It is shown that the number of infected people increases with increment in contact rate. So if we want to stop this outbreak pandemic we should be quaran-tine to reduce the contact rate. The effect of transmissibility multiple is shown by graphical representation. Effect on number on infected people and asymptotically infected people with different initial conditions.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.