Samuel Sindayigaya 1, Nyongesa L. Kennedy 2, Adu A.M. Wasike 3

Size: px

Start display at page:

Download "Samuel Sindayigaya 1, Nyongesa L. Kennedy 2, Adu A.M. Wasike 3"

Colin Barnett
5 years ago
Views:

1 Ieraioal Joural of Saisics ad Aalysis. ISSN Volume 6, Number (6, pp. -8 Research Idia Publicaios hp:// The Populaio Mea ad is Variace i he Presece of Geocide for a Simple Birh-Deah- Immigraio-Emigraio usig he Probabiliy Geeraig Fucio Samuel Sidayigaya, Nyogesa L. Keedy, Adu A.M. Wasike 3 Deparme of Saisics Applied o Ecoomy, Isiud EseigemeS uperiur de Ruhegeri/INES-Ruhegeri-Rwada Deparme of Mahemaics, Maside Muliro Uiversiy of Sciece ad Techology, Keya Deparme of Mahemaics, Maside Muliro Uiversiy of Sciece ad Techology, Keya Absrac This paper discusses o simple birh-deah-immigraio-emigraio (BDIE processes wih geocide o sochasic differeial equaios (SDE model. The geeral Soluio for he disribuio of he size of he populaio a ay isa i ime is obaied i he form of a probabiliy geeraig fucio (PGF.The exac soluio; mea ad variace are derived for cosa birh, deah, immigraio, emigraio ad caasrophe raes. Key words: Geocide, Caasrophe, BDIE, SDE, PGF. INTRODUCTION The works o birh-deah ype processes have bee ackled mosly by some scholars such as Yule, Feller, Kedal ad Gez amog ohers. These fellows have bee formulaig he processes o model he behavior of sochasic populaios.rece examples o birh-deah processes ad sochasic differeial equaios (SDE have also bee developed. Graia [], modeled liear growh birh ad deah processes wih immigraio ad emigraio usig he sochasic differeial equaio. Gez [], geerally modeled birh-deah processes wih posiive ad egaive corols usig he probabiliy geeraig fucio.the cosrucio of he rasie probabiliies for a simple birh-deah-immigraio process uder he ifluece of oal caasrophe was approached by Radall[3] ad i his paper a oal caasrophe ha wipes ou he oal

2 Samuel Sidayigaya e al populaio o size zero was cosidered. Di Crescezo [4] also worked o birh-deah process subjec o caasrophes usig he Laplace rasform of is probabiliy desiy fucio o obai he mea ad variace. As he applicaio of he birh-deah process, Sidayigaya [5]used he real daa of he populaio of Rwada o esimae he populaio dyamics usig he mea ad variace derived wih he probabiliy geeraig fucio. Moreover, Michael [6] modeled he immigraio-emigraio wih caasrophe ad foud he seady-sae soluio usig he classic recursive mehods. However, he geeraig fucio model of birh-deah-immigraio-emigraio process wih geocide as a parial caasrophe has o bee discussed i he previous works. I his paper, we shall cosider a geocide which occurs a a cosa rae ad whe i occurs, reduce he populaio o a cerai level. Addiioally, i appears ha o aemp has bee made o derive he mea ad variace o obai he explici imevaryig rajecory of he populaio. The purpose of his paper is o aalyze he BDIE processes by iroducig a geocide parameer ad upo he differeial differece equaio; he geeral soluio is obaied by usig he probabiliy geeraig fucio which fially leads o he deermiaio of he mea ad variace of he populaio.. MODEL DEVELOPMENT The process is formulaed by leig N( represe he size of he populaio a ime ad P ( PN( N( (. As i he simple birh-deah process, birhs ad deahs occur proporioal o he populaio size wih a birh rae > ad a deah rae μ >.Immigraio ad emigraio will occur idepede of he populaio size wih raesν > ad respecively >. Furher, he occurrece of a geocide is also idepede of populaio size ad will occur a a rae >. Thus, he process ca be described by he followig rasiio raes: Trasiio Rae + + ν, μ +,, By ha he rasie probabiliies which is he differeial differece equaio is give by P ( μ( + + P + ( + ( + νp ( ( + ν+ + P( (. Where P ( deoe he differeiaio of P ( wih respec o. Sice egaive corol whe, we have

3 The Populaio Mea ad is Variace i he Presece of Geocide 3 ( ( ( ( ( ( μ P( ( ν P ( P P P P + +. (.3 Leig, (.4 (, ( φ s P S be he probabiliy geeraig fucio(pgffor he sysem, i follows from he sadard geeraig fucio mehod ha φ ( s, saisfies he parial differeial equaio (PDE. φ φ s( s sμ ν ( s φ P ( + s + + s s (.5 s The mehod of soluio o (.5 is skeched ad leads o he form of geeraig fucio for he disribuio of he size of he populaio a ay ime, which here, saed i is mos geeral form for cosa parameers, appears o be a ew resul of he BDIE wih geocide/caasrophe. Fially he PDE (.5 has soluio φ ( s, ( s ( s e ( μ + ( s ( ( s e ( μ + s ( μ + ( ( ( ( μ s μ s e ϒ ( ( ( s + s e (.6 3. DETERMINATION OF THE MEAN AND VARIANCE FOR BDIE I is kow ha he wo mos impora momes of a disribuio are he mea ad variace, ad hese ca be foud quie easily if we pu s i (.4 ad derive wice. The, φ (, P ( E( (3. φ (, ( P ( E( ( (3. which are he meas ad secod facorial mome of he disribuio, respecively. Le ( deoe he mea a ime ad σ ( he variace a ime.sice he variace is he secod mome of he disribuio abou he mea, we have σ ( E( + ( ( (3.3 To faciliae he algebra ivolved i differeiaig (.6 oce ad he wice ad

4 4 Samuel Sidayigaya e al seig s afer each differeiaio, we ca approach he problem sep by sep i a similar maer of [] as follows: Le, f s s s e μ μ + + (3.4 ( ( ( ( ( ( μ ( μ ( ( g s s s e μ The, f (3.5 ( ( f ( s f ( e μ + f ( s f ( g ( g ( s g ( ( e ( μ + μ + ( ( g s g Le f ( s ψ ( s, (3.6 g( s κ ( s, s ( (3.7 g( s r( s, f ( s (3.8 The, ψ, ( ψ (, μ ( ( ψ, κ, ( ( e μ ( ( κ (, e μ ( ( (, e κ μ r(, (, ( r e μ ( ( ( r (, e μ Rewriig (.6 i erms of (3.6, (3.7 ad (3.8 usig (3.4 ad (3.5 we have

5 The Populaio Mea ad is Variace i he Presece of Geocide 5 φ( s, ψ κ ϒr (3.9 Havig i mid ha, if a ime, we kow ha he populaio has a disribuio P ϒ, i,,,... (3.a i i ( i ϒ i (3.b (because ( P is a disribuio, i mea (3.c Variace σ (3.d Upo usig (3.3 we have ha ϒ (3. ( ϒ σ + (3. We ca ow proceed o ge ( ad σ ( by differeiaig (3.9 oce ad he wice ad seig s i order o ge he mea ad variace respecively. (i Derivaio of he Mea φ μ + ψ ψκ ϒ r + ψ κ ϒ r + ψ κ ϒr r (3.3 s Usig Equaio (3.3 we obai, ν ( μ ( μ ( μ ( ( e + ( e + e μ ( ( e + ( e (3.4 Hece, ( ( ( e + ( e ; which is he explici soluio for ime-varyig rajecory of he mea for he BDIE process wih caasrophe. (ii Derivaio of he Variace To compue he variace, we ake he secod derivaive of (3.3 ad replace s.the by separaig he composig erms ad ake he secod derivaive separaely, we have φa ψ ψκ ϒr

6 6 Samuel Sidayigaya e al φ B ψ κ ϒr s φc ψ κ ϒ r r ad φ φa φb φc + +. I deail, we ge ν μ + ψ ( ψ κ r ψ ψκκ r φ ϒ + μ ϒ + A + ψ ψκ ϒr r ν ( ( μ ν ( ( e ( e e ( + ( ( ( + ( e ( e ( + ν + ν ( ν ( ν ( ( ( e + ( e ( e e ( ( ( ν ( + ( e μ + ψ ψ κ r ψ κ ( κ r ϒ + φ μ ϒ B + s + ψ κ κ ϒ r + ψ κ κ ϒr r ( ( μ ( ( e ( + ( e ( μ ( ( ( ( ( ( e ( ( e ( e μ ( μ ( ( ( e + ( e + ( e ( ( ( ( ( ( e ( e ( e (3.4a (3.4b

7 The Populaio Mea ad is Variace i he Presece of Geocide 7 μ + ψ ψκ r r ψ κ κ r r ϒ + μ ϒ φ C + μ + ( ψ κ ( ( + ϒ r r + ψ κ ϒr r ν ( ( ( ( ( e ( + ( e ( e ( + e μ μ ( ( + ( + ( σ + ( e + ( e ( ( ν ( ( ( ( ( e ( e + ( σ + ( e ( e ( + {( + + ( μ + ( ( e } Recall ha he variace is give as ( ( E ( E ( σ φ E( s or σ E ( + ( ( φ. The, σ + ( s bu (3.4c (3.4d hece, by assemblig (3, 4a, (3.4bad (3.4c ad apply io (3.4d we obai ν + ν ( ν ( ν ( ( ( e + ( e ( e ( e ( ( ν ( + ( e ( μ ( ( ( e + ( e + ( e ( μ ( μ + ( μ ( μ ( ( e ( e ( e σ ( ν ( ( ( ( ( e ( e + ( σ + ( e ( e + ( + {( + + ( ( ( e } ( ( ( ( ( e ( e ( e ( ( μ ( e ( ( ( μ + ( e ( e Ad fially, upo simplificaio we ge ( νμ ( + ( ( σ ( σ e + ( e + e ( e (3.5 ( Equaio (3.5 is he explici soluio for ime-varyig rajecory of he variace for he BDIE processes wih geocide or caasrophe. I he case ha, Equaios

8 8 Samuel Sidayigaya e al (3.4 ad (3.5 will ur up o BDIE wihou caasrophe ad will have a close relaioship wih he resuls obaied by [] ad []. SUMMARY I his work, we have esablished he parial differeial equaio (PDEfor he BDIE processes wih geocide/caasrophe from which he geeral form was obaied usig he probabiliy geeraig fucio (PGF.Upo derivaio of he PGF, he mea ad Variace fucios for BDIE processes wih geocide/caasrophe were effecively deermied. REFERENCES []. Graia ad A.Bahar, Asochasic Differeial Equaios Model for Liear Growh Birh ad Deah Processes wih Immigraio ad Emigraio, UTM ceer for Idusrial & Applied Mahemaics, AIP Publishig, 5. []. W.M.Gez, Opimal Corol of a Birh-ad-Deah Process Populaio Model, Mahemaical Bioscieces 3, 87-, 975 [3]. Radall J.Swif, Trasie Probabiliies for a Simple Birh-Deah-Immigraio Process uder he ifluece of oal Caasrophes, UJMMS 5, , [4]. Di Crescezo, V.Gioro, A.G.Nbile, L.M.Ricciardi, O he firs-visi-ime problem for Birh ad Deah processes wih caasrophes, Queuig Sysems 43, [5]. S.Sidayigaya, NyogesaL. Keedy, Applicaio of he Birh-Deah Processes for geeraig he Populaio Esimaes, Ieraioal Joural of Saisics ad Aalysis/IJSA, 37-49, 5. [6]. Michael L.Gree, The Immigraio-Emigraio wih Caasrophe Model, Califoria Sae Polyechic Uiversiy, Pomoa, CA 9768

The Moment Approximation of the First Passage Time for the Birth Death Diffusion Process with Immigraton to a Moving Linear Barrier

The Moment Approximation of the First Passage Time for the Birth Death Diffusion Process with Immigraton to a Moving Linear Barrier America Joural of Applied Mahemaics ad Saisics, 015, Vol. 3, No. 5, 184-189 Available olie a hp://pubs.sciepub.com/ajams/3/5/ Sciece ad Educaio Publishig DOI:10.1691/ajams-3-5- The Mome Approximaio of