Ci a ion: Das, M.; Saman a, G.; De la
Sen, M. A F ac ional O de Model o
S udy he E ec i eness o
Go e nmen Measu es and Public
Beha iou s in COVID-19 Pandemic.
Ma hema ics 2022,10, 3020. h ps://
doi.o g/10.3390/ma h10163020
Academic Edi o : Se gey A. Lashin
Recei ed: 26 July 2022
Accep ed: 19 Augus 2022
Published: 22 Augus 2022
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2022 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
ma hema ics
A icle
A F ac ional O de Model o S udy he E ec i eness o
Go e nmen Measu es and Public Beha iou s in
COVID-19 Pandemic
Meghad i Das 1, Gu up asad Saman a 1and Manuel De la Sen 2,*
1Depa men o Ma hema ics, Indian Ins i u e o Enginee ing Science and Technology, Shibpu ,
How ah 711103, India
2Ins i u e o Resea ch and De elopmen o P ocesses, Uni e si y o he Basque Coun y, 48940 Leioa, Spain
*Co espondence: [email p o ec ed]
Abs ac :
In his wo k, we emphasise he dynamical s udy o sp eading COVID-19 in Bangladesh.
Conside ing he unce ain y caused by he limi ed co ona i us (COVID-19) in o ma ion, we ha e
aken he modi ied Suscep ible-Asymp oma ic-In ec ious-Hospi alised-Reco e ed (SAIHR) compa -
men al model in a Capu o ac ional o de sys em. We ha e also in oduced public beha iou al
and go e nmen policy dynamics in ou model. The dynamical na u e o he solu ions o he sys-
em is analysed and we ha e also calcula ed he sensi i i y index o di e en pa ame e s. I has
been obse ed ha public beha iou and go e nmen measu es play an impo an ole in con ol-
ling he pandemic si ua ion. The go e nmen measu es (social dis ance, accina ion, hospi alisa-
ion, awa eness p og amme) a e mo e help ul han only public esponses o he e adica ion o he
COVID-19 pandemic.
Keywo ds:
Capu o ac ional di e en ial equa ion; COVID-19; SAIHR compa men al model;
s abili y
;
sensi i i y index
MSC: 9208; 26A33; 37C75
1. In oduc ion
Se e e Acu e Respi a o y Synd ome Co ona i us 2 (SARS-CoV-2) is he in ec ious
agen ha causes Co ona i us Disease 2019 (COVID-19), which was ini ially disco e ed in
China in ea ly Decembe 2019. Since hen, i has sp ead wo ldwide, des oying he heal h,
economy, and li es o billions o people. This has made i clea how impo an i is o
accu a ely ep esen in ec ious illnesses. In eali y, s a is ical s udies a e gene ally based
on nonlinea ma hema ical models, which deal wi h epidemiology, and mos ly de e mine
wo ldwide go e nmen policies.
The COVID-19 pandemic was i s con i med in Bangladesh on 8 Ma ch 2020. The
Heal h and Family Wel a e depa men , Go e nmen o he People’s Republic o Bangladesh,
has con i med a o al o 1,962,213 COVID-19 posi i e cases and 29,135 dea hs om 3 Jan-
ua y 2022 o 23 June 2022 [
1
]. A o al o 274,923,522 accine doses ha e been adminis a ed.
The go e nmen o Bangladesh ook ac ions, such as social dis ancing, mask-wea ing, a -
elling es ic ions, lockdowns, accina ion, and hospi alisa ion, o con ol he COVID-19
si ua ion [2].
F ac ional calculus is a pa allel b anch o calculus ha canno be conside ed a gene -
alised e sion o in ege o de calculus [
3
,
4
]. F ac ional o de sys ems a e mo e app op ia e
han in ege o de sys ems in many ields and can exp ess phenomena ha a e linked o
memo y and a ec ed by he edi a y p ope ies [
5
,
6
]. In endemic and epidemic a eas, peo-
ple’s awa eness o in ec ion will educe he a e o con ac be ween a ious compa men s,
such as be ween humans and mosqui oes in he dengue SIR-SI model [
7
], whe eas in he
Ma hema ics 2022,10, 3020. h ps://doi.o g/10.3390/ma h10163020 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2022,10, 3020 2 o 17
epidemic model wi h accina ion, people who ha e ecei ed accina ions ha e a s onge
endency o be awa e o p e ious epidemics han people who a e suscep ible. In o de
o specialis s o gain he mos knowledge om he a ailable da a be o e making majo
judgmen s, he e should be a sys ema ic way o combine he models and obse a ions.
This o e iew e e s o some obs acles in hese models and looks a some in iguing ap-
p oaches ocusing on he de elopmen o gene al s uc u es o such models, and p oposes
an al e na i e app oach, namely ac ional calculus, whose main con as o in ege o de
models is whe e such e ec s a e o e looked o di icul o in eg a e. The ac ional de i a-
i e o malism o epidemic models p o ides a use ul ool o inco po a ing memo y and
he edi a y ea u es o sys ems. Fu he mo e, he ac ional models ha e one mo e deg ee
o eedom han he in ege o de model o i ing da a. Examining nume ous pape s on
ac ional epidemic models and coun models based on dynamics wi h ac ional o de
de i a i es, we ha e p oposed se e al ma hema ical models on epidemiology and sugges
ha de eloping nume ical ools o i ing ma hema ical models o ac ual da a will assis
conce ned au ho i ies in a oiding o con ol in ec ious disease ou b eaks. Aside om hese
bene i s, he e a e some d awbacks o using he Capu o ac ional sys em.
1. Finding analy ical solu ions is di icul o Capu o di e en ial sys ems.
2. The e a e many concep s, such as bi u ca ion heo y, pa ame ic op imisa ion, pe sis-
ence, e c., ha ha e no ye been de eloped o Capu o ac ional o de sys ems.
3.
The nume ical algo i hms o delayed sys ems and s ochas ic ac ional sys ems ha e
no ye been de eloped.
Conside ing all o he abo e ac s, we ha e cons uc ed ou model in he Capu o di -
e en ial amewo k. In his con ex , he wo ks o Das e al. [
8
] and Das and Saman a
[9–11]
on ac ional o de dynamics may be men ioned. Signi ican con ibu ions ha e ecen ly
been made by se e al esea che s o he a ious COVID-19 models in bo h in ege and
ac ional o de sys ems [12–18].
F ac ional o de modelling is a use ul app oach o s udying he na u e o diseases
because i is an ex ension o he in ege -o de de i a i e. The ac ional o de sys em
also adds an ex a pa ame e ha can be used o imp o e nume ical simula ions. In his
model, we ha e conside ed a new in ec ion unc ion ha includes he s eng h o he
go e nmen ac ion and he s eng h o public esponse. This in ec ion unc ion has no
ye been used in p e ious wo ks o ac ional o de sys ems. We ha e also s udied he
public esponse o he sp eading o COVID-19 disease. Ou main objec i e is o s udy
he e ec o public beha iou and go e nmen al measu es on disease sp eading. We
ha e modi ied he con empo a y SAIHR model by cons uc ing a new in ec ion unc ion
and in oducing a new s a e a iable depic ing social beha iou al dynamics o public
awa eness. The e a e se e al models on COVID-19, bu ou model is uly di e en om
he o he s and may answe new que ies. We ha e aken Capu o ac ional de i a i e
because he p oposed sys em is au onomous in na u e, which has been made physically
meaning ul by dimensional homogenei y among he ac ional o de Capu o de i a i es
and he pa ame e s used in he sys em. The Capu o sys em can be simula ed easily using
Adams de ised echnique (FDE12) o inding app oxima e solu ions o ac ional Capu o
o dina y di e en ial equa ions.
In his wo k, a modi ied SAIHR model is o mula ed emphasising how he go e nmen
measu es and public beha iou con ol he disease sp ead. Sec ion 2con ains he p oposed
model on COVID-19 wi h non-nega i e ini ial condi ions. Sec ion 3desc ibes he dynamical
na u e o solu ions in wo di e en scena ios. Sec ion 4deals wi h he sensi i i y indices o
di e en pa ame e s. Sec ion 5shows nume ical e idence o he dynamical na u e o he
p oposed model suppo ing analy ical esul s. The wo k ends wi h a b ie conclusion.
2. Model Cons uc ion
A i e compa men al model unde a Capu o ac ional o de amewo k has been
cons uc ed, and he model is composed o suscep ible
(S)
, asymp oma ically in ec ed
(Ia)
,
symp oma ically in ec ed
(Is)
, hospi alised
(H)
and eco e ed
(R)
classes. In he con ex
Ma hema ics 2022,10, 3020 3 o 17
o COVID-19, he in ec ed class is di ided in o wo sub-classes, namely asymp oma ic
and symp oma ic, hey a e deno ed by
Ia
and
Is
, espec i ely. Acco ding o some epo s,
eco e y om he disease does no gua an ee pe manen eco e y, so some o he eco e ed
people e e back o he suscep ible class a a cons an a e ζ[19]. The model is gi en by
C
0Dε
S( ) = Λε−Φ−µεS+ζεR,
C
0Dε
Ia( ) = Φ−(µε+σε)Ia,
C
0Dε
Is( ) = σεIa−(µε
1+ρε
1+ρε
2)Is,
C
0Dε
H( ) = ρε
1Is−(µε
2+γε)H,
C
0Dε
R( ) = ρε
2Is+γεH−µεR−ζεR,
C
0Dε
Q( ) = dρε
2Is−λεQ
(1)
whe e we ha e de ined an in ec ion unc ion Φas ollows:
Φ= (1−α)[βε
1SIs(1−Q)κ+βε
2SIa] (2)
In his unc ion,
α
ep esen s he s eng h o go e nmen ac ion, and
κ
ep esen s he
s eng h o public esponse. I is wo h no ing ha
Q
is a new s a e a iable ha ep esen s
he social beha iou al dynamics. The e m
d
ep esen s he s eng h o he public pe cep ion
o isk,
λ−1
is he mean pe iod o public esponse, and he model akes in o accoun he
ac ha public eac ion will inc ease as mo e people become in ec ed and will na u ally
dec ease o e ime.
C
0Dε
deno es he Capu o ac ional de i a i e wi h ini ial ime
0
. He e
all equa ions o sys em (1) a e balanced wi h espec o he ime dimension. Fo he sake o
simplici y, we disca d all he powe s
ε
om he pa ame e s. All pa ame e s con aining
ε
as
powe ace a e aken in o accoun o an impac in nume ical analysis. We ha e disca ded
he powe
ε
o analy ical pu poses only. The desc ip ions o all pa ame e s a e gi en in
Table 1.
Table 1. Desc ip ion o biological in e p e a ion o model pa ame e s.
Pa ame e In e p e a ion Values (Range) Re e ence
Λ ec ui men a e o he human popula ion 0.001 [20]
β1 a e o in ec ion pe uni o ime by he symp oma ic in ec ed Is0.35 (0.005–0.34) [21]
β2 educ ion ac o o in ec ed popula ion by he Iaclass compa ed o Isclass 0.32 (0.005–0.34) [21]
σ a e a which asymp oma ic becomes symp oma ic 0.025 (0.02–0.1) [21]
ρ1 a e a which he symp oma ic in ec ed indi iduals a e hospi alised 0.07 Assumed
ρ2 a e o eco e y o he symp oma ic in ec ed indi iduals 0.14 Assumed
µ1 a e o mo ali y o symp oma ic in ec ed indi iduals 0.05 (0.05–0.1) [20]
ζ a e o e ea om eco e ed class o suscep ible class 0.1 [21]
µ2 a e o mo ali y o hospi alised indi iduals 0.07 [20]
γ a e o ans e o hospi alised indi iduals o eco e ed class 0.05 Assumed
εo de o ac ional de i a i e 0.95 (0–1) Assumed
Ma hema ics 2022,10, 3020 4 o 17
3. Basic Nonlinea Analysis
De ini ion 1
([
4
])
.
The Capu o ac ional de i a i e ope a o o o de
ε
o an absolu ely con inuous
unc ion g ∈Cn([0, ∞+),IR)is de ined as:
C
0Dε
g( ) =
1
Γ(n−ε)Z
0
g(n)(s)
( −s)ε−n+1ds,ε∈(n−1, n),n∈N
dn
d ng( ),ε=n.
whe e Γ(·)is he Gamma unc ion, ≥0, and n is a na u al numbe . In pa icula , o ε∈(0, 1):
C
0Dε
g( ) = 1
Γ(1−ε)Z
0
g0(s)
( −s)εds
Theo em 1 ([4]).Conside :
C
0Dε
x( ) = Ψ(x),
wi h
ε∈(
0, 1
)
,
x∈Rn
. The equilib ium poin s (o his sys em) a e solu ions o he equa ion
Ψ(x) =
0. I o all eigen alues (
λi
) o he Jacobian ma ix
J
,
|a g(λi)|>επ
2
, he equilib ium is
locally asymp o ically s able, whe e J =∂Ψ
∂xis calcula ed a he equilib ium poin .
3.1. Case 1: Model wi hou Con ol
In his case, we can omi he las equa ion o sys em (1), and he in ec ion unc ion is
aken as:
Φ= [β1SIs+β2SIa] (3)
The equilib ium poin s o he sys em (1) a e men ioned below.
1. Disease- ee equilib ium: E0=Λ
µ, 0, 0, 0, 0
2. Endemic equilib ium: E1= (S∗,I∗
a,I∗
s,H∗,R∗).
He e
S∗=(µ1+ρ1+ρ2)(µ+σ)
β1σ+β2(µ1+ρ1+ρ2)
I∗
a=(µ1+ρ1+ρ2)
σI∗
s
I∗
s=
µ(µ+σ)(µ1+ρ1+ρ2)
β2(µ1+ρ1+ρ2) + σβ1
(R0−1)
(µ+σ)(µ1+ρ1+ρ2)
σ−
ζρ2+γρ1
µ2+γ
µ+ζ
H∗=ρ1
µ2+γI∗
s
R∗=ρ2+γρ1
µ2+γ1
µ+ζI∗
s,
(4)
whe e
R0=λ
µ(µ+σ)hβ+σβ1
µ1+ρ1+ρ2i.(5)
Ma hema ics 2022,10, 3020 5 o 17
The necessa y and su icien condi ions o he exis ence o
E1
, in he easible egion
in
R5
, a e as ollows:
(µ+σ)(µ1+ρ1+ρ2)(µ+ζ)>ζσρ2+γρ1
µ2+γ
and
R0>
1, o ,
(µ+σ)(µ1+ρ1+ρ2)(µ+ζ)<ζσρ2+γρ1
µ2+γ
and
R0<
1 . He e,
R0
is he ep oduc ion
numbe o he uncon olled scena io calcula ed a disease- ee equilib ium by he nex -
gene a ion ma ix me hod. The nex -gene a ion ma ix
FV−1
a disease- ee equilib ium
E0is gi en as ollows [22]:
F=
β2Λ
µβ1Λ
µ
0 0
V=
µ+σ0
−σ µ1+ρ1+ρ2
Thus, we ge
R0=λ
µ(µ+σ)hβ+σβ1
µ1+ρ1+ρ2i
Theo em 2.
The disease- ee equilib ium
E0=Λ
µ, 0, 0, 0, 0
is asymp o ically s able i he oo s
(η) o he ollowing equa ion sa is y |a g(η)|>επ
2:
µη2+ηc1+c2=0,
c1= [2µ2+ρ1µ+ρ2µ−β2Λ−σµ],
c2=µ(µ+σ)(µ+ρ1+ρ2)−β2Λ(µ+ρ1+ρ2+σ).
(6)
P oo .
To s udy he local s abili y o disease- ee equilib ium poin
E0=Λ
µ, 0, 0, 0, 0
,
we ha e o compu e Jacobian ma ix Ja E0.
J(E0) =
−µ−β2Λ
µ−β1Λ
µ0ζ
0β2Λ
µ−µ−σ β1Λ
µ0 0
0σ−(µ1+ρ1+ρ2)0 0
0 0 ρ1(µ2+γ)0
0 0 ρ2γ−(µ+γ)
The cha ac e is ic equa ion o J(E0)is
(λ+µ)(λ+µ+ζ)(λ+µ2+γ)Υ(λ) = 0, (7)
whe e Υ(λ) = µλ2+λc1+c2,
c1= [2µ2+ρ1µ+ρ2µ−β2Λ−σµ],
c2=µ(µ+σ)(µ+ρ1+ρ2)−β2Λ(µ+ρ1+ρ2+σ).
(8)
The h ee oo s o he cha ac e is ic Equa ion (6) a e −µ,−(µ+ζ),−(µ2+γ), he disease-
ee equilib ium is s able i
|a g(λi)|>επ
2
, whe e
λi
,
i=
1, 2 a e he oo s o
Υ(λ) =
0.
Ma hema ics 2022,10, 3020 6 o 17
To analyse he local s abili y o endemic equilib ium E∗
1, we need he ollowing:
De ini ion 2
([
23
])
.
The disc iminan
∇( )
o a polynomial
(x) = xn+α1xn−1+α2xn−2+
... +αnis de ined as:
∇( ) = (−1)
n(n−1)
2|Sn( , 0)|,
Sn(
,
g)
is he Syl es e ma ix o
(x)
and
g(x)
o o de
(n+l)×(n+l)
, whe e
g(x) =
xl+β1xl−1+β2xl−2+... +βl.
Fo n=3, we ha e (x) = x3+α1x2+α2x+α3and 0(x) = 3x2+2α1x+α2.
|S3( , 0)|=
1α1α2α30
0 1 α1α2α3
3 2α1α20 0
0 3 2α1α20
0 0 3 2α1α2
=−18α1α2α3−(α1α2)2+4α2
1α3+4α2
2+27α2
3
Hence
∇( ) = −|S3( , 0)|=18α1α2α3+ (α1α2)2−4α2
1α3−4α2
2−27α2
3(9)
Theo em 3
([
24
])
.
I
∇(P)
is he disc iminan o he cha ac e is ic equa ion
P(λ)≡λ3+a1λ2+
a2λ+a3=
0o he Jacobian ma ix o sys em
(
1
)
e alua ed a he endemic equilib ium poin
E1= (S∗,I∗
a,I∗
s,H∗,R∗), whe e
a1=µ1+ρ1+ρ2+−β2S∗+2µ+σ+β1I∗
s+β2I∗
a
a2= (µ1+ρ1+ρ2)(µ+σ−β2S∗)−σβ1S∗+β1β2I∗
sS∗+β2
2I∗
aS∗
+(µ1+ρ1+ρ2−β2S∗+µ+σ)(µ+β1I∗
s+β2I∗
a)
a3= (µ+σ+β1I∗
s+β2I∗
a)[(µ1+ρ1+ρ2)(µ+σ−β2S∗)−σβ1S∗]
+ (β1I∗
s+β2I∗
a)(µ1+ρ1+ρ2+σβ1S∗)
hen he sys em is asymp o ically s able i any o he ollowing condi ions hold:
1.
∇(P)>0, a1>0, a3>0and a1a2>a3
2.
∇(P)<0, a1≥0, α2≥0, a3>0and α<2
3
3.
∇(P)<0, a1>0, a2>0, a1a2=a3and α∈(0, 1).
3.2. Case 2: Model wi h E ec s o Go e nmen al Ac ion and Addi ional Con ol
Le us conside he combined e ec s o go e nmen ac ion along wi h public pe cep-
i i y o isk ega ding se e e and c i ical cases. The a iable
Q
is added o he model
(sys em 1), which ep esen s he public pe cep ion o isk. The alue o
Q
inc eases when
mo e in ec ion occu s and dec eases na u ally. The in ensi y o pe cep ion o isk (
d
) is
connec ed o he in ensi y o popula ion (public) esponse
κ
. The in ec ion unc ion is
men ioned in (2).
Ma hema ics 2022,10, 3020 7 o 17
Theo em 4.
I he in ensi y o public pe cep ion has no impac , he s eady-s a e endemic s a e is
as ollows:
S∗∗ =(µ+σ)(µ1+ρ1+ρ2)
(1−α)[σβ1+β2(µ1+ρ1+ρ2)]
I∗∗
a=(µ1+ρ1+ρ2)
σI∗∗
s
I∗∗
s=
µ(µ+σ)(µ1+ρ1+ρ2)
(1−α)[β2(µ1+ρ1+ρ2) + σβ1](R0−1)
(µ+σ)(µ1+ρ1+ρ2)
σ−
ζρ2+γρ1
µ2+γ
µ+ζ
,
p o ided (µ+σ)(µ1+ρ1+ρ2)(µ+ζ)>ζσ(ρ2+γρ1
µ2+γ),R0>1
o (µ+σ)(µ1+ρ1+ρ2)(µ+ζ)<ζσ(ρ2+γρ1
µ2+γ),R0<1
H∗∗ =ρ1
µ2+γI∗∗
s
R∗∗ =ρ2+γρ1
µ2+γ1
µ+ζI∗∗
s,
Q∗∗ =dρ2I∗∗
s
λ
P oo . The s eady-s a e condi ions lead o
S∗∗(1−α)σβ11−dρ2I∗∗
s
λ+β2(µ1+ρ1+ρ2)= (µ+σ)(µ1+ρ1+ρ2) (10)
(1−α)[β1S∗∗ I∗∗
s(1−Q∗∗)κ+β2S∗∗ I∗∗
a]= (µ+σ)µ1+ρ1+ρ2
σI∗∗
s(11)
I∗∗
a=(µ1+ρ1+ρ2)
σI∗∗
s
H∗∗ =ρ1
µ2+γI∗∗
s
R∗∗ =ρ2+γρ1
µ2+γ1
µ+ζI∗∗
s,
Q∗∗ =dρ2I∗∗
s
λ
(12)
Subs i u ing he alue o
S∗∗
in o Equa ions (10) and (11), we a i e a a anscenden al
equa ion ha would no lead o an explici exp ession o
I∗∗
s
. In his con ex , we limi he
analysis o κ=0. In his si ua ion, we ha e ound he ollowing s eady-s a e:
Ma hema ics 2022,10, 3020 8 o 17
S∗∗ =(µ+σ)(µ1+ρ1+ρ2)
(1−α)[σβ1+β2(µ1+ρ1+ρ2)]
I∗∗
a=(µ1+ρ1+ρ2)
σI∗∗
s
I∗∗
s=
µ(µ+σ)(µ1+ρ1+ρ2)
(1−α)[β2(µ1+ρ1+ρ2) + σβ1](R0−1)
(µ+σ)(µ1+ρ1+ρ2)
σ−
ζρ2+γρ1
µ2+γ
µ+ζ
H∗∗ =ρ1
µ2+γI∗∗
s
R∗∗ =ρ2+γρ1
µ2+γ1
µ+ζI∗∗
s,
Q∗∗ =dρ2I∗∗
s
λ,
(13)
p o ided
(µ+σ)(µ1+ρ1+ρ2)(µ+ζ)>ζσ(ρ2+γρ1
µ2+γ),R0>1
o (µ+σ)(µ1+ρ1+ρ2)(µ+ζ)<ζσ(ρ2+γρ1
µ2+γ),R0<1
4. Sensi i i y Analysis
To examine he sensi i i y o
R0
o any pa ame e (say,
θ
), a no malised o wa d
sensi i i y index wi h espec o each pa ame e has been compu ed as ollows [22,25]:
ΩR0
θ=∂R0
∂θ
θ
R0
In a nume ical (o o he ) model, sensi i i y analysis (SA) is a echnique ha measu es how
he e ec s o unce ain ies in one o mo e inpu a iables can lead o unce ain ies in he
ou pu a iables. The alues o sensi i i y indexes o he pa ame e s
Λ
,
ρ1
,
ρ2
,
σ
,
β1
,
β2
co esponding o Table 1is gi en in Table 2. F om Table 2, i is clea ha
β2
is mo e sensi i e
han β1apa om Λ, and σis mo e sensi i e han ρ1,ρ2.
Table 2. Sensi i i y indices o di e en pa ame e s o sys em (1) co esponding o Table 1.
Pa ame e s Sensi i i y Index
Λ+1
σ−0.5502
ρ1−0.0127
ρ2−0.0141
β1+0.0748
β2+0.9252
Ma hema ics 2022,10, 3020 9 o 17
5. Nume ical Simula ions
In he nume ical s udy, he p edic o -co ec o PECE me hod (men ioned in
Appendix A)
o ac ional di e en ial equa ions has been applied in he MATLAB so wa e pla o m [
26
].
Table 3depic s he scena io o Bangladesh as a esul o COVID-19 om 1 Ma ch o 10
June 2022. We an nume ical simula ions o compa e ou model’s esul s o eal da a
om a ious epo s published by he WHO [
1
] and wo ldome e [
27
]. The alue o he
basic ep oduc ion numbe is 1.2610, acco ding o Table 1. Conside ing he demog a-
phy o Bangladesh [
28
] and p esen co id si ua ion, we ha e assumed
S(
0
) =
6,000,000,
Ia(0) = 600,000. Is(
0
) =
90,000,
H(
0
) =
80,000,
R(
0
) =
80,000,
Q(
0
) =
0.1. Ini ially, we
pe o med a simula ion o sys em (1) wi hou go e nmen measu es (
α=
0 bu
κ6=
0),
depic ed in
Figu es 1and 2
. We ha e aken
κ=
2000 o simula ions. I is ound ha he
symp oma ically in ec ed popula ion will be d as ically inc eased i no ac ion is aken
by he go e nmen (Figu e 1). The e o e, a la ge p opo ion o he popula ion needs o
be hospi alised, and his will c ea e a massac e in he Heal h depa men o Bangladesh
(Figu e 1). I is also obse ed ha he social beha iou a iable
Q
is diminished a ound
20 days om 1 Ma ch 2022 (Figu e 2). Figu es 3and 4depic he model wi h he con ol
(bo h go e nmen measu es and public beha iou ) scena io, which shows ha he model
i s well wi h eal-wo ld scena ios o he pandemic si ua ion in Bangladesh. Figu e 5
po ays he a ia ion o he ime se ies o
S
,
Ia
,
Is
,
H
,
R
,
Q
o di e en alues o he o de
o de i a i es (0.5, 0.6, 0.7, 0.8, 0.9). The cu e o he symp oma ically in ec ed popula ion
i s well wi h eal da a o
ε=
0.9. The ime se ies o di e en alues o go e nmen al
measu es (
α=
0.3, 0.5, 0.7) a e gi en in Figu e 6, and i is e ealed ha he cu e o
Is
is
an app oxima ion o ac i e in ec ed cases o he eal scena io in Bangladesh o
α=
0.7.
The a ia ion in he ime se ies o s a e a iables (wi h
κ
) is gi en in Figu e 7. The in e es -
ing obse a ion in Figu e 7is ha he change in ime se ies o all s a e a iables, including
Is
, is negligible o alues o
κ
anging om 0 o 80,000 bu inc eases o alues g ea e han
80,000. The ime se ies o he case
κ=
0 (Figu e 7) depic s he si ua ion in which only
go e nmen con ol is imposed, and no public beha iou is egula ed. I is also obse ed
ha he numbe o symp oma ic in ec ed indi iduals (
Is
) will inc ease i we educe he
o de o de i a i e close o 0. The alue o Q dec eases slowly i he o de o de i a i e ε
is ixed in a highe ange (Figu e 5).
Table 3. Numbe o ac i e cases be ween 1 Ma ch 2022 and 10 June 2022.
Day Ac i e Cases
1 Ma ch 2022 93,206
11 Ma ch 2022 62,302
21 Ma ch 2022 50,030
31 Ma ch 2022 42,010
10 Ap il 2022 25,650
20 Ap il 2022 31,241
31 Ap il 2022 27,005
10 May 2022 25,665
20 May 2022 21,616
30 May 2022 19,739
10 May 2022 20,011
Ma hema ics 2022,10, 3020 16 o 17
wi h
X( 0) =
S( 0)
Ia( 0)
Is( 0)
H( 0)
R( 0)
Q( 0)
,
in his con ex ,
0=
0. Then, sys em (A1) can be sol ed nume ically by using he ollow-
ing scheme:
G( j) = hε
Γ(ε+2)((j−1)ε+1−(j−ε−1)jε)G(X( 0)) + X( 0)
+hε
Γ(ε+2)
j−1
∑
i=1
((j−i+1)ε+1−2(j−i)ε+1
+(j−i−1)ε+1)G(X( i))
+hε
Γ(ε+2)GX( j−1) + hε
Γ(ε+1)G(X( j−1),
(A2)
wi h j+1= j+h, o j=0, 1.2....., N−1, we ha e used N=10,000, h=0.01.
The abo e scheme is used in he FDE12 unc ion, which is easily a ailable in he
MATLAB File exchange [29].
Re e ences
1.
WHO COVID-19 Dashboa d; Wo ld Heal h O ganiza ion: Gene a, Swi ze land, 2020. A ailable online: h ps://co id19.who.in /
(accessed on 23 June 2022). [C ossRe ]
2.
Islam, K.; Ali, S.; Akanda, S.Z.R.; Rahman, S.; Kam uzzaman, A.H.M. COVID -19 Pandemic and Le el o Responses in Bangladesh.
In . J. Ra e Dis. Diso d. 2020,3, 19.
3. Podlubny, I. F ac ional Di e en ial Equa ions; Academic P ess: San Diego, CA, USA, 1999.
4.
Pe as, I. F ac ional-O de Nonlinea Sys ems: Modeling Aanlysis and Simula ion; Highe Educa ion P ess: Beijing, China, 2011.
[C ossRe ] [PubMed]
5. Du, M.; Wang, Z.; Hu, H. Measu ing memo y wi h he o de o ac ional de i a i e. Sci. Rep. 2013,3, 3431.
6.
Podlubny, I. Geome ic and Physical In e p e a ion o F ac ional In eg a ion and F ac ional Di e en ia ion. F ac . Calc. Appl. Anal.
2002,5, 367–386. [C ossRe ] [PubMed]
7.
Sa da , T.; Rana, S.; Bha acha ya, S.; Al-Khaled, K.; Cha opadhyay, J. A gene ic model o a single s ain mosqui o- ansmi ed
disease wi h memo y on he hos and he ec o . Ma h. Biosci. 2015,263, 18–36. [C ossRe ]
8.
Das, M.; Mai i, A.; Saman a, G.P. S abili y analysis o a p ey-p eda o ac ional o de model inco po a ing p ey e uge. Ecol.
Gene . Genom. 2018,7–8, 33–46. [C ossRe ]
9.
Das, M.; Saman a, G.P. A p ey-p eda o ac ional o de model wi h ea e ec and g oup de ense. In . J. Dyn. Con ol.
2020
,9,
334–349.
10.
Das, M.; Saman a, G.P. A delayed ac ional o de ood chain model wi h ea e ec and p ey e uge. Ma h. Compu . Simul.
2020
,
178, 218–245. [C ossRe ]
11.
Das, M.; Saman a, G.P. Op imal Con ol o F ac ional O de COVID-19 Epidemic Sp eading in Japan and India 2020. Biophys. Re .
Le . 2020,15, 207–236. [C ossRe ]
12.
Das, M.; Saman a, G.P. S abili y analysis o a ac ional o de ed COVID-19 model. Compu . Ma h. Biophys.
2021
,9, 22–45.
[C ossRe ]
13.
Das, M.; Saman a, G.P.; De la Sen, M. A F ac ional O de ed COVID-19 Model Inco po a ing Como bidi y and Vaccina ion.
Ma hema ics 2021,9, 2806. [C ossRe ]
14.
Khan, M.A.; A angana, A. Modeling he dynamics o no el co ona i us (2019-nCo ) wi h ac ional de i a i e. Alex. Eng. J.
2020
,
59, 2379–2389. [C ossRe ] [PubMed]
15.
Liu, Z.; Magal, P.; Seyd, O.; Webb, G. A COVID-19 epidemic model wi h la ency pe iod. In ec . Dis. Model.
2020
,5, 323–337.
[C ossRe ] [PubMed]
16.
Shaikh, A.S.; Shaikh, I.N.; Nisa , K.S. A ma hema ical model o COVID-19 using ac ional de i a i e: Ou b eak in India wi h
dynamics o ansmission and con ol. Ad . Di e . Equ. 2020,373, 1–19. [C ossRe ]
Ma hema ics 2022,10, 3020 17 o 17
17.
Xie, Y.; Wang, Z.; Meng, B.; Xia Huang, X. Dynamical analysis o a ac ional-o de p ey–p eda o model wi h Holling III ype
unc ional esponse and discon inuous ha es . Appl. Ma h. Le . 2020,106, 106342. [C ossRe ]
18.
Wang, X.; Wang, Z.; Xia, J. S abili y and bi u ca ion con ol o a delayed ac ional-o de eco-epidemiological model wi h
incommensu a e o de s. J. F ankl. Ins . 2019,356, 8278–8295. [C ossRe ]
19.
Shen, M.; Peng, Z.; Xiao, Y.; Zhang, L. Modelling he epidemic end o he 2019 no el co ona i us ou b eak in China. bioRxi
2020
.
20.
Wikipedia Con ibu o s. S a is ics o he COVID-19 Pandemic in Bangladesh. In Wikipedia, The F ee Encyclopedia. Re ie ed
19:19. 12 June 2022. A ailable online: h ps://en.wikipedia.o g/wiki/S a is ics_o _ he_COVID-19_pandemic_in_Bangladesh
(accessed on 16 June 2022). [C ossRe ] [PubMed]
21.
Kwuimy, C.A.K.; Naza i, F.; Jiao, X.; Rohani, P.; Na a aj, C. Nonlinea dynamic analysis o an epidemiological model o COVID-19
including public beha io and go e nmen ac ion. Nonlinea Dyn. 2020,101, 1545–1559. [C ossRe ]
22.
Van den D iessche, P.; Wa mough, J. Rep oduc ion numbe s and sub- h eshold endemic equilib ia o compa men al models o
disease ansmission. Ma h. Biosci. 2002,180, 29–48.
23.
Gel , I.M.; Kap ano , M.M.; Zele insky, A.V. Disc iminan s, Resul an s, and Mul idimensional De e minan s; Bi khäuse : Bos on, MA,
USA, 1994; ISBN 978-0-8176-3660-9. [C ossRe ]
24.
Ahmed, E.; El-Sayed, A.M.A.; El-Saka, H. On some Rou h–Hu wi z condi ions o ac ional o de di e en ial equa ions and hei
applica ions in Lo enz, Rössle , Chua and Chen sys ems. Phys. Le . A 2006,358, 1–4. [C ossRe ]
25.
A iola, L.; Hyman, J. Sensi i i y Analysis o Unce ain y Quan i ica ion in Ma hema ical Models. In Ma hema ical and S a is ical
Es ima ion App oaches in Epidemiology; Chowell, G., Hyman, J.M., Be encou , L.M.A., Cas illo-Cha ez, C., Eds.; Sp inge ,
Do d ech , The Ne he land, 2009. [C ossRe ]
26.
Ga appa, R. On linea s abili y o p edic o -co ec o algo i hms o ac ional di e en ial Equa ions. In e na . J. Compu . Ma h.
2010,87, 2281–2290.
27.
COVID-19 Co ona i us Pandemic. 2022. A ailable online: h ps://www.wo ldome e s.in o/co ona i us/coun y/
i aly (accessed on 3 Sep embe 2020).
28.
Wo ld-Popula ion. Bangladesh. A ailable online: h ps://www.wo ldome e s.in o/wo ld-popula ion/bangladesh-popula ion
(accessed on 26 June 2022).
29.
Ga appa, R. P edic o -Co ec o PECE Me hod o F ac ional Di e en ial Equa ions. MATLAB Cen al File Exchange. A ail-
able online: h ps://www.ma hwo ks.com/ma labcen al/ ileexchange/32918-p edic o -co ec o -pece-me hod- o - ac ional-
di e en ial-equa ions (accessed on 16 Augus 2022).
