Approximation for Collective Epidemic Model

Transcription

1 Advaces Appled Mathematcal Bosceces. ISSN Volume 5, Number 2 (2014), pp Iteratoal Research Publcato House Approxmato for Collectve Epdemc Model Dr.Mrs.T.Vasath ad A.Mart Departmet of Mathematcs, ADM College, Nagapattum Taml Nadu, Ida [Emal:svhadvt@gmal.com] Departmet of Mathematcs, Avvayar Govermet College for Wome Karaal , Puducherry, Ida [Emal:mart292033@gmal.com] Abstract The collectve epdemc model s a qute flexble that descrbes the spread of a fectous dsease of the Susceptble-Ifected-Removed type a closed populato. I the preset paper a ecessary ad suffcet codto s derved that geerates the wea covergece of the law of ths varable to a mxed posso dstrbuto whe the tal susceptble populato teds to fty provded that the out brea s severe a certa sese. Keywords: Collectve epdemc model,, Mxed Posso law, Brachg Process, PLT 2000 Mathematcs Subject Classfcato Numbers: 60G20 Itroducto I the mathematcal theory of epdemcs, a mportat class of models s cocered wth fectous dsease of the S(Susceptble)-I(Ifected)-R(removed) type. A closed populato cotas tally susceptble dvduals ad m fected dvduals. All the fectve tal or subsequet are supposed to behave depedetly.each of them stays fectous durg a certa perod of tme of radom legth. After that perod t s mmue a permaet way ad may thus be regarded as removed from the fectous process. Whle fected a dvdual s able to trasmt the fecto to others.a susceptble f ever cotacted by a fectve s fected ad becomes mmedately fectous. Cosder ay fxed subject of susceptble amog the tal oes 1. It s assumed that ay fxed fectve does ot trasmt the fecto durg ts whole fectous perod wth such a group of susceptble wth a probablty q() that depeds oly o the sze of the group(ad possbly o ).These q() s are fxed ad costtute the parameters of the model. Paper Code: AAMB

2 98 Dr.Mrs.T.Vasath ad A.Mart The collectve epdemc termates at some fte tme A as soo as there are o more fectve preset the populato. The the ultmate umber of susceptble survvg the dsease deoted by S(A) say. For smple fecto the parameters q() ca be wrtte as fuctos of a small umber of epdemc compoets. Ay gve par of dvduals s ow assumed to mae cotacts at the pots of a Posso process wth rate all these processes beg depedet. Moreover ay fectve fectous durg a perod of radom legth D all the D s beg..d ad dstrbuted as the varable D say. We the get q( ) E[exp( D)], 1 I partcular the so called geeral epdemc correspods to the specal case where D s expoetally dstrbuted wth parameter, here thus q( ) / ( ). Moreover aslog as S(A) s cocered the model covers aother stadard model ow as the Reed-Frost epdemc whch s obtaed by Supposg that D s equal to some costat d; ths yelds q( ) q wthq exp( d). For the collectve model the exact dstrbuto of S(A) was obtaed ad studed by Pcard ad Lefevre(1990). Mart-Lof(1986)establshed the exstece of a threshold pheomeo together wth a brachg or Gaussa lmt approxmato. The problem examed the preset paper s the alteratve approxmato of S(A) by a mxed Posso law. Also Daels(1967) showed for the geeral epdemc uder some codtos S(A) ca have a Posso-le behavor. Later Ball ad Barbour(1990) appled the Ste- Che methodology to derve a Posso approxmato wth a order of magtude for the model of Mart-Lof(1986). Recetly Lefevre ad Utev(1995) obtaed a ecessary ad suffcet codto that guaratees the valdty of such a Posso lmt for thegeeralzed epdemc. Our purpose here s to go further ths subject by dervg ow a mxed Posso approxmato for the fal state of the collectve epdemc. Brachg process Defto1 Let the radom varables x 0, x 1, x 3,..., x... deote the sze of (or the umber of th st d objects ) the o,1,2... geerato respectvely. Let the probablty that a object(rrespectve of the geerato to whch t belogs) geerates smlar objects be deoted by p where p 0, 0,1, 2,3... p 1. The sequece{ x, 0,1,2,3...} costtutes a Galto-Watso brachg process wth offsprg dstrbuto { p }. Propertes of geeratg fuctos of Brachg process. We have

3 Approxmato for Collectve Epdemc Model 99 x 1 x r where r are..d varables wth dstrbuto{ p }. r1 r be the p.g.f of { r } Let P( s) Pr{ } S P S ad let P ( s) Pr{ X } s, 0,1, 2... be the p.g.f of { x }. We assume that x0 1, clearly P0 ( s) S ad P1 ( s) P( s). We start by assocatg wth the collectve epdemc a equvalet model of brachg type. By equvalet we mea that the fal susceptble state S( ) ca be obtaed as the frst-crossg level of some decreasg brachg model a learly decreasg barrer. { z, z,... z } be a famly of exchageable Bereoull radom varables Let.1,2, wth parameters q( ),1. For t=1, 2, 3,. Let { z,1 (t),...z, ( t )} be..d copes of that famly. The Marov Cha { x ( t), t 0} defed by x (0) ad x ( t) z ( t), t 1. ( t1), 1 Ths cha s decreasg overtme.its traset dstrbuto s gve below. Lemma1 Foe each t 1 x( t) t E q( t),1 K e x (t) s dstrbuted as the sum of exchageable Beroull radom varables wth t parametersq (),1. Proof ( j) Let z deote the dcator of the evet ( susceptble escapes fecto from, fectve j durg the epdemc spread). From the defto of the model we see that ( j) ( j ) each vector { z,... z } s a famly of exchageable Beroull radom,1, varables ad all the vectors are..d copes of the famly { z,1,... z, } say wth q ( ) P(z z... z 1),1 (1),1,2, At ths pot we emphasze that the probabltes (1) correspods to the parameters usually troduced whe costructg the jot law of exchageable Beroull radom varables { z,1,... z, }. I partcular cosder the partal sums x, u z,1... z, u 1 u. We have the dstrbuto x, u whe specfed by ts x, u u bomal momets s gve by E q( ) 1 u. x Hece for t=1 the

4 100 Dr.Mrs.T.Vasath ad A.Mart statemet of lemma s true. For t 2 we obta that X ( t) X (t) X ( t 1) E E E / X (t1) E q ( ) Whch leads to the proof of the lemma by ducto. To mae the l wth the epdemc model cosder the decreasg le m 1 ad let T be the frst tme whe the brachg model crosses the le T f{ t 0: x ( t) m t} Clearly 1 T m. Proposto 1 The process { x( t), t 0} s a decreasg Marov cha wth X (t) MB(, Q, s), t 1 (2) d s1 t At tme T, the state X ( T ) has the same law as the varable s( ) whch s provded by the followg relato X ( T ) ( ) { m / [ ( )] m E E Q X T } 1 (3) Proof The frst asserto s obvous from X ( t1) X ( t) Z, t 1 1 ( t ), The law (2) for X ( t ) s obtaed from d,t X ( t) MB( X ( t 1), Q ), t 1by ducto ad well ow fact that MB( B( l, u), v) B( l, uv) from (2) we get 1 that d X ( t) t E E( Q ), t 0 Whch shows that the process X ( t) t E( Q ), t 0 Forms a martgale. Now from T f t : t X ( t) m T s a Marov tme ad applyg the optoal stoppg theorem that yelds the relato (3). These costtutes a tragular set of lear equatos the ultmate state P S ( T ), 1. The probablty for =0 follows. Fally we probabltes

5 Approxmato for Collectve Epdemc Model 101 ote that the system (3) s detcal wth relato provdg the law of S( ). Ths leads to the followg proposto. Proposto 2 The brachg model tersects the barrer at T. That s X (T ) m T (4). Furthermore X ( T ) has the same dstrbuto as S( ) whch s provded by relato S( ) m S ( ) E / q( ),1 Ths represetato has a smple terpretato. Returg to the epdemc model we mae a chage of tme scale ad we defe a ew artfcal tme t=1, 2,.as the cumulatve umber of removals the course of real tme. Put X (0) ad let X ( t), t 1 deote the umber of dvduals that escape fecto cotacts wth the frst t fectve removed. Moreover put I (0) m ad let I ( t), t 1 be the umber of fected dvduals stll preset after th r removal.. By costructo t X ( t) I( t) m. the Thus the frst tme whe there are o more fectve preset the populato s qute detcal wth T ad (4) does ot hold true. We the feel tutvely that X ( T ) ad S ( ) are equdstrbuted. Cocluso Hece we coclude that X ( T ) does satsfy a Posso Lmt Theorem. Refereces [1] Baley.N.T.J., 1975 The Mathematcal Theory of Ifectous Dseases ad ts Applcatos, Grff, Lodo. [2] Daels, H.E, The dstrbuto of the total sze of a epdemc.proc. Ffth Bereley Symp.Math.Stat. probab.iv, [3] Kedall, W.S Cotrbuto to the Dsscusso of Epdemc Models ad data by D.Mollso, V.Isham ad B.T.Grefell. J.Roy.Statst. Soc. Ser. A [4] Mart-Lof, A Symmetrc Samplg Procedures, geeral epdemc processes ad ther threshold lmt theorems. J. Appl. Probab [5] Scala-Tomba, G Asymptotc fal sze dstrbuto for some chabomal processes. Adv. I Appl. Prob [6] Star, J., Iaell, P. ad Baget, S, 2001 A olear dyamcs perspectve of momet closure for stochastc proceses. Nolear Aalyss 47,

6 102 Dr.Mrs.T.Vasath ad A.Mart