pandemic COVID-19 spread in Indonesia Stability analysis and numerical simulation of SEIR model for Chaos, Solitons and Fractals

(1)

ContentslistsavailableatScienceDirect

Chaos, Solitons and Fractals

NonlinearScience, andNonequilibrium andComplexPhenomena journalhomepage:www.elsevier.com/locate/chaos

Stability analysis and numerical simulation of SEIR model for pandemic COVID-19 spread in Indonesia

Suwardi Annas

^a

, Muh. Isbar Pratama

^b

, Muh. Rifandi

^b

, Wahidah Sanusi

^b

, Syafruddin Side

^b^,^∗

aDepartment of Statistics, Faculty of Mathematics and Natural Science, UniversitasNegeri Makassar, ParangTambung, South Sulawesi 90244, Indonesia

bDepartment of Mathematics, Faculty of Mathematics and Natural Science, UniversitasNegeri Makassar, ParangTambung, South Sulawesi 90244, Indonesia

a rt i c l e i n f o

Article history:

Received 20 April 2020 Revised 28 June 2020 Accepted 1 July 2020 Available online 3 July 2020 Keywords:

SEIR model COVID-19 Simulation model Basic reproduction number Lyapunov function

a b s t r a c t

TheAimofthisresearchisconstructtheSEIRmodelforCOVID-19,StabilityAnalysisandnumericalsimu- lationoftheSEIRmodelonthespreadofCOVID-19.ThemethodusedtoconstructthemodelistheSEIR modelbyconsideringvaccinationand isolationfactors asmodelparameters,theanalysisofthemodel usesthegeneration matrix methodto obtainthebasic reproductionnumbersand theglobal stability oftheCOVID-19 distributionmodel.Numerical simulationmodels usesecondarydataonthe number ofCOVID-19casesinIndonesia.TheresultsobtainedaretheSEIRmodelforCOVID-19;model analysis yieldsglobalstabilityfromthespreadofCOVID-19;Theresultsoftheanalysisalsoprovideinformation ifnovaccine, IndonesiaisendemicCOVID-19. Thenthesimulationresults provideapredictionpicture ofthenumberofCOVID-19inIndonesiainthefollowingdays,thesimulationresultsalsoshowthatthe vaccinecanaccelerateCOVID-19healingandmaximumisolationcanslowthespreadofCOVID-19.The resultsobtainedcanbeusedasareferenceforearlypreventionofthespreadofCOVID-19inIndonesia

1. Introduction

According tothe WorldHealthOrganization (WHO),COVID-19 is transmitted through people who have been infected with the corona virus. The virus can easily spread through small droplets from the nose or mouth compilation of someone infected with thisvirus tosneeze orcough. The drops then landon objects or surfaces which are touched andthe healthy person adjusts their eyes,noseormouth.Howtospreadthecorona viruscompilation of smalldroplets inhaled by someone compiling switch withthe one supported by corona [1]. It’s importantto spend 1 m more distance than people who are sick. Untilnow there has beenno researchthat statestheCOVID-19coronaviruscanbetransmitted through the air, "explained WHO as quoted from its website on March23,2020[2].

The number of cases of corona virus infection that causes Covid-19 continuestoincrease in variousparts oftheworld. The rateof increase,both forthe numberofcases ofinfection,death andcure,variesineachregion.Eachcountryalsohasitsownpol- icytocurbthespreadofvirusesthatoccurinitsterritory.Accord- ing todata collectedby JohnHopkinsUniversity,asofMarch 23,

∗ Corresponding author

E-mail addresses: [email protected] (S. Annas), [email protected] (Muh. Isbar Pratama), [email protected] (W. Sanusi), [email protected] (S. Side).

2020,the totalnumberofCovid-19casesworldwidehadreached 331,273 cases, with 14,450 deaths, and 97,847 patients declared cured. The highest number of cases is still recorded in China, namely81,397cases,followedbyItaly with59,138cases,andthe UnitedStatesasmanyas33,073cases.In termsof mortality,the largest numberis in Italy, with 5476cases. The number exceeds thedeath ratethat occurredinChina,which is3265.Meanwhile, in terms of recovery, the largest number is in China, which is 72,362 patients. Both the progress of the number of infections, deaths,patientsrecovereduntilcertainpoliciescontinuetobere- portedinvariouscountries[2].

The number of COVID-19 cases in Indonesia continues to increase, until March 22, 2020 the number of positive cases of COVID-19 numbered 514 people with 29 people (5.64%) recov- eredand the numberof deaths48 people (9.34%)or the largest in Southeast Asia. The disease that has become a pandemic and causes a fairly high death in this world has not beenfound the cure[3].TrendofthenumberofcasesandspreadmapofCOVID- 19inIndonesiaispresentedinFigs.1and2.

Mathematical modelingof SIR, SIRS, SEIR andSEIRS on transmission of diseases such as dengue fever, tuberculosis, diabetes, HIV-AIDShas been done by [4–19],then mathematical modeling on thespread of COVID-19 has beencarried out by [20] namely SEIRVmathematicalmodelingintheWuhan,Chinatakingintoac- countenvironmentalfactors, whiletheanalysisandsimulationof themodeluseddataofthenumberofCOVID-19casesinWuhan.

(2)

Fig. 1. Trend of the number of positive cases for COVID-19 in Indonesia on March 2020 Negative Positive.

Source: [2]

Fig. 2. Spread map of COVID-19 in Indonesia on March 2020 Source: [2] .

ThisSEIRVmodeldoesnot considervaccinationandisolationfac- tors as parameters in the model. Therefore, this research build, analyzeusing generationmatrices methodandsimulate theSEIR modelthroughisolationandvaccination onthespreadofCOVID- 19inIndonesiausingMatlabsoftware.

2. Method

The SEIRmathematicalmodelingonthespreadofCOVID-19 is a theoretical study. The method used to construct the model is theSEIRmodel [12]by consideringvaccination andisolationfac- torsasmodelparameters,themodelanalysisusesthegeneration matrixmethod [9] to obtain the basicreproduction number and theglobalstabilityforCOVID-19 spreading. Numericalsimulation ofmodelusesecondarydataonthenumberofCOVID-19casesin Indonesia[21]byusingMatlabsoftwaretopredictthenumberof COVID-19casesinIndonesiaasameasuretoprevent thenumber ofCOVID-19casesinIndonesia.

3. Resultsanddiscussion

3.1.SEIRmodelformulationforCOVID-19

TheSEIRmodelonthespreadofCOVID-19isdividedintofour compartmentsnamelySuspected(S),Exposed(E),Infected(I),and

Fig. 3. Scheme of the Model SEIR transmission for COVID-19.

Recovered(R).Individualsinaninfectedclasscancauseotherindi- vidualstobecomeinfected.Thechangesthatoccurineachhuman populationinCovid-19transmissionfortheSEIRmodelcanbein- terpretedbyFig.3

DeﬁnitionofvariablesandparametersofmodelSEIRforCOVID- 19presentedinTable1.

BasedonthepopulationschemeinFig.3,therateofchangein thenumberofpeopleSuspected,Exposed,InfectedandRecovered overtimeintheSEIRmathematicalmodelofthespreadofCovid- 19canbeinterpretedasfollows:

dS

dt =

μ

^N−

( α

^I+

μ

+

v )

S (1)

(3)

Table 1

Deﬁntion of variable/parameter.

Variable/Parameter Deﬁnition

N Number of human population

S Number of Suspected population

E Number of Exposed population

I Number of Infected Population R Number of Recovered Population μ The rate of birth/death population α Probably of changing from S to E β Probably of changing from E to I μi The rate of death population by COVID-19 δ Probably of changing from I to R v Vaccine of Suspected Population

dE

dt =

α

^IS⁻

( β

⁺

μ )

^E ⁽²⁾

dI

dt =

β

^E−

( μ

i+

δ

+

μ )

^I ⁽³⁾

dR

dt =

δ

^I+

v

S−

μ

^R ⁽⁴⁾

Let S= N^S; E= ÊN; I=NÎ; and R= ^RN, simplified model be- comes:

dS

dt =

μ

⁻

( α

^I⁺

μ

⁺

v )

S (5)

dE

dt =

α

^IS⁻

( β

⁺

μ )

^E ⁽⁶⁾

dI

dt =

β

^E⁻

( μ

i+

δ

⁺

μ )

^I ⁽⁷⁾

dR

dt =

δ

^I+

v

S−

μ

^R ⁽⁸⁾

3.2. AnalysisofSEIRmodelforCOVID-19

3.2.1. Equilbriumanalysis

Based onEqs.(5)–(8),stabilityanalysisiscarriedouttodeter- mine thedisease freeequilibriumpoint andendemicequilibrium point. Todeterminethe two equilibriumpoints, eachequation in Eqs.(5)–(8),mustbeequaltozero,or ^dS_dt =0, ^dE_dt =0, ^dI_dt =0,and

dR

dt =0,thusobtained:

μ

−

( α

^I+

μ

+

v )

S=0 (9)

α

^IS⁻

( β

⁺

μ )

^E⁼⁰ ⁽¹⁰⁾

β

^E⁻

( μ

i+

δ

⁺

μ )

^I⁼⁰ ⁽¹¹⁾

δ

^I+

v

S−

μ

^R=0 (12)

Then,wefoundtheequilibriumpointofS,E,I,andR.

3.2.2. Free-diseaseequilibriumforCOVID-19

Equilibrium pointsfordisease-free areconditions wherethere isnospreadofCOVID-19then,E=I=0

FromEq.(9):

μ

−(

α

^I+

μ

+

v

₎S=0thusobtained S=

μ

( μ

+

v )

FromEq.(12):

δ

^I+

v

S−

μ

^R=0thusobtained:

R=

v ( μ

⁺

v )

Then,the Equilibriumpointsof disease-freeforCOVID-19 are:

K0=

(

^S,^E,^I,^R

)

⁼

μ

( μ

+

v )

^,⁰^,⁰^,

v ( μ

+

v )

(13)

3.2.3. EndemicequilibriumforCOVID-19

Endemicequilibrium points are used to indicate the possibil- ityofdisease spread.Because in endemic conditionsand disease spread,thepopulationS=0,E=0,I=0,andR=0.FromEqs.(9)– (12)obtainedendemicequlibriumpointsfromCOVID-19are:

S=

( μ

i+

δ

+

μ ) ( β

+

μ ) αβ

E=

αβμ

−

( μ

i+

δ

+

μ ) ( μ

+

v ) αβ

I=

αβμ

⁻

( μ

i+

δ

⁺

μ ) ( μ

⁺

v ) ( β

⁺

μ ) α ( μ

i+

δ

⁺

μ ) ( β

⁺

μ )

R=δαβ²μ−β(μi+δ+μ)(μ+v)(β+μ)−v((μi+δ+μ)(β+μ))² βα²(μi+δ+μ)(β+μ)

Then,theEquilibriumpointsofendemicforCOVID-19are:

Ke=

(

^S,E,I,R

)

=

⎛

⎜ ⎝

(μi+δ+μ)(β+μ)

αβ ,^αβμ⁻⁽^μⁱ⁺_αβ^δ⁺^μ)⁽^μ⁺^v⁾, αβμ−(μi+δ+μ)(μ+v)(β+μ)

α(μi+δ+μ)(β+μ) ,

δαβ²μ−β(μi+δ+μ)(μ+v)(β+μ)−v((μi+δ+μ)(β+μ))² βα²(μi+δ+μ)(β+μ)

⎞

⎟ ⎠

(14) BaseontheEqs.(5)–(8),foundtheJacobianmatrices

J=

−

( α

^I+

μ

+

v )

0

α

^S ⁰

α

^I ⁻

( μ

⁺

β ) α

^S ⁰

0

β

⁻

( μ

ⁱ⁺

δ

⁺

μ )

⁰

v

0

δ

−

μ

(15)

ThenﬁndtheeigenvalueoftheJacobianmatrixinEq.(15)

det (λI −J )=

λ+ (αI + μ+ v) ⁰ ⁻αS 0

−αI λ+ (μ+ β) −αS 0

0 −β λ+ (μi+ δ+ μ) ⁰

−v 0 −δ λ+ μ

= 0

SubtitutionthevalueS= ₍_μ^μ₊_v₎ andI=0,obtained:

λ

+

( μ

+

v )

0 −

αμ ( μ

+

v )

⁻¹ 0 0

λ

⁺

( μ

⁺

β )

⁻

αμ ( μ

⁺

v )

⁻¹ 0 0 −

β λ

+

( μ

i+

δ

+

μ )

⁰

−

v

0 −

δ λ

+

μ

=0

(λ+μ)(λ+(μ+v))

(λ+(μ+β))(λ+(μi+δ+μ))− αβμ (μ+v)

=0

( λ

+A

) ( λ

+B

) ( ( λ

+C

) ( λ

+D

)

−E

)

=0 (16) with

A=μ;B=(μ+v);C=(μ+β);D=(μi+δ+μ)^and^E= αβμ (μ+v) IfEq.(16)isresolved,then:

λ

⁴+

(

^A+B+C+D

) λ

³+

(

^AB+

(

^A+B

) (

^C+D

)

+CD−E

) λ

²

+

( (

^A⁺^B

) (

^CD⁻^E

)

⁺^AB

(

^C⁺^D

) ) λ

+ABCD−ABE=0 (17) Basedon Descartes’rule [14]the numberofnegative rootsof thecharacteristicEq.(17)is equaltothe numberofvariationsin

(4)

thechangeinthecoeﬃcientsign,soEq.(17)hasfournegativeval- ues ifitisintheformof:

K=L1

λ

⁴⁻^L²

λ

³⁺^L³

λ

²⁻^L⁴

λ

⁺^L⁵

with L1 =1

L2 =A+B+C+D

L3 =AB+

(

^A⁺^B

) (

^C⁺^D

)

⁺^CD⁻^E L4 =

(

^A⁺^B

) (

^CD⁻^E

)

⁺^AB

(

^C⁺^D

)

L5 =ABCD−ABE

WithconditionL₁,L₂, L₃, L₄, L₅ >0

ThisconcludesthatifCD>EandL₁,L₂, L₃, L₄, L₅>0,then:

λ

1,

λ

2,

λ

3,

λ

4<0

Because thecharacteristic values of theequation system in theCOVID-19 model arenegative, the equilibriumpoint is stable globalasymptotic.

3.2.4. Thebasicreproductionnumber(R₀)ofSEIRmodelfor COVID-19

The basic reproduction number (R₀)is determined using the matricesgeneration method,Based on Eqs.(5)–(8), todetermine (R₀).

LetF=

0

α

^S

β

⁰

andV=

( β

+

μ )

⁰ 0

μ

i+

δ

⁺

μ

Then,wefound:

V⁻¹= 1

( β

⁺

μ ) ( μ

i+

δ

⁺

μ )

μ

i+

δ

+

μ

⁰

0

( β

⁺

μ )

V⁻¹=

₁

(β+μ) 0

0 _μ ¹

i+δ+μ

andthen, P=FV⁻¹=

0

α

^S

β

⁰

1

(β+μ) 0

0 _μ ¹

i+δ+μ

P=

₀ _αS (μi+δ+μ) (ββ+μ) 0

Then,bythematricesgenerationmethod[9],wefound:

R₀=

αβμ

( μ

+

β ) ( μ

+

v ) ( μ

i+

δ

+

μ )

⁽¹⁸⁾

3.2.5. StabilityanalysisofSEIRmodelforCOVID-19 Theorem1

1 IfR₀ ≤ 1 then system in Eqs. (5)–(8) of the model is global asymptoticstable.

2 If R₀ > 1, then system in Eqs. (5)–(8) is unstable in D=

{

(^S,E,I,R)∈R⁴₊, S+E+I+R≤S₀

}

^.

Proof

LetX=(Ê,I)^T^,ît^can^be^stated^that ^dX_dt ≤(^F−V)^X,^with^Fând VarematricesonsearchR₀.then,letuisaeigenvectorofF.

Letthat:

R₀=

ρ

FV⁻¹

=

ρ

V⁻¹F

IfV⁻¹F=R₀, thenuV⁻¹F=R₀u LetLyapunovfunction:

L0=uV⁻¹X

dL0

dt =uV⁻¹dX dt

dL0

dt =uV⁻¹

(

^F⁻^V

)

^X

dL0

dt =

uV⁻¹F−u

X

dL0

dt =u

(

^R⁰⁻¹

)

^X

IfR₀≤1,then^d_dt^L⁰ =0,sothatuX=0,asaresultE=I=0with u>0

IfR₀ <1thentheEquation ^dE_dt +_dt^dI=0andwefound:

α

^IS⁻

( β

⁺

μ )

^E⁺

β

^E⁻

( μ

ⁱ⁺

δ

⁺

μ )

^I⁼⁰

α

^IS⁻

( μ

ⁱ⁺

δ

⁺

μ )

^I⁼⁰^with^E⁼⁰ R0

( μ

⁺

β ) ( μ

⁺

v ) ( μ

i+

δ

⁺

μ ) βμ

IS−

( μ

i+

δ

⁺

μ )

^I⁼⁰

( μ

i+

δ

+

μ )

R0

( μ

+

β ) ( μ

+

v ) βμ

^S⁻¹

=0

( μ

ⁱ⁺

δ

⁺

μ )

R0

( μ

⁺

β ) β

⁻¹

=0

Itmeans,(

μ

i+

δ

+

μ

)^[^R0−1]≈0

Thenitcanbeseenthat,ifR₀<1,then ^d_dt^L⁰ <0 Thisprovesthatthesystemisglobalasymptoticstable.

Incontrast,ifR₀<1,thenitfollowsfromthecontinuityofthe vector ﬁelds that ^d_dt^L⁰ >0 ina neighborhood of the systemin D. Thus the systemin Eqs.(5)–(8)is unstableby the Lyapunovsta- bilitytheory. The last part ofthe theorem can be proved by the persistenttheory[22]whichissimilartotheproofofTheorem2.5 in[23].

3.3. NumericalsimulationoftheSEIRmodelofCOVID-19in Indonesia

Modelsimulationsare performedusingMathematicasoftware.

The initial values S(0), E(0), I(0), R(0) and parameter values of the modelsused inthis simulationare presented inTables 2–4, withthebasicreproductionnumbervaluesR₀ obtainedbasedon Eq.(18),isR₀= ₍_μ₊_β)(_μ^αβμ₊_v₎₍_μ

i+δ+μ)

Table 2

Initial Value of SEIR Model for COVID-19 in Indonesia.

Variable Estimated value Source N (0) 269.6 juta [24]

S (0) 37,538 [21]

E 13,923 [21]

I 23,191 [21]

R 13,213 [21]

(5)

Fig. 4. Stability phase portrait of the SEIR Model for COVID-19 in Indonesia .

Table 3

Parameter value of SEIR Model for COVID-19 in In- donesia.

Parameter Estimated value Source

μ ^6.25^×10⁻³ ^[25]

α 0.62 ×10 ⁻⁸/person/day [11]

δ 0.0006667 per day [11]

μi 7.344 ×10 ⁻⁷ [21]

Table 4

Assumption parameter values of the SEIR Model for COVID-19 in Indonesia.

Parameter Simulation 1 Simulation 2 Simulation 3

v 1% 50% 100%

β 3 days 7 days 14 days

3.3.1. NumericalsimulationresultsoftheSEIRmodelofCOVID-19in Indonesia

The value of the equilibriumpoints of the SEIR model is de- terminedbysubstitutingtheparametervalues (simulation1)in Tables 3and4in Eqs.(5)–(8)whichare equatedwithzero,then thefollowingEqs.(5)–(8)systemisobtained:

⎧ ⎪

⎨

⎪ ⎩

548.60625−

7.0634x10⁻⁴I

(

^t

)

+0.5063

S=0

7.0634x10⁻⁴I

(

^t

)

⁺⁰^.⁵⁰⁶³

S−0.07777E=0 0.0714E−0.00692I=0

0.00692I+0.5S−0.00625R=0

(19)

Theequationsystem(19)providestheequilibriumpointsofthe endemicSEIRmodelofthespreadofCOVID-19,namely:

(

^S^∗, E^∗, I^∗,R^∗

)

=

(

⁰.00982,0.01793, 0,01,811,0.89008

)

Theseequilibriumpointsexplainthat thenumberofCOVID-19 suspectedpopulationswas982people,exposedto1793people,in- fected1811 peopleandthoserecovered were89,008people from thetotal100,000humanpopulation.

The eigenvalues based on Eq. (17)with the parameter values inTables2and3fortheCovid-19transmissionSEIRmodelare:

λ

¹⁼⁻⁰^.¹³⁸⁷⁵^,

λ

²⁼⁻⁰^.⁰⁰⁶²⁵^,

λ

³⁼⁻⁰^.⁰⁷²⁹²^,

λ

⁴⁼⁻⁰^.³³⁶²⁵

In thesame way,the equilibriumvalues andeigenvalue for simulation 2and simulation3 are writtenin Table5. The eigenvalues obtainedare realand negative, basedon [17], thetype of stabilityatthisequilibriumpointisasymptoticstable.Thestabil- ityphaseofthesystemcan beillustratedinFig.4andtheﬁtting dataoftheSEIRmodelusingRungeKuttamethodversusthereal dataforcovid19inIndonesiacanbeillustratedinFig.5.

20000 22000 24000 26000 28000 30000

I I real

Fig. 5. Fitting infected data of SEIR model for covid19 versus real data in Indonesia.

AccordingtoFig.5,TheﬁttingdataofSEIRModelandthereal dataofcovid19inIndonesiaisasimilar,thisshownthattheSEIR modelonthespreadofcovid19canbeusedtopredictthenumber cases ofcovid 19 in Indonesia, so that the government can take strategicstepstopreventthispandemic.

The basic reproduction number R0 for the endemic case of Covid-19 from Eq. (18) with only 1% vaccination is R₀= 3.2094.

Thismeansthat,ifapersonisinfectedwithCovid-19itwillinfect 3otherpeople.WhereastheR₀valueforsimulation2andsimula- tion3presentedinTable 5explainsthat50% vaccinewillreduce transmission ofCOVID-19 and100% doesnot couse spreading of Covid-19inIndonesia.

3.3.1.1.Simulation 1 of the SEIR model for COVID-19 in Indonesia.

Numericalsimulationtodeterminetheeffectofvariationsinpop- ulationisolationtime onthedynamicsofthenumberofExposed andInfectedCovid-19.The simulationusesthe initialvalues in Table2andtheparameters inTable 3withvaccination

ν

= 50%.

ThenumericalsimulationresultsoftheCovid-19spreadingmodel arepresentedinFigs.6and7

Based on Fig.6, the numberof Exposed populations immedi- atelydecreasesifthey areisolated for3days,butthenumberof COVID-19infectedpopulationsinIndonesiahasincreaseddramat- icallyandonlytakesthreedaystoreachapeakof34,000people.

Then,ifisolationofSuspectedCOVID-19iscarriedoutfor7days, thenumberofExposedpopulationsdecreasesafterthefourthday withthe population infectedwithCOVID-19 in Indonesiaexperi- encinga notvery highincrease andreachinga peakonthe sixth day withthe number sufferers of COVID-19 around 32,000 people.Furthermore,ifisolationofSuspectedCOVID-19iscarriedout for 14 days, the number of Exposed populations decreases after theseventhdaywiththepopulationinfectedwithCOVID-19inIn- donesiaexperiencingasmallerincreaseandreachingapeakonthe

(6)

Table 5

The Values of equlibrium point, eigen value and R 0.

Simulation λ1 λ2 λ3 λ4 S ^∗ E ^∗ I ^∗ R ^∗ R 0

1 −0 . 13875 −0 . 00625 −0 . 07292 −0 . 33625 0.00982 0.01793 0,01,811 0.89008 3.20949 2 −0 . 07292 −0 . 00625 −0 . 14911 −0 . 50387 0.00555 0.02307 0.04521 0.92616 0.10057 3 −0 . 07292 −0 . 00625 −0 . 07768 −0 . 98387 0.00389 0.03106 0.03042 0.93463 0.04955

Fig. 6. Variation in the number of population Exposed (E) for different values of β.

Fig. 7. Variation in the number of population Infected (I) for different values of β.

eighthdaywiththe numberofpatientsCOVID-19 around29,000 people.This showsthat isolationis very inﬂuentialin increasing thenumberof populationinfectedwithCOVID-19, soisolation is verynecessary to do withSuspected COVID-19 population in In- donesia.

3.3.1.2.Simulation 2 of the SEIR model for COVID-19 in Indonesia.

Numericalsimulationtodeterminetheeffectofvariationsinpop- ulationisolationtimeonthedynamicsofthenumberofSuspected andRecoveredCovid-19inIndonesia.Thesimulationusestheini- tialvalues in Table 2 and the parameters in Table 3 with the value

β

= 14days.ThenumericalsimulationresultsoftheCovid- 19spreadingmodelarepresentedinFigs.8and9.

Based on Fig. 8, by giving 3% vaccine, the number of Sus- pectedpopulationsrequiresalongtimetodecrease,aswellasthe numberofpopulations recoveredfromCOVID-19inIndonesia,the numberisverysmall andtheduration isquitelong asshownin Fig.9.Then, ifthevaccine is50%,then thenumber ofSuspected populationsrequiresashortamountoftimetodecrease,withthe

Fig. 8. Variation in the number of population Suspected (S) for different values of ν.

Fig. 9. Variation in the number of population Recovered (R) for different values of ν^.

numberofpeoplerecoveredofCOVID-19inIndonesiabeinglarge, around 60,000people withashort durationoftime asshownin Fig.8.Furthermore,ifbygivingavaccineof100%,thenthenum- ber ofSuspected populations decreases in a very shorttime and thenumberof peoplerecovering ofCOVID-19in Indonesiais be- comingmorethat around6200peoplewithaveryshortduration oftime asinFig.8.Thisshowsthat thegivingofvaccinesinthe Suspected group is very inﬂuential in increasing the total popu- lationof RecoveredCOVID-19 isso necessaryto provide vaccines conductedonSuspendedCOVID-19populationsinIndonesia.

(7)

3.4. Discussion

TheSIRandSEIRmodels[8,10,17]developedatuberculosis(TB) transmission model, conductedanalyzes andpredicted the number of TB cases in South Sulawesi. Then [18,19] built a Dengue Fever (DHF) transmission model, conductedan analysisand pre- dicted the numberofDengue Fevercases inSouthSulawesi. The COVID-19 spreading model conducted by [20] is a SEIRV model in the Wuhan, China by considering environmental factors, anal- ysisandsimulationmodelsusingdataonthenumberofCOVID-19 cases inWuhan. ThisSEIRV modeldoes not considervaccination andisolationfactorsasparametersinthemodel,whiletheresults ofthisstudybuild the COVID-19spreading modelby considering vaccinationandisolationperiodsintheCOVID-19populationinIn- donesia.Analysis ofthemodelalsousesthemethod[9]toobtain thebasicreproductionnumbers.Incontrastto[17,18,19]whoused Maple topredict thenumber ofTBandDHFcases,theresults of thisstudyuseMathematicatopredictthenumberofpatientswith COVID-19 in Indonesia in the form of numerical simulation. Re- sultsofthe simulationoftheSEIR modelforCOVID-19spreading providedinformationandpredictionsonthenumberofCOVID-19 cases in Indonesia. The results of this simulation is very helpful inimprovingtheabilityofthestrategyinguardingthenumberof COVID-19casesinIndonesia.

4. Conclusion

Based on the results of the study, it wasconcluded that the SEIRmodelcouldbeareferencemodelforthespreadofCOVID-19 inIndonesia.Analysisofthemodelprovidesanoverviewofglobal stabilityinthespreadofCOVID-19andalsoprovidesinformationif IndonesiaisinCOVID-19 endemicstatus.Then thesimulationre- sultsprovideapredictivepictureofthenumberofCOVID-19cases inIndonesiaandalsoshowthatvaccines canaccelerateCOVID-19 healingandtheisolationperiodcanslowthespreadofCOVID-19 in Indonesia.The results obtainedcanbe used asa referencefor earlypreventionofthespreadofCOVID-19inIndonesia.

DeclarationofCompetingInterest

Allauthorsdeclarenoconﬂictsofinterestinthispaper.

CRediTauthorshipcontributionstatement

Suwardi Annas: Conceptualization, Methodology. Muh. Isbar Pratama:Software,Writing-review&editing.Muh.Rifandi:Data curation. Wahidah Sanusi: Supervision, Validation. Syafruddin Side:Conceptualization,Writing-originaldraft.

Acknowledgements

We wouldlike thankto UNM2020 forﬁnancial support,also thankstoDiktiforsupportthisresearch.

References

[1] Anonim, 2020. Ungkap cara penyebaran virus Cororna di Dunia.

https://www.cnbcindonesia.com/tech/20200323104158- 37- 146860/simak- nih- who- ungkap- cara- penyebaran- virus- corona- di- dunia. , [Accessed 24th March, 2020].

[2] Briantika A. Positif COVID-19 di Indonesia Jadi 514 Kasus. Mening- gal 2020;48. https://tirto.id/positif- covid- 19- di- indonesia- jadi- 514- kasus- 48- meninggal-eGUa [Accessed, 23rd March, 2020] .

[3] M.F. Vina2020. 331.273 Orang Recovered, 97.847 Orang Sembuh", https:

//www.kompas.com/tren/read/2020/03/23/072649465/update- virus- corona- di- dunia- 331273- orang- Recovered- 97847- orang- sembuh?page=2 . [Accessed, 23rd, March 2020].

[4] Egonmwan1 AO , Okuonghae1 D . Analysis of a mathematical model for tuberculosis with diagnosis. J Appl Math Comput 2018;59:129–62 .

[5] Abdallah S Waziri , Estomih S Massawe , Oluwole Daniel Makinde . Mathemati- cal modelling of HIV/AIDS dynamics with treatment and vertical transmission.

Appl Math 2012;2(3):77–89 .

[6] T. Ashley, S. Jacqueline S. John. 2010. Modeling the spread of tuberculosis in a closed population. http://educ.jmu.edu/strawbem/math _ 201/ﬁnal _ reports/

Scotti _ Takahashi _ Spreadbury _ Final.pdf [Diakses, 20 April 2018].

[7] Tang B , Wang X , Li Q , Bragazzi NL , Tang S , Xiao Y , et al. Estimation of the transmission risk of 2019-nCoV and its implication for public health interventions.

J Clin Med 2020;9:462 .

[8] Syafruddin S , Mulbar U , Sidjara S , Sanusi W . A SEIR Model for transmission of tuberculosis. AIP conference proceedings, 1830; 2017 .

[9] Diekmann O , Heesterbeek JAP , Roberts MG . The construction of next-gen- eration matrices for compartmental epidemic models. J R Soc Interface Vol 2010;7:873–85 .

[10] Dontwi IK , Obeng WD , Andam EA , Obiri LA . A mathematical model to predict the prevalence and transmission dynamics of tuberculosis in amansie west dis- trict, Ghana. Br J Math Comp Sci 2014;4(3):402–4025 .

[11] Spencer JA , Shutt DP , Moser SK , Clegg H , Wearing HJ , Mukundan H . et al., Epi- demiological parameter review and comparative dynamics of inﬂuenza, respi- ratory syncytial virus,rhinovirus, human coronvirus, and adenovirus. medRxiv 2020 .

[12] Rusliza A , Budin H . Stability analysis of mutualism population model with time delay. Int J Math Comput Phys Electr Comput Eng 2012;6(2):151–5 .

[13] Ofosuhene O . Modelling the spread of HIV and AIDS epidemic trends in male and female populations. World J Model Simul 2017;13(3):183–92 (2017) . [14] Elif D , Arzu U , Nuri O . A fractional order SEIR model with density dependent

death rate. Hacet J Math Stat 2011;40(2):287–95 .

[15] Rangkuti YM , Side S , Noorani MSM . Numerical analytic solution of SIR model of dengue fever disease in south Sulawesi using homotopy perturbation method and variational iteration method. ITBC J Sci (J Math Fundam Sci) 2014;46A(1):91–105 .

[16] Syafruddin Side , Irwan T , Usman Mulbar , Wahidah Sanusi . SEIR model simulation for Hepatitis B, 1885. AIP Conference Proceedings; 2017. p. 20185 . [17] Syafruddin S . A susceptible-infected-recovered model and simulation for trans-

mission of tuberculosis. Adv Sci Lett Vol 2015;21(2):137–9 .

[18] Syafruddin S , Noorani MSM . Lyapunov function of SIR and SEIR model for transmission of dengue fever disease. Int J Simul Process Model (IJSPM) 2013;8:177–84 (2/3) .

[19] Syafruddin S , Noorani MSM . A SIR model for spread of dengue fever disease (simulation for south sulawesi Indonesia and selangor Malaysia). World Journal Modeling Simulation 2013;9(2):96–105 .

[20] Yang C , Wang J . A mathematical model for the novel coronavirus epidemic in Wuhan, China. Math Biosci Eng 2020;17(3):2708–24 .

[21] Anonim, 2020. Situasi Kasus Indonesia. https://covid19.kemkes.go.id/ . [diakses 5 April 2020]

[22] Thieme HR . Persistence under relaxed point-dissipativity (with application to an endemic model). SIAM J Math Anal 1993;24:407–35 .

[23] Gao D , Ruan S . An SIS patch model with variable transmission coeﬃcients.

Math Biosci 2011;232:110–15 .

[24] Anonim, 2020. Proyeksi Jumlah Penduduk Indonesia 2020. https:

//databoks.katadata.co.id/datapublish/2020/01/02/inilah- proyeksi- jumlah- penduduk- indonesia- 2020 . [diakses 29 maret 2020]

[25] Anonim, 2020. Daftar Negara Menurut Pertambahan Jumlah Penduduk. https://

id.wikipedia.org/wiki/Daftar _ negara _ menurut _ pertambahan _ jumlah _ penduduk [diakses 29 maret 2020]