Journal Pre-proofs

A novel fractional mathematical model of COVID-19 epidemic considering

quarantine and latent time

Prashant Pandey, Yu-Ming Chu, J.F. Gómez-Aguilar, Hadi Jahanshahi,

Ayman A. Aly

PII: S2211-3797(21)00420-4

DOI: https://doi.org/10.1016/j.rinp.2021.104286

Reference: RINP 104286

To appear in: Results in Physics

Received Date: 26 February 2021

Revised Date: 2 May 2021

Accepted Date: 3 May 2021

Please cite this article as: Pandey, P., Chu, Y-M., Gómez-Aguilar, J.F., Jahanshahi, H., Aly, A.A., A novel

fractional mathematical model of COVID-19 epidemic considering quarantine and latent time, Results in Physics

(2021), doi: https://doi.org/10.1016/j.rinp.2021.104286

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© 2021 Published by Elsevier B.V.

A novel fractional mathematical model of

COVID-19 epidemic considering quarantine and

latent time

Prashant Pandey1,2

, Yu-Ming Chu3,4∗

, J.F. G´omez-Aguilar5,6∗∗

,

Hadi Jahanshahi7

, Ayman A. Aly8

1Department of Mathematical Sciences

Indian Institute of Technology (BHU), Varanasi, 221005, India.

2Department of Mathematics

Government M.G.M. P.G. College, Itarsi, 461111, India.

3Department of Mathematics, Huzhou University,

Huzhou 313000, P.R. China.

4Hunan Provincial Key Laboratory of Mathematical Modeling

and Analysis in Engineering, Changsha University

of Science & Technology, Changsha 410114, P.R. China.

5CONACyT-Tecnol´ogico Nacional de M´exico/CENIDET,

Interior Internado Palmira S/N, Col. Palmira, C.P. 62490,

Cuernavaca Morelos, Mexico.

6Universidad Tecnol´ogica de M´exico – UNITEC MEXICO ´

– Campus En L´ınea.

7Department of Mechanical Engineering, University of Manitoba,

Winnipeg, R3T 5V6, Canada.

8Department of Mechanical Engineering, College of Engineering,

Taif University, P.O.Box 11099, Taif 21944, Saudi Arabia.

*Email: [email protected]

**Email: [email protected]

Abstract

In this paper, we investigate the fractional epidemic mathematical

model and dynamics of COVID-19. The Wuhan city of China is considered as the origin of the corona virus. The novel corona virus is continuously spread its range of effectiveness in nearly all corners of the world.

Here we analyze that under what parameters and conditions it is possible

to slow the speed of spreading of corona virus. We formulate a transmission dynamical model where it is assumed that some portion of the

people generates the infections, which is affected by the quarantine and

1

latent time. We study the effect of various parameters of corona virus

through the fractional mathematical model. The Laguerre collocation

technique is used to deal with the concerned mathematical model numerically. In order to deal with the dynamics of the novel corona virus we

collect the experimental data from 15th −21st April, 2020 of Maharashtra

state, India. We analyze the effect of various parameters on the numerical

solutions by graphical comparison for fractional order as well as integer

order. The pictorial presentation of the variation of different parameters

used in model are depicted for upper and lower solution both.

Keywords

Covid-19; Mathematical model; Laguerre collocation technique; Caputo derivative; Operational Matrix.

1 Introduction

Nowadays the mankind is continuously disobeys the basic and fundamental laws

of Nature. In order to secure the future of their survival and to make a strong,

positive, and respectful impact on other mankind they developed scientific research very rapidly. As a result of strong scientific research they invented and

developed many scientific tools which make the life of mankind more wealthy,

smarter, and easier. This new scientific way of tackling the existing and upcoming problems enhance the power of mankind so that in this race of advanced

research they tackle many strong and dangerous problems. The fundamental

and basic laws of nature have been made in such a way that the balance of

the environment is being balanced for a longer time. Mankind has succeeded

more enough in these new scientific developments and research. During these

advanced invention many laws of nature are continuously being violated and led

society to face many diseases and natural disasters.

At the end of December 2019, there was a case of pneumonia in Wuhan

city of China with undermined cause and for this case the available vaccines

were ineffective. Later on, it was found that the cause of the case is the novel

corona virus. The World Health Organization (WHO) has been declared a public health emergency and this new disease as a pandemic due to its exponential

worldwide growth [1]. The COVID-19 disease is now a global health threat [12].

Till date there is no any available vaccine, medical treatments are registered

to cure from this disease [10]. Many symptoms of COVID-19 were found, the

majority of the cases have symptoms of fever and dry cough whereas some cases

have symptoms which like joint and muscle pain, fatigue, sore throat, shortness

of the breath and headache. Due to the unavailability of an accurate treatment and vaccine, social distancing and quarantine are accepted by the world

as an effective strategy to reduce the spreading of the transmission of corona

virus. Many countries adopted the quarantine strategy for a large portion of

the community and lockdowns of cities and states. Although there is a longtime

2

lock-down across many countries, there is a class of doctors, sweepers, police

and others who provides their essential services in this lock-down. Because of

their dedication in service, the Govt. of India named these classes as Corona

Warriors.

The World Health Organization has been defined four main stages of the

transmission of COVID-19 disease. These stages are the areas or countries

with:

1 : Stage first is reached, when there are no any cases reported in the particular

areas or countries.

2 : Stage two is reached, when there are some sporadic cases reported.

3 : Stage three is reached, when the cases are come in clusters at any places.

4 : The stage four is a more sensitive stage where the cases start reported due

to community transmission.

All the countries may be looking for which measures are to be adopted at

different stages and analyze the situation of this pandemic regularly [15].

The first case of COVID-19 disease in India was reported on 30th January

2020. The starting few cases were reported in students who had a recent travel

history from Wuhan, China. In the month of March, many cases were reported

throughout India due to contact with some existing cases of COVID-19 [2].

Indian government realize the dangerous effect of the disease and takes some

important decisions to prevent peoples from COVID-19. The Indian Prime

Minister Shri Narendra Modi announced a lock-down for 21 days on 24th

March throughout the country.

The field of investigation of biological is now a day encountered by many

researchers [5, 8]. In many cases, the formulation of biological models by mathematical modeling based on ordinary differential equations i.e., on classical

derivatives have some limitations and may not be able to describe the biological

phenomenon accurately. To deal with these biological models very efficiently

and accurately, fractional mathematical models are being in used [13, 19, 20].

The mathematical formulation of the dynamics of many infectious diseases has

a very deep history. Many researchers have developed a successful formulation

of biological models using fractional calculus theory [4, 22]. The fuzzy logic

approach for various epidemic mathematical models was also discussed and analyzed by many researchers [6, 7, 16].

In the present paper, we formulate the COVID-19 effect on society in a systemic way. This mathematical model will able to explain the rate of transmission

of disease, the impact of disease in the rate of susceptible cases, exposed cases,

infectious cases with different parameters viz., average latent time, contact rate,

etc. Here we perform the examination of model simulations and investigate the

sensitive analysis of the concerned COVID-19 model for effective outcomes of

the model. Here, We will also analyze the nature and behavior of the solution of

the concerned COVID19 model in the fuzzy environment under the interaction

of various parameters.

This paper is organized as: Section 2 contains some definitions and properties

of fractional derivative. The fractional mathematical formulation of dynamics

of COVID-19 is given in section 3. A brief outline of the Laguerre polynomial

3

is given in section 4. The approximation of the real state variables present in

the mathematical model is given in section 5. In section 6 the novel operational

matrix is derived. Section 7 contains the application of the derived fractional

operational matrix to formulated model. The obtained numerical results are

discussed in section 8. The effect of different parameters present in the model

is analyzed in section. The overall outcomes of the research paper are given in

section 10.

2 Preliminaries and Notations

2.1 Caputo Fractional Derivatives

In this section of the manuscript, we have given basic definitions and properties

of fractional order derivatives. The Riemann-Liouville’s derivatives have some

drawbacks in modeling the mathematical modeling some real and physical phenomenon. Fractional order derivative operator Dη of the given order η > 0 of a

function k(t) in the Caputo sense is given by [14, 21]

(Dη

k)(t) = (d

rk(t)

dtr , if η = r ∈ N,

1

Γ(r−η)

R t

0

(t − ρ)

r−η−1k

(r)

(ρ) dρ, if r − 1 < η < r.

(1)

In the above expression r be the integer number. Few properties of Caputo

fractional derivatives are:

DηP = 0, (2)

where P is an arbitrary constant.

Dη

t

ε =

(

0, if ε ∈ N ∪ {0}, ε < ⌈η⌉,

Γ(ε+1)

Γ(ε−η+1) t

ε−η

, if ε ∈ N ∪ {0}, ε ≥ ⌈η⌉ or ε /∈ N, ε > ⌊η⌋,

(3)

where ⌈η⌉ be the ceiling function and ⌊η⌋ be the floor function.

2.2 Fuzzy Set Theory:

The idea of fuzzy sets has been introduced by L. Zadeh [23] in 1965 to tackle the

uncertainty arises because of imprecision and vagueness. Consider a nonempty

set Z which is called as base a set and every member z ∈ Z is associated to a

membership grade ζ(z). A nonempty subset of Z ×[0, 1] is considered as a fuzzy

subset of Z by L. Zadeh.

The definition of a fuzzy set B is given as: B ⊂ {(z, ζ(z)) : z ∈ Z} where ζ is

a function from Z to [0, 1]. The symbol ζ is commonly used for the notation of

the fuzzy set B.

Definition 1. (Fuzzy Numbers): Now we are going to provide the definition of a fuzzy number ˜ω. A real valued function ˜ω from R to unit interval

[0, 1] i.e. ˜ω : R −→ [0, 1] is said to be a fuzzy number if it satisfies the following

4

basic properties:

• The function ˜ω should be upper semi-continuous.

• The function ˜ω should satisfy the normality properties i.e., ∃ a real number

z0 such that ˜ω(z0) = 1.

• The convexity property should be satisfied by the function ˜ω i.e., ∀k ∈ [0, 1]

and ∀z1, z2 ∈ R, we have

ω˜(kz1 + (1 − k)z2) ≥ min{ω˜(z1), ω˜(z2)}. (4)

• The closure set of the support of function ˜ω is a compact set. The support of

the function ˜ω is defined as supp(˜ω) = {z ∈ R : ˜ω(z) > 0}.

Definition 2. (v-Level Set of Fuzzy Numbers): Consider the collection

of all fuzzy numbers defined on set of real number R is denoted by RF . Let the

fuzzy number ˜ω ∈ RF for some v ∈ [0, 1] then for every v ∈ [0, 1] the v-level set

of fuzzy number [˜ωv] is defined as

[˜ω] = (

{z ∈ R : ˜ω(z) ≤ v}, v ∈ (0, 1],

closure(supp(˜ω)), v = 0.

(5)

From the above definition we can find that the v−level set [˜ω] is a closed and

bounded set. Let ˜ω

−(v) and ˜ω

+(v) are the end points of the v-level fuzzy

interval then we can write the v-level fuzzy interval as [˜ω] = [˜ω

−(v), ω˜

+(v)].

The following definition of the fuzzy number can be very useful in embedding

the set of real number to the set of fuzzy number for all z, y ∈ R:

ω˜(y) = (

1, y = z,

0, y 6= z.

(6)

Definition 3. (Parametric Interval Form): For any fuzzy number ˜ω ∈ RF ,

the parametric interval form can be given as:

ω˜[v] = [˜ωl(v), ω˜u(v)], v ∈ [0, 1]. (7)

The above form of the fuzzy number satisfies the following properties:

• For every v ∈ [0, 1] The functions ˜ωu(v) and ˜ωl(v) satisfy the inequality

ω˜l(v) ≤ ω˜u(v).

• The function ˜ωu(v) is a non-increasing and left continuous function of v.

• The function ˜ωl(v) is a non-decreasing and left continuous function of v.

The arithmetic operations i.e. vector addition and scalar multiplication of any

two arbitrary fuzzy numbers ˜ω1(v) and ˜ω2(v) are defined for v ∈ [0, 1] as:

(˜ω1 ⊕ ω˜2)[v] = [˜ω1l + ˜ω2l

(v), ω˜1u + ˜ω2u

(v)],

(k ⊙ ω˜)[v] = (

[kω˜l(v), kω˜u(v)], k ≥ 0,

[kω˜u(v), kω˜l(v)], k < 0.

(8)

5

Definition 4. (gH-difference): Let M and N are two nonempty compact set

then there gH-difference (generalized Hukuhara difference) as the compact set

P is given as:

M ⊖gH N = P ⇔

(

(a) M = N + P,

or (b) N = M − P.

(9)

Definition 5. (gH-derivatives): Here we will provide the definition of fuzzy

derivatives (gH-derivatives) of any arbitrary fuzzy valued function. Consider a

point z0 in (l, m) and a fuzzy valued function ζ such that ζ : (l, m) −→ RF .

Then the function ζ is H-differentiable at point z0 and is equal to a fuzzy number ζ

′

(z0) if it is satisfy the following equations:

(i) Case 1: if the H-difference for two fuzzy number ζ(z0 + h ⊖ ζ(z0)) and

ζ(z0) ⊖ ζ(z0 − h) exists then we have:

ζ

′

(z0) = lim

h→0+

ζ(z0 + h ⊖ ζ(z0))

h

= lim

h→0+

ζ(z0) ⊖ ζ(z0 − h)

h

. (10)

This definition of differentiation is called as 1-differentiation of function ζ at

(l, m).

(ii) Case 2: if the H-difference for two fuzzy number ζ(z0 ⊖ ζ(z0 + h)) and

ζ(z0 − h) ⊖ ζ(z0) exists then we have:

ζ

′

(z0) = lim

h→0+

ζ(z0 ⊖ ζ(z0 + h))

−h

= lim

h→0+

ζ(z0 − h) ⊖ ζ(z0)

−h

. (11)

This definition of differentiation is called as 2-differentiation of function ζ at

(l, m).

The gH-derivative can also be given in same manner as:

ζ

′

(z0) = lim

h→0

ζ(z0 + h ⊖gH ζ(z0))

h

. (12)

In order to establish the fractional derivatives in Caputo and Riemann-Liouville

we are going to provide the definition of Lebesgue integration of any function

ζ

′

(t) in parametric fuzzy interval form as:

[

Z t

0

ζ

′

(z)dz]v =

Z t

0

[ζ

′

(z)]vdz =

(

[

R t

0

ζ

′

−(z; v)dz, R t

0

ζ

′

+(z; v)dz], for case-1,

[

R t

0

ζ

′

−(z; v)dz, R t

0

ζ

′

+(z; v)dz], for case-2.

(13)

2.3 Fuzzy Fractional Derivatives

Here we are going to present fuzzy fuzzy fractional derivatives of a fuzzy differential function ζ(t). The fuzzy fractional derivatives are the generalization of

6

classical fractional differentiation in crisp sense.

Definition 6. (Caputo fractional g-derivatives:) The fractional g-derivatives

of any fuzzy valued measurable continuous function ζ(t) of any arbitrary fractional order in Caputo sense at point t is given as:

gD

µ

a+ ζ(t) = lim

h→0

λ(t + h) ⊖g λ(t)

h

, (14)

where the function λ is given by

λ(t) = 1

Γ(1 − µ)

Z t

a

(t − ρ)

−µ

ζ(ρ)dρ. (15)

Let the function ζ(t) is absolutely continuous fuzzy valued function then Caputo

fractional fuzzy derivatives is defined for both previously cases as:

[

C

a+ D

i,µ

t

ζ(t)] = [C

a+ D

i,µ

t

ζ−(t; v),

C

a+ D

i,µ

t

ζ+(t; v)] for case-1,

[

C

a+ D

ii,µ

t

ζ(t)] = [C

a+ D

ii,µ

t

ζ+(t; v),

C

a+ D

ii,µ

t

ζ−(t; v)] for case-2.

(16)

where [C

a+ D

i,µ

t

ζ−(t; v),

C

a+ D

i,µ

t

ζ+(t; v)], [

C

a+ D

ii,µ

t

ζ−(t; v) and C

a+ D

ii,µ

t

ζ+(t; v)] are

given by the following equations:

C

a+ D

i,µ

t

ζ−(t; v) = 1

Γ(1 − µ)

Z t

a

(t − ρ)

−µ

ζ

′

−(ρ)dρ,

C

a+ D

i,µ

t

ζ+(t; v) = 1

Γ(1 − µ)

Z t

a

(t − ρ)

−µ

ζ

′

+(ρ)dρ,

C

a+ D

ii,µ

t

ζ+(t; v) = 1

Γ(1 − µ)

Z t

a

(t − ρ)

−µ

ζ

′

+(ρ)dρ,

C

a+ D

ii,µ

t

ζ−(t; v) = 1

Γ(1 − µ)

Z t

a

(t − ρ)

−µ

ζ

′

−(ρ)dρ.

(17)

Definition 7. (Riemann Integrability:) Here we are going to provide the definition for Riemann integrability in fuzzy approach. A fuzzy valued function ζ

such that ζ : [l, m] −→ RF is said to be Riemann integrable if for any ρ > 0

there exist τ > 0 such that for every partition E = {[k1, k2], τ} of the given

domain [l.m] we have

Υ(

X∗

E

(k1 − k2) ⊙ ζ(τ )) < ρ, (18)

with norms of partition E less than τ . The symbol P∗

E

denotes the summation

for addition under fuzzy calculus. We can also rewrite the Riemann integrability

in fuzzy sense as

J = (F R)

Z m

l

ζ(k)dk. (19)

7

The symbol Υ in equation (18) denotes the metric on set of all fuzzy numbers

RF which is defined as

Υ(˜ω1, ω˜2) = Sup

v∈[0,1]

M ax{| ω˜1l

(v) − ω˜2l

(v) |, | ω˜1u

(v) − ω˜2u

(v) |}. (20)

With respect to metric Υ, the set of fuzzy numbers RF forms a complete metric

space.

3 Fractional COVID-19 Model Formulation

In this scientific contribution, authors have reformulated a modified mathematical model observing the threshold behavior in recovery rate and quarantined

cases concerning the latent time and quarantine time with a significant difference

to SEIR model [9] w.r.to Caputo fractional derivative.

In order to characterize the effect of COVID-19 disease in state Maharashtra

of India, here we consider the main six various parameters viz., Sm(t), Em(t), Im(t),

Qm(t), Dm(t) and Rm(t) at some time t. Let Nm denotes the total population in

Maharashtra in current time, Sm(t) denotes total the number of susceptible cases

at time t, Em(t) denotes the total number of exposed cases at time t i.e. total

number of infected case which not yet be infectious, Im(t) denotes the total number of infectious cases which is not quarantined, Qm(t) denotes the total number

of quarantined case, Dm(t) denotes the death cases and Rm(t) denotes the recovered cases. We can write Nm(t) = Sm(t)+Em(t)+Im(t)+Qm(t)+Dm(t)+Rm(t)

at time t for any state or place.

Figure 1: Relation of interaction among parameters of COVID-19 model in

Maharashtra.

8

The relation of interaction among these parameters in the epidemic fractional

mathematical model of COVID-19 is demonstrated graphically in the Fig. 1.

In this pictorial presentation the constant α is a rate of natural mortality rate

in the state, β is contact rate among peoples, 1/γ denotes the average latent

time, 1/δ is average quarantined time, ϑ(t) is cure rate at a time t, ξ(t) is

mortality rate, ω denotes the recovery rate of infectious peoples without going

through the quarantine state and ω denotes the death rate of infectious peoples

without going through the quarantine state. The term αSm(t) is the natural

death rate of susceptible cases and the term βSm(t)Im denotes the number of

susceptible peoples who got infected (but not infectious yet) through the contact

with infected peoples. The term (γβ)Em(t) arises in the average latent time. In

the Fig. 1 it can be seen that the death rate ξ(t) and the cure rate ϑ(t) is time

dependent variable coefficients. In an instance by the treatment of medical team

and uses of newly tested drugs it can be found that the recovery rate is increases

so that death rate decreases very quickly. The term αRm(t) denotes the death

cases for recovered peoples by natural cause. The number of recovered peoples

among the infectious class without going through the quarantined are expressed

by term ωIm(t). Here it is assumed that the time dependent coefficients ξ(t)

denotes the mortality rate and the total closed cases (mortality cases) during

the quarantine state is given by the term ξ(t)Qm(t). Assuming that the there

is quarantine system of infectious persons, in that case the quarantine case is

computed by the average quarantine time with infectious class i.e. by the term

δIm(t). This fractional mathematical model of the dynamics of corona virus

in the Maharashtra (India) is formulated on the assumption that peoples need

some average time to come in to the class of infectious peoples, quarantine class,

recovery class or death class. The total number of quarantined cases, death

cases, infected cases and the recovered cases are available of common man by

the Govt. of India. Because of the time dependence of cure and death rate the

term Qm(t) plays an important role in the modeling of such epidemic disease.

The average latent time is calculated within many days of observations.

The formulated experimental fractional model for corona virus disease in

Caputo sense is given by the following system of differential equations:

C

0 D

µ

t Sm(t) = −β

Sm(t)Im(t)

Nm

− αSm(t),

C

0 D

µ

t Em(t) = β

Sm(t)Im(t)

Nm

− (α + γβ)Em(t),

C

0 D

µ

t

Im(t) = γβEm(t) − (α + δ + ω + σ)Im(t),

C

0 D

µ

t Qm(t) = δIm(t) − (ϑ(t) + ξ(t))Qm(t),

C

0 D

µ

t Dm(t) = σIm(t) + ξ(t))Qm(t),

C

0 D

µ

t Rm(t) = ϑ(t)Qm(t) + ωIm(t) − αRm(t),

(21)

with the non-negative initial conditions

Sm(0) = a, Em(0) = b, Im(0) = c, Qm(0) = d, Dm(0) = e, Rm(0) = f.

(22)

9

In the above fractional epidemic COVID-19 model the term µ denotes the fractional order of Caputo sense derivatives. To find the numerical solution of this

experimental model we adopt the novel numerical scheme developed in the literature [18].

Although there are wide application of fractional PDEs but the accurate mathematical model of many complex physical processes can not be found. Zadeh developed the concept of fuzzy theory to overcome this lackness of fractional PDEs

and the application of fuzzy theory with fractional PDEs can be able to mathematical modeling of such complex physical process. Fuzzy DEs are very efficient

tools which explain the many dynamical process accurately where the nature of

dynamical process is uncertain with vague information [11]. There we consider

COVID19 model (21) under fuzzy environment with ˜k(v) = [0.9+0.1v, 1.1−0.1v]

as

C

0 D

µ

t S˜m(t) = −β

S˜m(t)

˜Im(t)

Nm

− αS˜m(t),

C

0 D

µ

t E˜m(t) = β

S˜m(t)

˜Im(t)

Nm

− (α + γβ)E˜m(t),

C

0 D

µ

t

˜Im(t) = γβE˜m(t) − (α + δ + ω + σ)

˜Im(t),

C

0 D

µ

t Q˜m(t) = δIm(t) − (ϑ(t) + ξ(t))Q˜m(t),

C

0 D

µ

t D˜m(t) = σ ˜Im(t) + ξ(t)Q˜m(t),

C

0 D

µ

t R˜m(t) = ϑ(t)Q˜m(t) + ω ˜Im(t) − αR˜m(t),

(23)

with the non-negative initial conditions

Sm(0) = ˜k(v)a, Em(0) = ˜k(v)b, Im(0) = ˜k(v)c,

Qm(0) = ˜k(v)d, Dm(0) = ˜k(v)e, Rm(0) = ˜k(v)f.

(24)

4 Laguerre Polynomials

Before the computation of concerned COVID-19 model we are going to introduce

the important results and definitions of few key tools.

The involvement of Laguerre polynomial in several aspects of engineering and

applied mathematics seeks the attention of many researchers and established it

as a strong tool for finding the numerical solution of integer as well as fractional

order PDEs. The l

th degree Laguerre polynomial is defined as [18]

Ll(t) = 1

l!

e

t

∂

l

t

(t

l

e

−t

), l = 0, 1, 2, · · · ; (25)

The Laguerre polynomial of degree l on the interval I is simplified from above

as:

Ll(t) = X

l

k=0

l!(−1)k

(−k + l)!(k!)2

x

k

, k = 0, 1, 2, · · · ; (26)

10

The following recurrence relations are satisfied by Laguerre polynomial

∂t(te−t

∂tLl(t)) + le−tLl(t) = 0, t ∈ I,

Ll(t) = ∂tLl(t) − ∂tLl+1(t), l ≥ 0.

Ll+1(t) = (2l + 1 − t)Ll(t) − lLl−1(t)

l + 1

, ∀l ≥ 1.

(27)

The set of all Laguerre polynomials form and orthogonal system in L

2

r

(I) i.e.,

Z

I

La(t)Lb(t)r(t)dt = δlm, ∀a, b ≥ 0, (28)

where r(t) = e

−t

is weight function and δlm is the Kronecker delta function.

5 Approximation of the function ζ(x)

The collection of Laguerre polynomial form a basis set for L

2

r

(I) so any function

ζ(t) ∈ L

2

r

(I) can be expanded as:

ζ(t) = X∞

g=0

bgLg(t), m = 0, 1, 2, · · · ; (29)

where the unknown constants bg can be calculated as:

bg =

Z ∞

0

ζ(t)Lb(t)r(t)dt. (30)

In the terms of first r+1-Laguerre polynomials the above approximation reduced

to

ζr(t) = Xr

g=0

bgLg(t) = B

T σ(t), (31)

where BT = [b0, b1, ….., br] is the matrix of unknown constants and σ(t) =

[L0(t), L1(t), ….., Lr(t)]T

is the Laguerre vector.

5.1 Approximation of fuzzy valued function ˜ζ(t)

Here we use the orthogonal Laguerre polynomial in order to approximate a

fuzzy valued measurable and continuous function ˜ζ(t). The approximation of

the fuzzy valued function in terms of first r + 1- Laguerre polynomials is given

as:

˜ζr(t) = Xr

g=0

˜bg ⊙ Lg(t) = B˜T ⊙ σ(t), (32)

where σ(t) is the Laguerre vector and the unknown fuzzy coefficients are given

by the following equation:

˜bg =

Z ∞

0

˜ζ(t) ⊙ Lb(t) ⊙ r(t)dt. (33)

11

Here the summation is taken in accordance with fuzzy algebraic addition ⊕

and ⊙ denotes the fuzzy scalar multiplication. The unknown fuzzy coefficients

matrix B˜ = [˜bg] can be calculated with the help of equation (33) where all the

operation will be taken as fuzzy set algebra.

6 Fuzzy Operational Matrix for Fractional Derivative

In order to derive the novel fuzzy operational matrix we have to provide following lemma.

Lemma 1. Consider the fractional fuzzy derivative operator Dµ of fractional

order µ in Caputo sense. Let σ(t) be the Laguerre vector defined as in equation

(31) then we have following results:

C

0 D

µ

t Lr(t) = 0, r = 0, 1, · · · , ⌈µ⌉ − 1, 0 < µ ≤ 1. (34)

Proof: This result can be easily proved by using the basic properties of fractional Caputo derivative in the equation (26).

We are going to derive the operational matrix of orthogonal Laguerre polynomial [18]. On solving the equations (29)-(30) and equation (26) together we

get the following expressions

C

0 D

µ

t Ll(t) = X

l

g=0

(−1)g

l!Dµt

g

(l − g)!(g!)2

=

X

l

g=⌈µ⌉

(−1)g

l!t

g−µ

(l − g)!Γ(g − µ + 1)g!

, l = ⌈µ⌉, · · · , r.

(35)

As set of Laguerre polynomials is a basis so we can write the term t

g−µ as

t

g−µ =

Xr

h=0

bhLh(t), (36)

where the constant coefficient bh is given by

bh =

X

h

a=0

(−1)ah!Γ(g − µ + a + 1)

(h − a)!(a!)2

. (37)

Using the equations (36) − (37) into the equation (35), we have

C

0 D

µ

t Ll(x) = Xr

h=0

Pµ(l, h)Lh(t), l = ⌈µ⌉, · · · , r. (38)

In the above equation the term Pµ(l, h) is given by the following equation

Pµ(l, h) = X

l

g=⌈µ⌉

X

h

a=0

(−1)g+a

l!h!Γ(g − ν + a + 1)

(l − g)!Γ(g − µ + 1)g!(h − a)!(a!)2

. (39)

12

The vector for of the equation (38) is written as as

C

0 D

µ

t Ll(t) = [Pµ(l, 0), Pµ(l, 1), …, Pµ(l, r)]σ(t), l = ⌈µ⌉, · · · , r. (40)

In the view of lemma 1, we have

C

0 D

µ

t Ll(t) = [0, 0, …, 0]σ(x), l = 0, 1, · · · , ⌈µ⌉ − 1. (41)

Using the equations (40) − (41), we can write in matrix form as

C

0 D

µ

t σ(t) = H(µ)σ(t), (42)

where Hµ is an operational matrix fractional derivatives of r + 1 × r + 1 order

and is given by the following equation:

H(µ) =

0 0 0 . . . 0

.

.

.

.

.

.

.

.

. . . .

.

.

.

0 0 0 . . . 0

Pµ(⌈µ⌉, 0) Sµ(⌈µ⌉, 1) Pµ(⌈µ⌉, 2) . . . Pµ(⌈µ⌉, r)

.

.

.

.

.

.

.

.

. . . .

.

.

.

Pµ(i, 0) Pµ(i, 1) Pµ(i, 2) . . . Pµ(i, r)

.

.

.

.

.

.

.

.

. . . .

.

.

.

Pµ(r, 0) Pµ(r, 1) Pµ(r, 2) . . . Pµ(r, r)

. (43)

7 Implementation of Laguerre operational matrix on COVID-19 model

Here we are going to apply the Laguerre operational matrix and spectral collocation techniques to our concerned non-linear fractional COVID-19 model.

In view of equation (29), the function ζ(t) can be expressed in terms of initial

r + 1 Laguerre polynomials as:

ζr(t) = Xr

g=0

bgLg(t) = B

T

.σ(t), (44)

where bg is the unknown coefficients to be determined and σ(t) is a Laguerre

vector.

We operate the fractional derivatives in the equation (44) with respect to time

of fractional order µ and using the equation (43), we get

∂

µζ(t)

∂tµ

= Hµ

ζ(t) = B

T

.Hµσ(t). (45)

13

Now in view of equation (44), the initial conditions (22) can be rewritten as:

B

T

1

.σ(0) = a,

B

T

2

.σ(0) = b,

B

T

3

.σ(0) = c,

B

T

4

.σ(0) = d,

B

T

5

.σ(0) = e,

B

T

6

.σ(0)= f.

(46)

In view of the equation (45) the concerned model’s variables can be approximated as:

C

0 D

µ

t Sm(t) = B

T

1

.Hµσ(t),

C

0 D

µ

t Em(t) = B

T

2

.Hµσ(t),

C

0 D

µ

t

Im(t) = B

T

3

.Hµσ(t),

C

0 D

µ

t Qm(t) = B

T

4

.Hµσ(t),

C

0 D

µ

t Dm(t) = B

T

5

.Hµσ(t),

C

0 D

µ

t Rm(t) = B

T

6

.Hµσ(t).

(47)

Definition 8. (Fuzzy system of linear equation:) Consider a system of

linear equations as

b11z1 + b12z2 + · · · + b1nzn = d1,

b21z1 + b22z2 + · · · + b2nzn = d2,

.

.

.

.

.

.

.

.

.

bn1z1 + bn2z2 + · · · + bnnzn = dn.

(48)

We can write the above system of equations in the matrix form as

BZ = D. (49)

In the above equation the constant numbers d

′

j

s are fuzzy numbers and Cn×n

is crisp matrix of order n × n. This system of linear equations can be solved by

the techniques given in the literature [3].

Now collocating equation (21) with the help of equation (46) at points ta=

a

r

for a = 0, 1, 2, · · · , r. After collocating, we get a system of equations. Further by simplifying this algebraic system of equations and finding the unknown

matrix coefficients Bi

′s , we obtain numerical solution of our given problem by

substituting Bi

′s in equation (44).

Now we are going to investigate the proposed Covid-19 model into fuzzy environment.

14

8 Numerical Results

In this section of the article the adopted numerical scheme is applied to our

concerned nonlinear fractional corona virus model (21) with the non-negative

initial conditions (22) which is estimated and collected for the total population

of Maharashtra Nm(0) = 12486220. The total susceptible case for is Sm(0) =

12113755, exposed cases Em(0) = 300000, infectious cases Im(0) = 2916, death

cases Dm(0) = 186, quarantine cases Qm(0) = 69068 and the recovered cases

Rm(0) = 295. The values of other coefficients parameter is given by

Natural Mortality Rate(α) = 1

69.73 × 365

= 3.93056 × 10−5

,

Contact Rate(β) = 0.05,

Average Latent Time(1/γ) = 0.00042914,

Average Quarantine Time(1/δ) = 0.005,

Recovery Rate of Im (ω) = 0.09648,

Death Rate of Im (σ) = 0.913012,

Cure Rate ϑ(t) = 1.4,

Death Rate ξ(t) = 0.6.

(50)

The plot of the infected cases between the experimental infected cases and the

estimated, collected data fitting for the Maharashtra between the 15th − 21th

April 2020 is shown through the Fig. 2 for µ = 1. This figure ensure that the

model fitting curve quite agree with the experimental outcomes. Fig. 3 is the

plot for the recovered cases between the data fitting and experimental recovered

cases for µ = 1. A good agreement between the fitting curve and experimental

outcomes can be easily noticed from the Fig. 3. The effectiveness and variation

of the degree of approximation r is discussed in the literature [17]. One can easily

understand that the order of convergence of the numerical techniques provided

in this article [17] is increases as we increase the degree of approximation and the

errors in computing the numerical solution is decreases as we increase the degree

of approximation. In the present article we consider the degree approximation

r = 5 for each numerical computation and pictorial presentation.

15

Figure 2: Plot of number of infected case between reported case vs. experimental

result of COVID-19 in Maharashtra.

Figure 3: Plot of number of recovered case between reported case vs. experimental result of COVID-19 in Maharashtra.

The comparison between the variation of fuzzy solution and normal solution

for susceptible cases at different values of fractional order µ = 0.6, 0.8, 1 for the

16

obtained experimental results are shown through the Fig. 4. In Figure 5 the

comparison between the variation for number of exposed cases are depicted for

the different value of the contact rate β at fractional order µ = 0.8 for both fuzzy

solution as well as normal solution. The comparison between the variation of

number of infected cases at fractional order µ = 0.8 for different contact rate

is shown through the Fig. 6 for both the solution. For the different fractional

order µ = 0.6, 0.8, 1 the variation of quarantined cases is also depicted through

the Fig. 7. The pictorial representation of susceptible cases for upper and lower

solutions are depicted through Fig. 8 and Fig. 9 respectively at different values

of fractional order µ = 0.6, 0.8, 1. The variation of exposed cases for upper and

lower solutions are depicted through Fig. 10 and Fig. 11 respectively at different

values of contact rate β = 0.05, 15, 30. The variation of infected cases for upper

and lower solutions are depicted through Fig. 12 and Fig. 13 respectively at

different values of contact rate β = 0.05, 15, 30.

Figure 4: Comparison between numerical fuzzy solution and normal solution

for susceptible cases at different values of fractional order µ.

17

Figure 5: Comparison between numerical fuzzy solution and normal solution

for exposed cases at different values of contact rate β.

Figure 6: Comparison between numerical fuzzy solution and normal solution

for infected cases at different values of contact rate β.

18

Figure 7: Comparison between numerical fuzzy solution and normal solution

for quarantined cases at different values of fractional order µ.

Figure 8: Variation of susceptible cases for upper numerical solution at different

values of fractional order µ.

19

Figure 9: Variation of susceptible cases for lower numerical solution at different

values of fractional order µ.

Figure 10: Variation of exposed cases for upper numerical solution at different

values of contact rate β.

20

Figure 11: Variation of exposed cases for lower numerical solution at different

values of contact rate β.

Figure 12: Variation of infected cases for upper numerical solution at different

values of contact rate β.

21

Figure 13: Variation of infected cases for lower numerical solution at different

values of contact rate β.

9 Results and Discussions

Here we are going to analyze the sensitiveness and effect of the different parameters present in the experimental COVID-19 model on the numerical outcomes

for different variables. From the above plots between the reported cases and

experimental cases for different unknowns variables, we see that our proposed

method is effective and valid. The plot for the susceptible cases for fractional

order cases advances towards the plot for integer order as we move fractional

order system to integer order system which justifies the modeling of the dynamics of concerned experimental model from the Fig. 4. The enhancement

of the exposed cases can be easily calculated for the different contact rate β.

We can notice that the exposed cases increases as we increase the contact rate

this justifies the fact of social distancing to prevent our self from this epidemic

through Fig. 5. Through the Fig. 6 the variation in infected cases can be

visualized for different contact rate. The theory behind the lock-down is well

justified from the variation in infected cases for different contact rate as infected

cases increases as we increase the contact rate. The graph for quarantined cases

is shown through the Fig. 7 for different fractional order system. Fig. 8 and

Fig. 9 are the plots between susceptible cases and time for the case of upper

and lower solutions of concerned fuzzy fractional COVID19 model. Fig. 10 and

Fig. 11 are the plots between exposed cases and time for the case of upper and

lower solutions. Fig. 12 and Fig. 13 are the plots between infected cases and

22

time for the case of upper and lower solutions.

10 Conclusion

This scientific work achieve the many novel consequences. The first one is the

accurate modeling of the dynamics of COVID-19 by fractional order system

of differential equations and by fuzzy fractional order systems for the state

Maharashtra of India. The Laguerre operational matrix is derived to tackle the

concerned fractional mathematical model numerically. Here we shows the effect

of different parameters for the concerned system. The numerical outcomes of

the concerned model has good agreement with the available data for the both

cases of concerned model i.e. in fractional differential environment as well as

in fractional fuzzy environment. The whole calculations and simulations are

done by using the fractional derivatives in Caputo sense. The model fitting

curve and the numerical outcomes have been plotted for the value µ = 1. In

order to explore the concerned model for complete factor of epidemic few more

important parameters will be added in future after a justification with available

data sources.

The authors of this scientific work are optimistic to use the concerned fractional COVID19 model with more data sets for India and other countries in their

future work. They are also interested to deal financial and economic factors in

the concerned model with adaptive fuzzy logic approach. In addition, authors

are planning to use neural network schemes to analyze various predictions with

different factors.

Author Statement

Prashant Pandey: Conceptualization, Methodology, Writing- Original draft

preparation, Software; Yu-Ming Chu: Conceptualization, Methodology; Shaban

Aly: Data curation, Writing original draft preparation; J.F. G´omez-Aguilar:

Conceptualization, Methodology, Writing- Original draft preparation, Software,

Supervision; Hadi Jahanshahi: Conceptualization, Methodology, Software, WritingOriginal draft preparation; Ayman A. Aly: Conceptualization, Methodology;

Shaban Aly: Data curation, Writing original draft preparation.

Availability of data and materials

Not applicable

Competing interests

The authors announce that there are not any competing interests.

23

Authors contributions

All authors have equally contributed to the manuscript, and read and approved

it.

Acknowledgements

The research was supported by the Taif University Researchers Supporting

Project number (TURSP-2020/77), Taif University, Taif, Saudi Arabia.

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