ANALYSIS OF FRACTIONAL SUSCEPTIBLE-EXPOSED-INFECTIOUS (SEI) MODEL OF COVID-19 PANDEMIC FOR INDIA

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The purpose of this article is to develop and analyse COVID-19 pandemic for India in terms of mathematical equations. We consider the basic Susceptible-Exposed-Infectious (SEI) epidemic model and develop the SEI model of COVID-19 for India. We use Adomian decomposition method to find solution of the group of fractional differential equations. We discuss the stability by using Routh-Hurwitz criterion for disease-free equilibria point and endemic equilibrium point. We obtain approximate solution of the group of fractional differential equations and its solution represented graphically by Mathematica software, that will be helpful to mininize the infection.


INTRODUCTION
The Pandemic of a new human corona virus, named as COVID-19 by WHO officially is an ongoing Pandemic. The first case was detected in the month of December 2019 in Wuhan, Hubei, Chiana. As of 03 : 27 UTC on 30 th April 2020, a total of 3, 193, 886 cases are confirmed, in more than 185 countries and 200 territories, including 26 cruise ships and 227, 638 deaths.
The first Patient of Corona-Virus was found on 30 th January 2020 in India. Ministry of Health and Family Welfare (MHFW) have confirmed 33050 cases upto 30 th April 2020. Now a days, many real world problems in the field of biology, physics, engineering, financial and sociological can be represented in terms of group of non-linear ordinary and fractional order differential equations. Most of the time researchers are not able to find analytical solution of non-linear ordinary and fractional differential equations; due to that we use numerical methods to obtain their approximate solutions. Recently, there are several methods have been studied by many researchers to find solution of ordinary and fractional differential equations.
Mathematical system of the transmitted diseases play an important role in understanding spread of disese and taking measures to controll disease. After the start of the emanation in Wuhan, many researchers modelled various mathematical structures for the purpose of estimations and predictions for the Corona virus. In the paper, [21] researcher studied Age structured impact of social distancing on the COVID-19 of India. More literature pertaining to this can be refer in ( [28,29,30]   Thus above SEI model can be written as system of ordinary differential equations as follows Motivated by the above literature applications of epidemic mathematical models, in this paper we are studying dynamics of novel coronavirus SEI model derived in (1) in the from of system of the nonlinear differential equations involving Caputo fractional derivative operator of order α such that α ∈ (0, 1] as follows We arrange the paper as per folowing sequence: In section 2, we discuss few basic definitions of fractional calculus. In Section 3, we discuss about Adomian Decomposition Method to solve fractional SEI model. In Section 4, we comment about equilibrium points and stability and calculate basic reproduction number R 0 . In section 5, we find the solution of fractional SEI model and represent their solutions graphically by Mathematica software. Section 6 is devoted to conclusions.

PRELIMINARIES
In this section, we study some basic definition of fractional integral, fractional derivative and their properties for further development. We use the Caputo's definition due to its convenience for initial conditions of the differential equations.

ADM FOR THE SYSTEM OF FRACTIONAL ORDINARY DIFFERENTIAL EQUATIONS
Consider the system of fractional ordinary differential equation where D α is the Caputo fractional differential operator.
We consider the series solution of equation (7) is where A i j (u i0 , u i1 , · · ·, u in ) are called Adomian polynomials.
By substituting (8) and (9) in (7), we get From this we define, In order to determine the Adomian polynomial, we introduce a parameter λ and (9) becomes, In view of (12) and (13), we get Hence, the equation (10) and (14) leads to following recurrence relation , n = 0, 1, 2, 3, · · · We can approximate the solution u i by the truncated series

EQUILIBRIUM POINTS, STABILITY
respectively.
Proof: Now, substituting s = S N , e = E N , i = I N in to the system of equations (1), we obtain Putting i * = 0 in third equation of system (17) we get e * = 0.
We obtain the endemic equilibrium point, from third equation of system (17) as follows From the second equation of (17), we get β σ E From the first equation of (17), we get Thus, we obtain

Theorem 4.2. Stability [18]
If J is the Jacobian matrix of order (k × k) for a nonlinear system of k first order equations, then trajectory of the system that is equilibrium point will have stable behavior when real part of all eigenvalues is negative.
Proof: The Jacobian matrix of SEI model is evaluated as follows The Jacobian Matrix of SEI model at DFE is as below To find its eigenvalue, we must have to solve det[J(1, 0, 0) − λ I 3 ] = 0. Therefore Therefore, eigenvalues are According to (4.2), DEF is stable if In addition to stability of DFE, we have to discuss stability of EE.
Let us find eigenvalues of EE of SEI model.
To find eigenvalues, characteristic equation will be considered as Thus, Characteristic equation is Now, compare with λ 3 + a 1 λ 2 + a 2 λ + a 3 = 0, we get Threfore, from lemma (4.1), EE is stable if a 1 > 0, a 2 > 0, a 3 > 0, a 1 a 2 − a 3 > 0. Now, we have to calculate basic reproductive number (R 0 ) and discuss stability according to that The basic reproductive number (R 0 ) is define as the number of secondary infectious that one infectious individual create over the duration of the infectious period, provided that every one else is susceptible.
The biological interpretation of R 0 is that if To compute basic reproductive number R 0 of our model, we employ the NGM as applied by Diekmann et. al. [19]. We will refer the second and third equations of (17) as the linearized infection subsystem as it only describe the production of new infected and changes in the states of already existing infected. The matrix corresponding to transmissions and the matrix As per the NGM, R 0 is the dominant eigenvalue of matrix F V −1 , hence .
So authors will recommend to the government to increase measures for reducing R 0 (i.e. < 1) which control Pandemic situation. This is possible by reducing contact rate between peoples through effective social distancing, lockdown, by taking safety measures.

NUMERICAL SIMULATIONS
In this section, we obtain the solution of fractional SEI model (2) by Adomian Decomposition Method as discussed in previous section.
Consider the series solution of system (2) as By using (14), we obtain the Adomian polynomial as follow The Adomian decomposition series given in (10), leads to following result We obtain, the first three iterations of the solution of fractional SEI model of COVID-19 for India as follows Γ(3α + 1) +··· Next, We have analysed the above model by considering the assumptions given below and further estimating or fitting various parameters given by table (i) µ I = µ 0 = µ that is the emigration from the population is equal to emigration into population.
(ii) 1 D E = σ E and 1 D I = σ I . (iv) Average exposed(Incubation) period is 14 days and average infection period is 5 days [24].
(vi) Initial case are only from emigrated people (i.e. persons having foreign travel history).
(vii) Initial rate of transmission is 1.7    2, 3 and 4 shows the graphical representation of suspectable population S(t), exposed population E(t) and infected population I(t) for various values of α(= 1, 0.9, 0.8) using Adomian Decomposition method which predicts that this method can foresee the conduct of said variables precisely for the considered region. We observe that infection dies out slowly as value of α decreases. Hence, the non-integer order has a significant effect on the dynamics of SEI model of COVID-19 for India.

CONCLUSIONS
In this work, we have developed fractional SEI mathematical models and analysed dynamics of COVID-19, which is Pandemic throughout the world. Further, we have obtained series solutions of this model by Adomian Decomposition Method. We found basic reproduction number for this disease is R 0 = 2.92, which is unstable because it is greater than one. We also observe that R 0 is depend upon β and µ(note that σ E and σ I are constant). Results of above mathematical model are agree with recommendation of WHO that "It is still possible to interrupt virus spread, provided that countries put in place strong, measures to direct disease only". To keep R 0 < 1 that is asymptotically stable, we have to control β specially. This can be done by promoting social distancing measures, avoid large social gatherings, applying lock down and travel ban. To control µ, we may increasing effectiveness of passenger screening at airports. To stop spread of virus Indian government has already suspended all commercial passenger flight from 23 rd March 2020. Also, declare lock down from 25 th March 2020. This steps will help to reduce β and µ and hence R 0 . We further Also, we represented solution graphically by Mathematica software. However, the result obtained above is depends on the limited data available in various research paper and note that real situation at initial stage of transmission is may be different.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.