Age-Structured Population Projection of Bangladesh by Using a Partial Differential Model with Quadratic Polynomial Curve Fitting

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1 Ope Jourl of Applied Scieces, 5, 5, Published Olie September 5 i SciRes. Age-Structured Popultio Projectio of Bgldesh by Usig Prtil Differetil Model with Qudrtic Polyomil Curve Fittig Shiri Sult, Mhmudul Hs, Lek Szzd Adllh Deprtmet of Nturl Scieces, Dffodil Itertiol Uiversity, Dhk, Bgldesh Deprtmet of Mthemtics, Jhgirgr Uiversity, Dhk, Bgldesh Emil: shir.s@dffodilvrsity.edu.bd, smmhs@juiv.edu, dllh@juiv.edu Received Jue 5; ccepted September 5; published 3 September 5 Copyright 5 by uthors d Scietific Reserch Publishig Ic. This work is licesed uder the Cretive Commos Attributio Itertiol Licese (CC BY). Abstrct I this pper, the ge-specific popultio of Bgldesh bsed o lier first order (hyperbolic) prtil differetil equtio which is kow s Vo-Foerster Equtio is studied. Applyig qudrtic polyomil curve fittig, the totl popultio d popultio desity of Bgldesh re projected for the yers to 5 bsed o the explicit upwid fiite differece scheme for the ge-structured popultio model bsed o give dt (source: BBS & ICDDR, B) for iitil vlue i the yer. For ech ge-group, the future birth rtes d deth rtes re estimted by usig qudrtic polyomil curve fittig of the dt for the yers to. Qudrtic polyomil curve fittig is lso used for the boudry vlue s the ( - 4) ge-group popultio bsed o the popultio size of the ge-group for the yers to. Keywords Vo-Foerster Equtio, Birth Rte, Deth Rte, Curve Fittig. Itroductio The fst growth of popultio durig the pst decdes hs frustrted the developmet efforts i Bgldesh. I 97 the popultio of the coutry ws roud 75 millio. Accordig to the 5th cesus of Bgldesh Bureu of Sttistics (BBS) i, the totl popultio of Bgldesh is 5 millio. The re of Bgldesh is 47,57 squre kilometers oly d it is oe of the most desely populted coutries ll over the world. To hdle such How to cite this pper: Sult, S., Hs, M. d Adllh, L.S. (5) Age-Structured Popultio Projectio of Bgldesh by Usig Prtil Differetil Model with Qudrtic Polyomil Curve Fittig. Ope Jourl of Applied Scieces, 5,

2 S. Sult et l. mss popultio i such short ld is huge problem for the govermet to tke y developig steps. Due to scrcity of resources, it is ot possible to provide eductiol, helth, medicl, trsport d housig fcilities to the etire popultio. A rpidly icresig popultio plugs the ecoomy ito mss uemploymet d uder employmet. As result, the ctul developmet is just gettig beig delyed hmpered lot. With the help of popultio model, we c predict wht the umber of popultio will be by the yer of 5. Therefore, the govermet should tke immedite steps to keep the popultio uder cotrol d the people themselves should dopt fmily plig for their ow beefit. I order to mke efficiet plig for the demds of differet ge-group, it is importt to predict the ge-structured popultio of the coutry. Therefore, i this pper we project the future ge-structured popultio of Bgldesh bsed o prtil differetil equtio model. The iformtio we get from ge-specified popultio group c help us i future plig of socil d ecoomicl developmet. For exmple, if we c predict the popultio of childre of - 4 yers old, we c provide ecessry medicl cre d bby food to reduce the mortlity rte d keep childre helthy d ourished. After 5 yers of libertio of Bgldesh, the govermet of Bgldesh is goig to celebrte the yer s potheosis of libertio. The govermet hs lredy declred the yer s visio. To educte the people of Bgldesh withi, we c fford ecessry support for childre of 4-5 yers old. We c provide hum resource developmet progrm for youg people to reduce the uemploymet problem d lso proper helth cre to elder people.. Age-Structured Popultio Model For the predictio of ge-structured popultio, vrious differetil equtio models hve bee formulted by differet Mthemticis i differet time. I [], Murry studies the lier ge structure popultio model where he used time-idepedet deth rte. A extesive study of lier d olier ge-depedet popultio dymics c be foud i the works of Webb [], Gurti [3] d Ielli [4]. I this pper, we study lier first order hyperbolic prtil differetil equtio to predict ge depedet popultio. We project the future ge-structured popultio i Bgldesh bsed o this lier model of popultio which is kow s Vo-Foerster equtio, give s follows: u x, = u x, x with ( ) ( ) (, ) u( xt, ) u xt t ( xt) u( xt) + = µ,,, xt, x λ the ge specific fertil- x is the iitil ge distributio. u(, t) = λ ( xt, ) u( xt, ) d x, t where, u( xt, ) is the ge desity fuctio, u( xt, ) the ge specific deth rte, ( xt, ) ity rte so tht λ ( xt, ) u( xt, ) dx is the birth rte d u ( ) 3. Numericl Scheme for the Model The ge-structured popultio model c be writte s with iitil coditio ( ) ( ) () u+ u = µ txutx,,, x, t () t x ( ) ( ) u x, = u x, x (3) u ( x ) oegtive fuctio, d boudry coditio t x = (the iflow boudry of the domi) u(, t) bt ( ), t = (4) Bsed o [5] d [6] we discretize the time derivtive by forwrd differece formul, d the ge derivtive with bckwrd differece o discrete mesh xi = i xi, N + d t = t, N +. The forwrd differece u pproximtio for obtied by the Tylor series formul is t 543

3 S. Sult et l. d the bckwrd differece pproximtio for (, ) (, ) (, ) u t x u t+ k x u t x t k u x is (, ) (, ) (, ) utx utx utx h x h We cosider uiform grid spcig with step size h d k for spce d time respectively, x = i x + + i h d t + = t + k. Usig the pproximtio u x, t i (5) d (6), Equtio () pproximtes s U i for ( i ) + Ui Ui Ui Ui + = µ ( xt, ) U t x ( ) U = U vu U tµ U i + i i i i i which is the explicit upwid differece scheme for the ge-structured popultio model. t Here v : =. The boudry vlue U x, ; must be obtied from the boudry coditio (4) t x = d iitil vlue U i for i N +, obti from the iitil coditio (3) t t =. 4. Numericl Experimets We implemet the explicit upwid differece scheme d itroduce qudrtic polyomil curve fittig procedure for the ge-structured popultio model. 4.. Curve Fittig Curve fittig is the process of costructig curve, or mthemticl fuctio tht hs the best fit to series of dt poits, possibly subject to costrits. Curve fittig c ivolve either iterpoltio, where exct fit to the dt is required, or smoothig, i which smooth fuctio is costructed tht pproximtely fits the dt. 4.. Qudrtic Polyomil Models Give dt poits ( x, y ),( x, y ),,( x, y ) The residul t ech dt poit is give by The sum of the squre of the residuls is give by. Cosider the Qudrtic Polyomil Modelss = + + (8) y x x e = y x x (9) i i i i ( ) i i i i i= i= () S= e = y x x To fid the costts of the polyomil regressio model, we put the derivtives successively with respect to, d to zero, tht is, S = ( yi x i x i )( ) = () i= S = ( yi x i x i )( xi) = () i= S = ( yi x i x i )( xi ) = (3) i= (5) (6) (7) 544

4 S. Sult et l. Settig those equtios i mtrix form gives The bove re solved for,,. xi x i yi i= i= i= 3 xi xi x i = xy i i i= i= i= i= 3 4 x x i xi x i i yi i= i= i= i= (4) 4.3. Icorportio of Dt ito Explicit Upwid Differece Scheme To predict the Age Distributed Popultio we icorporte the iitil d boudry dt ito the Explicit Upwid Scheme with respect to the ssumptios d cosidertios below: We ssume tht utx (, ) is the totl popultio distributio fuctio rther th desity distributio fuctio. We hve cosidered the ge of people of Bgldesh i betwee to 85+ yers. We divide this ge-group ( to 85 yers) ito 8 sub-groups, ech sub-group cotis 5 yers itervl i.e. - 4, 5-9, - 4,, 8-84, 85+. We hve used ge distributed popultio of s iitil dt u ( x ) d we hve ssumed the popultio of the ( - 4) ge-group s the ewbors bby (popultio) d the popultio of this first ge-group of the yers of to of dt (Tble ) [7] is the bsis to formulte the birth rte bt ( ). We hve estimted the birth bt, we hve bee fittig qudrtic polyomils rte by fittig qudrtic polyomils. To evlute birth rte ( ) Tble. Mid-yer popultio distributio i percetge by ge-group from to. Age Popultio i percetge i the yers (Source: ICDDR, B). 545

5 S. Sult et l. ( ) ( ) ( * ) ( * ) The we hve used bt ( ) s the boudry coditio u ( ) bt = b + b t+ b t (5) t. For this we hve used dt of birth rte from Tble [7]. We hve estimted deth rte, which hs bee prmeterized by µ i the ge specific popultio model, by µ xt,, we hve bee fittig qudrtic polyomils, fittig qudrtic polyomils. To evlute deth rte ( ) ( t) ( ) ( * ) t ( * ) t µ = µ + µ + µ (6) For this we hve used dt of deth rte from Tble [8]. Usig theses ge d time depedet deth rte, iitil vlue d boudry coditio o Explicit Upwid Fiite Differece Scheme, we c forecst the ge specific popultio distributio Totl Popultio Projectio Figure shows our estimted totl popultio projectio mrked by Age-Structured Popultio Model (By Qudrtic Polyomil) for the yers to 5. Here it is observed tht the iitil popultio i our model is 3 millio for the yer d the predicted popultio for the yer 5 is millio. I the yer, i our model the predicted popultio will be 7.44 millio wheres the totl popultio ws 75 millio i 97. So the totl popultio i will be.6 multiple of the totl popultio of 97. Figure shows the compriso of the predicted popultio of Bgldesh from the yers to 5 with the projectio of Bgldesh Bureu of Sttistics (BBS) which is mde by Md. Kbir [9], the predicted popultio by Dutt d Adllh [] d the predicted popultio by Md. Mirul Hque, Fruque Ahmed, Syedul Am, Md. Rshed Kbir []. Here, the popultio of Bgldesh is clculted by three mthemticl methods. Dutt d Adllh hve used Lier Equtio, Md. Mirul Hque, Fruque Ahmed, Syedul Am, Md. Tble. Deth rte by ge-group d yer (per popultio). Age Deth rte (per popultio) i the yers (Source: ICDDR, B). 546

6 S. Sult et l. 6 Age-Structured Popultio Model(By Qudrtic Polyomil) 4 Popultio i Millio Time i Yer Figure. Exmple of figure cptio (figure cptio). 3 8 Age-Structured Popultio Model(By Qudrtic Polyomil) Bgldesh Bureu of Sttistics (By Md. Kbir) Age-Structured Popultio Model(By Lier Profile) Popultio Projectio (Usig Logistic Popultio Model) 6 Popultio i Millio Time i Yer Figure. Compriso of our popultio projectio with differet methods. Rshed Kbirhve used Logistic Popultio Model d we hve used Qudrtic Polyomil Curve Fittig. Our curve is goig to most erer to the curve of Logistic Popultio Model with the icresed period of time. Here it is observed tht the iitil popultio is 3 millio i i our projectio s well s sme i the projectio by Lier Equtio where s it is 3. i the projectio of BBS d 9.9 millio i the projectio 547

7 S. Sult et l. by Logistic Popultio Model. It is lso observed tht, for the yer 5 our predicted popultio is millio, but it is millio correspodig to BBS d correspodig to the projectio by Lier Equtio. I 35, i our model the predicted popultio will be millio wheres the predicted popultio will be millio correspodig to the projectio of BBS, millio correspodig to the projectio by Lier Equtio d millio correspodig to the projectio by Logistic Popultio Model. By Figure oe c observe tht the iclitio of the icresig popultio of our Age-Structured Popultio Model is more cosistet th the demogrphicl method of BBS d the popultio model by Lier Equtio. From the compriso it is lso observed tht the predictio by the Qudrtic Polyomil Curve Fittig method is much closer to the predictio by Logistic Popultio Model Popultio Projectio for Differet Age-Groups Figure 3 is showig the popultio projectio for differet ge-groups of Bgldesh from the yers to 5. It is clerly observed tht, the ge of the people i the scle ( - 5) will icrese rpidly. The middle ged people will lso icrese but the icresig rte will be prtilly sloth d the icresig rte of ged people will be very sloth. Figure 4 shows the compriso of our estimted popultio per-squre kilometer projectio mrked by Age- Structured Popultio Desity (By Qudrtic Polyomil) with tht of mrked by Bgldesh Bureu of Sttistics Desity (by Md. Kbir) for the yers to 5. Here it is observed tht the iitil popultio persqure kilometer is 888 i i our projectio wheres it is 88 ccordig to the projectio of BBS. It is lso observed tht, for the yer 5 our predicted popultio per-squre kilometer is 669, but it is 995 ccordig to BBS. I, i our model the predicted popultio per-squre kilometer will be 53 wheres the predicted popultio per-squre kilometer will be 7 ccordig to the projectio of BBS. The totl predicted popultio for the yers to 5 is preseted i Tble 3 d the predicted popultio per-squre kilometer is preseted i Tble Coclusio We hve cosidered cotiuous d determiistic mthemticl model kow s Vo-Foerster model, which is lier first order prtil differetil equtio used to predict popultio distributio by ge t y time, give the iitil distributio d the vritio of birth d deth rtes with ge d time. Although this is lier equtio, it is ot esy to solve the difficulty of eforcig boudry coditio. For this, we hve used fiite differece Popultio i Millio Popultio i Millio Time i Yer Figure 3. Popultio projectio of differet ge-groups Time i Yer 548

8 S. Sult et l. Tble 3. Popultio projectio of Bgldesh from to 5 (i millios). Yer Projectio by Qudrtic Polyomil Curve Fittig Projectio by BBS Projectio by Lier Equtio Projectio by Logistic Popultio Model

9 S. Sult et l. Tble 4. Popultio per-squre kilometer of Bgldesh from to 5. Yer Our desity Desity by BBS

10 S. Sult et l. Age-Structured Popultio Desity(By Qudrtic Polyomil) Bgldesh Bureu of Sttistics Desity(By Md. Kbir) 8 Poultio Per Squre Kilometer Time i Yer Figure 4. Compriso of our popultio desity projectio with Bgldesh Bureu of Sttistics for the yers to 5. method for the umericl solutio of the ge-structured popultio model. For the umericl experimet, the yer is used s the iitil time. We hve predicted the ge distributio popultio up to the yer 5. I this experimet we hve provided the dt for birth rte d deth rte up to from [7] d [8] respectively. The results hve showed very good greemet with the ge distributio popultio size up to give i [7]. These hve justified the correctess of our implemettio of the explicit upwid fiite differece scheme for the ge-structured popultio projectio i Bgldesh d i other coutries s well. We hve predicted the totl popultio d popultio desity up to 5 d these results hve showed very good greemet with the results ccordig to the projectio by Logistic Popultio Model [] which hve bee predicted up to 35. We hve observed tht, i the yer 5, our popultio will be millio wheres it is millio ccordig to Kbir [9] d millio ccordig to the projectio by Lier Equtio []. I [], the clcultio is bsed o lier modelig of birth d deth rte d our clcultio is bsed o qudrtic polyomil curve fittig of birth d deth rte. We hve estimted the birth rte d deth rte by o-lier profile. So, the results re more relistic. This motivtes us for further study of the model. Refereces [] Murry, J.D. (989) Mthemticl Biology. Spriger-Verlg, Berli Hiedelberg. [] Webb, G.F. (985) Theory of Nolier Age-Depedet Popultio Dymics. Mrcel Dekker Ic., New York. [3] Gurti, M.E. d McCmy, R.C. (984) Nolier Age-Depedet Popultio Dymics. Archive for Rtiol Mechics d Alysis, 54, 8-3. [4] Ielli, M. (995) Mthemticl Theory of Age-Structured Popultio Dymics. Applied Mthemtics Moogrphs, 7, Cosiglio Nzioledelle Ricerche, Pis. [5] Burde, R.I. d Firs, J.D. (997) Numericl Alysis. 6th Editio, Brookscole. [6] Abi, L.M., Agulo, O. d Lopez-Mrcos, J.C. (5) Age-Structured Popultio Models d Their Numericl Solutio. Ecologicl Modellig, 88, [7] ICDDR, B Documet, Tble.3 Mid-Yer Popultio Distributio by Age Group. [8] ICDDR, B Documet, Tble 3.3 Deth Rte by Age d Yer (per Popultio). [9] Kbir, Md. d Ahmed, T. (6) Poverty-Agig Popultio Projectio. BBS, Miistry of plig, Bgldesh. [] Dutt d Adllh, L.S. (8) Age-Structured Popultio Projectio i Bgldesh Bsed o Numericl Solutio of Determiistic Model. Bgldesh Jourl of Scietific Reserch,, [] Hque, Md.M., Ahmed, F., Am, S. d Kbir, Md.R. () Future Popultio Projectio of Bgldesh by Growth Rte Modelig Usig Logistic Popultio Model. Als of Pure d Applied Mthemtics,,