View of Modeling Dynamics of Spread of Prostitution due to Poverty in the Society

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Annals of R.S.C.B., ISSN:1583-6258, Vol. 25, Issue 6, 2021, Pages. 6916 - 6924 Received 25 April 2021; Accepted 08 May 2021.

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Modeling Dynamics of Spread of Prostitution due to Poverty in the Society

G.Divya

^1*

S.,Athithan

²

,

DepartmentofMathematics,FacultyofEngineeringandTechnology SRMInstitute ofScienceandTechnology, Kattankulathur-

603203,KanchipuramDistrict.

E-mail: 1*

[email protected],2

[email protected] ABSTRACT

Womenprostitution isbeingpracticedsincefromancienttimes.Povertyis oneofthemajormotivatorforwomentobecomeprostitutes.Itisconsidered

a sinfectiousdiseasetothesociety.Hereadeterministicmodelhasformulatedand

itexhibitstwoequilibriumpointsnamelyprostitutionfreeequilibriumpointand prostitution presentequilibriumpointandalsoanalyzedtheirstability locally. The basic reproduction number R0has computed. Further, this deterministic modelhasextendedtostochasticdifferentialequation. Finally wecomparedboththedeterministicandstochasticmodelsusingnumericalsimulation

.

Keywords

:

Poverty, P r o s t i t u t i o n ,Stability a n a l y s i s ,stochastic differential equation

1 Introduction

Prostitution,sometimesreferredtoastheworld’soldestprofession,hasbeen practiced sinceprehistoric era.Thetermprostitutioniscommonlyusedtorefertothetrade of sexualservicesformonetaryorin- kindremuneration,andhencetoatypeof social interactionthatisbothsexualandeconomic [4].Inessentialaspects,sextrafficking andprostitutionoverlap.Bothtargetedforcommercialsexualexploitationhavesome criticaldemographictraits,includingpoverty,young,minoritystatusinthecountryof

exploitation,ahistoryofabuse,andlimitedfamilysupport.Bothpreyonvulnerablewomenandgirlsasaresultofwomen andgirlsasaresultofpoverty,discrimination,andabuse,leavingthemtraumatized,unwell,andpoor.Bothsexualandmone taryrewardsaregiventopredators, henceincreasingdemandandcriminalactivitiesthatassuresupply [10].

Ifamarriedcouplearebothimpoverished,hecansellhiswife’sbodyinthebazaar.

Thisdoesnotfunctionintheoppositedirection.Maledominancecreatesprostitution,

andinequalityexposesspecificgroupsofwomentosexualexploitation [8].Womenin thesextradearerapedandmurderedatthegreatestratesof anygroupof womenon theglobe [6].AssexualbehaviorisanimportantdeterminantintransmittingHIVand

sexuallytransmitteddiseases(STDs),sexworkers(SWs),transgenderandclientsare oftenlabeled asa”highriskgroup”inthecontextofHIVandSTDs[7].

TheCOVID-19hasimpacted

onalltypeofpeoplegloballybysystemicpoverty.Duringthiscrisisthesexworkersalsofacedmanyhindrances.Preventionfro mCOVID-19 andprotectionofsexworkershavediscussedin[13][3].Mostimportantly,sexworkers needbetterskilltraining,opportunitiestoleadtheirlifepeacefullytoovercome poverty aswell asprostitution.

Consideringthisasareallife problemweconstructedamathematicalmodeltoreduce thespreadofprostitutionduetopoverty.Thereareexistingmathematical modelson several socialissuessuchas[2],[11].Also in [1] the authors discussed prostitution caused by poverty in Nigeria.Inthisarticleweframedamathematical

modelwiththepopulationofpovertywomen,populationofprostitutesandrehabilitationwomenpopulation.

Thisarticlehasorganized a s follows: InSection2weconstructeda deterministic modeland analyzedits equilibrium points inSection3.Thebasicreproductionnumberforthismodelhas found. Stability analysis has presented in Section 4.FurtherthisO D E modelhasextendedtostochasticdifferentialequation inSection5.

NumericalsimulationhasdoneintheSection6.FinallyweconcludedinSection7

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2. TheModel

WedevelopedthedeterministicmodelusingthecompartmentsPovertywomenpopulation𝑃𝑉,Prostitutewomenpopulatio n 𝑃𝑆,Rehabilitationwomenpopulation(R). Alsothetotalpopulationis 𝑇 = 𝑃𝑉+ 𝑃𝑠 + 𝑅.Themodelhasframedbyconsideringthe followingassumptions. The povertywomenpopulationhasrecruitedintheregionat

therateofΛ.Whenthesepovertywomenindividualsinteractwiththeprostitutesthen they becomeprostitutestofulfilltheirfinancialrequirements.Thishasrepresented

a s bilineartypeincidence𝛽𝑃_𝑉𝑃_𝑆.Alsointherate∝₁womenindividualsmoveto rehabilitationtoeradicatetheirpoverty.Oncetheprostitutesrealizedtheirphysical

andmentalhealththeymovetorehabilitationtolead properlifewithoutprostitution. Ashumanlifeisambiguous s o ithasconsidered as,atthereducedrate 𝛿women w h o

_𝑆

(1) 𝑑𝑅

.

Table1:Tableof Parameters Parameter Description

Λ Recruitmentrate of populationofwomen

𝛽 Rateof interactionbetweenpoverty womenandprostitutes 𝛼1 Rateof poverty womenmovefrom𝑃_𝑉toR

𝛼2 Rateof prostitutesmovefrom𝑃_𝑉toR 𝛿 𝛽𝑃𝑆𝑅Reducedrate progressionofRbacktoPs

µ Naturaldeathrateofwomen

µ1 Rateof deathduetosexuallytransmitteddisease

3. .Existence ofequilibrium p o i n t s &Basicreproductionnumber

Themodel(2)hasexhibitedtwo equilibriumpointsnamelyprostitutionfreeequilibrium pointandprostitutionpresentequilibriumpoint.

3.1 Prostitution free equilibrium point

The prostitution free equilibrium point is ℰ₀= (𝑃_𝑉⁰, 0, 𝑅⁰) where𝑃_𝑉⁰= ^Λ

𝑘₁, 𝑅⁰ =^𝛼¹^Λ

𝑘₁𝜇for the system (2).

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Received 25 April 2021; Accepted 08 May 2021.

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Figure1:Schematicdiagramofmodel (1)

3.2....Basicreproduction nu mb er 𝓡

_𝟎

Next generation method [15] as adopted to find the basic reproduction number at prostitution free equilibrium pointℰ0. Here ℱ𝑖(𝑡) be the primary women prostitute,𝒱𝑖+(𝑡)be the inflow of transfer rate of individuals into the compartment and 𝒱_𝑖⁻(𝑡)be the outflow of the compartment with the index 𝑖. Using (2) we constructed a column matricesℱ(𝑡),𝒱⁺ 𝑡 and,𝒱⁻(𝑡) as follows

ℱ 𝑡 =

0 𝛽𝑃_𝑉𝑃_𝑆+ 𝛿 𝛽𝑃_𝑆𝑅

0

𝒱⁺ 𝑡 = Λ 0 𝛼1𝑃𝑉+ 𝛼2𝑃𝑆

𝒱⁻ 𝑡 =

Λ𝛽𝑃𝑉𝑃𝑆+ 𝑘1𝑃𝑉

𝑘₂𝑃_𝑆 𝛿 𝛽𝑃𝑆𝑅 + 𝜇𝑅

Also

𝑃_𝑉 𝑃_𝑆 𝑅 ^𝑇= ℱ 𝑡 − 𝒱 𝑡 Where𝒱 𝑡 = 𝒱⁻ 𝑡 − 𝒱⁺ 𝑡 The spectral radius of the matrix 𝐹𝑉⁻¹is ℛ₀has given below

ℛ₀=^{Λ𝛽 (1+𝛿)}

𝑘₁𝑘₂ = ^{Λ𝛽 (1+𝛿)}

(𝛼₁+𝜇 )(𝛼₂+𝜇 +𝜇₁) (3)

Where 𝐹 represents Jacobian matrix of ℱ 𝑡 evaluated at ℰ₀and V represents Jacobian matrix of 𝒱 𝑡 evaluated at ℰ0

3.3.ProstitutionPresentEquilibriumPoint

Theprostitutionpresentequilibriumpointforthe system (2) is ℰ1= 𝑃𝑉∗, 𝑃𝑆∗, 𝑅^∗ where𝑃_𝑉^∗= ^𝑘²

𝛽 (𝛿+1), 𝑃𝑆∗=^{Λ𝛽 1+𝛿 −𝑘}¹^𝑘²

𝑘₂𝛽 𝑎𝑛𝑑 𝑅^∗=

Λ𝛼₂𝛽 1+𝛿 ²−Λ𝛽 𝑘₂𝛿 1+𝛿 −𝛼₂𝑘₁𝑘₂ 𝛿 +1 +𝑘₂²(𝛼₁+𝑘₁𝛿)

𝛽𝜇 𝑘₂(𝛿+1) ,

4.StabilityAnalysis

The Jacobianmatrixforthesystem(2)isgivenby 𝒥 =

−𝑃𝑆𝛽 − 𝑘1 −𝑃𝑉𝛽 0 𝑃_𝑆𝛽𝛿 + 𝑃_𝑆𝛽 𝑃_𝑉𝛽𝛿 + 𝑃_𝑉𝛽 − 𝑘₂ 0

−𝑃𝑆𝛽𝛿 + 𝛼1 −𝑃𝑉𝛽𝛿 + 𝛼2 −𝜇

The local stability of these two equilibrium points have presented below

4.1 LocalStabilityofProstitutionFreeEquilibriumPoint

The stability of prostitution free equilibrium point ℰ0 has examined from the Jacobian matrix ℐ atℰ0. Then we have

𝒥0=

−𝑘₁ −𝑃_𝑉⁰𝛽 0 0 𝑃𝑉0𝛽𝛿 + 𝑃𝑉0𝛽 − 𝑘2 0 𝛼₁ −𝑃_𝑉⁰𝛽𝛿 + 𝛼₂ −𝜇

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The above matrix has produced three eigen values −𝜇, −𝑘1and ^{Λ𝛽 1+𝛿 −𝑘}¹^𝑘²

𝑘₁ = 𝑘2 ℛ₀− 1 < 0 since ℛ0< 1. Therefore all three eigen values are negative. This proves the prostitution free

equilibrium point is

stable when ℛ₀< 1. Further, we have checked the local stability of prostitution present equilibrium point ℰ₁.

4.2 Localstability o f ProstitutionPresentEquilibriumPoint

The stability of prostitution free equilibrium point ℰ₁ has examined from the Jacobian matrix ℐ atℰ₁. Then we have

𝒥₁=

−𝑃𝑆∗𝛽 − 𝑘1 −𝑃𝑉∗𝛽 0 𝑃_𝑆^∗𝛽𝛿 + 𝑃_𝑆^∗𝛽 𝑃_𝑉^∗𝛽𝛿 + 𝑃_𝑉^∗𝛽 − 𝑘₂ 0

−𝑃_𝑆^∗𝛽𝛿 + 𝛼₁ −𝑃_𝑉^∗𝛽𝛿 + 𝛼₂ −𝜇

The eigen values of the above matrix are – 𝜇 , −Λ𝛽 1+𝛿 − ℛ₀²𝑘₁²𝑘₂²−4𝑘₁𝑘₂³(ℛ₀−1)

2𝑘₂ and

−Λ𝛽 1+𝛿 + ℛ₀²𝑘₁²𝑘₂²−4𝑘₁𝑘₂³(ℛ₀−1)

2𝑘₂ . Here the eigen values are either negative or have negative real part. Therefore the prostitution present equilibrium point is locally asymptotically stable.

5 Stochasticmodel

Tostudythestochasticnatureofthedeterministicmodelweextendourdeterministic modeltostochasticmodel[16],[5][12][14].Considerthecontinuousrandomvariable

𝑋(𝑡) = 𝑋1 𝑡 , 𝑋2 𝑡 , 𝑋3 𝑡 ^𝑇for 𝑃_𝑉 𝑡 , 𝑃_𝑆 𝑡 , 𝑅 𝑡 ^𝑇where𝑇denotesthetransposeof the matrix.

Let ∆X=X(t+∆t)−X(t)=(∆X1,∆X2,∆X3)T

betherandomvector for the change in randomvariablesduringtimeinterval∆𝑡.All t he possible changes

betweenstatesintheSDEmodelcanbedefinedbythetransitionmap[9].Fromour model(1),thereexist10possiblechangesbetweenstatesinasmall-

timeinterval∆𝑡. StatechangesandtheirprobabilitiesareelucidatedinTable2.Forillustration,when

apovertywomaninteractwiththeprostitute. Then thestatechange∆Xisdenotedby Δ𝑋 = −1,1,0 ^𝑇 and its probability is given by

𝑃𝑟𝑜𝑏{ ΔX₁, ΔX₂, ΔX₃ = −1,1,0 |(𝑋₁ 𝑡 , 𝑋₂ 𝑡 , 𝑋₃ 𝑡 )} = 𝑃₂=

𝛽𝑋1𝑋2Δ𝑡 + 𝑂(Δ𝑡) Forstatechange∆𝑋,𝐸𝑥𝑝(∆𝑋)and𝑉𝑎𝑟(∆𝑋)areexpectationchangeandits covariancematrixrespectivelywithneglectedtermshigherthan𝑂(∆𝑋).Theexpectationchange

𝐸𝑥𝑝 Δ𝑋 = 𝑃_𝑖 Δ𝑋 _𝑖Δ𝑡

10

1

=

Λ − 𝛽𝑋1𝑋2− 𝛼1𝑋1− 𝜇𝑋1

𝛽𝑋₁𝑋₂+ 𝛿𝛽𝑋₂𝑋₃− 𝛼₂𝑋₂− 𝜇₁𝑋₂− 𝜇𝑋₂ 𝛼1𝑋1+ 𝛼2𝑋2− 𝛿𝛽𝑋2𝑋3− 𝜇𝑋3

Δ𝑡 (4)

= 𝑓(𝑋1 𝑡 , 𝑋₂ 𝑡 , 𝑋₃ 𝑡 )Δ𝑡 (5) From the above calculation the expectation vector and the function 𝑓 are of same form as in deterministic system (1). Further we find the covariance matrix

𝑉𝑎𝑟 𝑋 = 𝐸𝑥𝑝( Δ𝑋 Δ𝑋 )^𝑇− 𝐸𝑥𝑝(Δ𝑋)𝐸𝑥𝑝( Δ𝑋 ^𝑇) and𝐸𝑥𝑝 Δ𝑋 𝐸𝑥𝑝 Δ𝑋 ^𝑇 = 𝑓 𝑋 𝑓 𝑋 ^𝑇 , it can be approximated with diffusion matrix Ω times Δ𝑡 by neglecting the term of Δ𝑡 ² such that 𝑉𝑎𝑟(Δ𝑋) ≈ 𝐸𝑥𝑝(Δ𝑋)𝐸𝑥𝑝( Δ𝑋 ^𝑇)

𝐸𝑥𝑝 Δ𝑋 𝐸𝑥𝑝 Δ𝑋 ^𝑇 = 𝑃_𝑖( Δ𝑋 𝑖 Δ𝑋 _𝑖^𝑇)

10

1

Δ𝑡 =

𝑉₁₁ 𝑉₁₂ 𝑉₁₃ 𝑉₂₁ 𝑉₂₂ 𝑉₂₃ 𝑉₃₁ 𝑉₃₂ 𝑉₃₃

Δ𝑡 = Ω. Δ𝑡

Where the above diffusion matrix is symmetric, positive definite and each component of this 3×3 diffusion matrix are given by

𝑉11= Λ + 𝛽𝑋1𝑋2+ 𝛼2𝑋2+ 𝜇𝑋1= 𝑃1+ 𝑃2+ 𝑃3+ 𝑃6 , 𝑉12= 𝑉21= −𝛽𝑋1𝑋2= −𝑃2 , 𝑉13= 𝑉31= −𝛼1𝑋1=

−𝑃₃, 𝑉₂₂= 𝛽𝑋₁𝑋₂+ 𝛿𝛽𝑋₂𝑋₃+ 𝛼₂𝑋₂+ 𝜇𝑋₂+ 𝜇₁𝑋₂= 𝑃₂+ 𝑃₄+ 𝑃₅+ 𝑃₇+ 𝑃₈, 𝑉₂₃= 𝑉₃₂= −𝛿𝛽𝑋₂𝑋₃− 𝛼₂𝑋₂= −𝑃₄− 𝑃₅, 𝑉₃₃= 𝛼₂𝑋₂+ 𝛿𝛽𝑋₂𝑋₃+ 𝛼₂𝑋₂+ 𝜇𝑋₃= 𝑃₃+ 𝑃₄+ 𝑃₅+ 𝑃₉. Further we used the method in [16] and constructed a matrix 𝑀 such that 𝑉 = 𝑀𝑀^𝑇, where 𝑀 is 3×6 matrix

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𝑃1+ 𝑃6

0 0

𝑃2

− 𝑃2

0

𝑃3

0

− 𝑃3

0 𝑃4+ 𝑃₅

− 𝑃4+ 𝑃5

0 𝑃7+ 𝑃₈

0

0 0 𝑃9

Then, the Ito stochastic differential model has the following form:

𝑑 𝑋 𝑡 = 𝑓 𝑋1, 𝑋2, 𝑋3 𝑑𝑡 + 𝑀𝑑𝑊(𝑡)

With initial condition 𝑋 0 = 𝑋₁ 0 , 𝑋₂ 0 , 𝑋₃ 0 ^𝑇 and a Wiener process, 𝑊 𝑡 = 𝑊₁ 𝑡 , 𝑊₂ 𝑡 , 𝑊₃ 𝑡 ^𝑇. In view of the above facts, we construct the SDE model as follows: 𝑑𝑃_𝑉= Λ − 𝛽𝑃_𝑉𝑃_𝑆− 𝛼₁𝑃_𝑉− 𝜇𝑃_𝑉 𝑑𝑡 + Λ + 𝜇𝑃𝑉 𝑑𝑊1+ 𝛽𝑃𝑉𝑃𝑆 𝑑𝑊2+ 𝛼1𝑃𝑉 𝑑𝑊3

𝑑𝑃𝑆=

𝛽𝑃_𝑉𝑃_𝑆+ 𝛿 𝛽𝑃_𝑆𝑅 − 𝛼₂𝑃_𝑆− 𝜇𝑃_𝑆− 𝜇₁𝑃_𝑆 𝑑𝑡 − 𝛽𝑃𝑉𝑃_𝑆 𝑑𝑊₂+ 𝛿 𝛽𝑃𝑆𝑅 + 𝛼₂𝑃_𝑆 𝑑𝑊₄+ 𝜇 + 𝜇₁ 𝑃_𝑠 𝑑𝑊₅ (6)

𝑑𝑅 = 𝛼₁𝑃_𝑉+ 𝛼₂𝑃_𝑆+ 𝛿 𝛽𝑃_𝑆𝑅 − 𝜇𝑅 𝑑𝑡 − 𝛼1𝑃_𝑉 𝑑𝑊₃− 𝛿 𝛽𝑃_𝑆𝑅 + 𝛼₂𝑃_𝑆 𝑑𝑊₄− 𝜇𝑅𝑑𝑊6

Table2:Tableof Parameter

Possiblestatechange Probabilityofstatechange

∆𝑋 ₁= 1, 0, 0 ^𝑇Recruitmentrate of poverty women Population

𝑃₁= 𝛬∆𝑡 + 𝑂(∆𝑡)

∆𝑋 ₂= −1, 1, 0 ^𝑇Changewhenpoverty woman becomeprostitutewoman

𝑃₂= 𝛽𝑋₁𝑋₂∆𝑡 + 𝑂(∆𝑡)

∆𝑋 ₃= −1,0,1 ^𝑇Changewhenpoverty woman joinrehabilitationclass

𝑃₃_ = 𝛼1𝑋¹∆𝑡 + 𝑂(∆𝑡)

∆𝑋 ₄ =

0,1, −1 ^𝑇Changewhenwomaninrehabilitationclass againjointoprostituteclass

𝑃4= 𝛽𝛿𝑋2𝑋3∆𝑡 + 𝑂 ∆𝑡

∆𝑋 ₅ = 0, −1,1 ^𝑇Changewhenprostitutewoman joinrehabilitationclass

𝑃₅= 𝛼2𝑋2∆𝑡 + 𝑂(∆𝑡) ∆𝑋 ₆= −1,0,0 ^𝑇Naturaldeathof

povertywomenpopulation

𝑃₆= µ𝑋₁∆𝑡 + 𝑂(∆𝑡)

∆𝑋 ₇= 0, −1,0 ^𝑇Naturaldeathof prostitutewomenpopulation

𝑃₇= µ𝑋₂∆𝑡 + 𝑂(∆𝑡)

∆𝑋 ₈=

0, −1,0 ^𝑇Changeduetosexuallytransmitteddisease ofprostitutes

𝑃8 = µ1𝑋2∆𝑡 + 𝑂(∆𝑡)

∆𝑋 ₉= 0,0, −1 ^𝑇Naturaldeathof womeninrehabilitationclass

𝑃₉= µ𝑋₃∆𝑡 + 𝑂(∆𝑡)

∆𝑋 ₁₀= 0,0,0 ^𝑇

There is no change 𝑃10= 1 − 𝑃𝑖

9

1

∆𝑡 + 𝑂(∆𝑡)

6.Numerical Simulation

In this section, we have done numerical simulation for our deterministic model (1) forboth the equilibrium points namely prostitution free equilibrium point and prostitutionpresent equilibrium point. For prostitution free equilibrium point the parametric set:Ʌ = 100, 𝛽 = 0.00001, 𝛼₁= 0.002, 𝜇 = 0.0143, 𝜇₁= 0.025, 𝛿 = 0.02, 𝛼2= 0.04 For the corresponding parametric values the ℛ0= 0:67 (Figure 2) and the prostitution freeequilibrium point is ℰ0= (2915: 45; 0; 4077: 55). For prostitution present equilibriumpoint the parametric set: Ʌ = 500, 𝛽 = 0.00002, 𝛼₁= 0.02, 𝜇 = 0.0143, 𝜇₁= 0.025, 𝛿 = 0.01, 𝛼₂= 0.6 For the corresponding parametric values theℛ0 = 2:96 (Figure 2) and the prostitution present equilibrium point is ℰ1 = (4915:84; 3370:59; 20785:93). Further, we simulated our SDE model (6) by using Euler-Maruyama method for the following set of parameters:Ʌ = 500, 𝛽 = 0.00002, 𝛼1= 0.02, 𝜇 = 0.0143, 𝜇1= 0.025, 𝛿 = 0.02, 𝛼2=

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0.06 We compare the mean of 100 runs of stochastic model simulation with the results of corresponding deterministic model. And this fact is shown in Figures 4-6.

7.Conclusion

In this article a nonlinear mathematical model to study the dynamics of the spread of prostitution in the society is formulated and analyzed. The threshold (basic reproduction number ℛ0) is obtained which determines whether prostitution will persist in the society or will die out. The existence and stability of different equilibria are discussed. Finally, thedeterministic model is converted to stochastic model and the results ofstochastic model are compared with corresponding deterministic model. It is observed that the level of rehabilitation women population in stochastic simulation is slightly higher than the simulation result of corresponding deterministic model. Additionally, it is found that increase in the parameters 𝛼₁ and 𝛼₂ decreases the poverty women population and prostitute women population in both deterministic and stochastic simulation.

Figure2:Variationofwomenpopulationunderprostitutionfreeequilibriumpoint ℰ

₀

Figure3:Variationofwomenpopulationunderprostitutionpresentequilibriumpointℰ

₁

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Figure4:Variationofpoverty womenpopulationwithtime.

Figure5:Variationofprostitutewomenpopulationwithtime.

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Figure6:Variationofrehabilitationwomenpopulationwithtime.

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