BAYESIAN ESTIMATION METHOD FOR PARAMETER OF EPIDEMIC SIR REED-FROST MODEL. Puji Kurniawan M

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1 BAYESAN ESTMATON METHOD FOR PARAMETER OF EPDEMC SR REED-FROST MODEL Puji Kuriawa M447 ABSTRACT. fecious diseases is a impora healh problem i he mos of couries, belogig o doesia. Some of ifecious diseases ca spread i he populaio ad cause epidemic. Someoe ca be ifeced if appear adequae coac wih ifeced. Oe of probabilisic model o describe spread of diseases is epidemic SR ( Suscepible- feced-recovery ) Reed-Fros model. Rae of ifecio i he SR Reed-Fros model be defied as probabiliy caac a ifeced ad suspecible (p). his research, he Bayesia mehod is applied o esimae probabiliy rae of ifecio. The esimaio wih Bayesia mehod eed prior disribuio ad likehood fucio. Prior disribuio ad likelihood fucio are used o deermie poserior disribuio. The poserior disribuio is used o deermie probabiliy rae of ifecio. As he resul, his research gives coclusio ha probabiliy rae of ifecio is p=-q, wih ( α ) E[q]= qˆ = ( α + β ) qˆ is expeced values of he poserior disribuio. Key words : SR Reed-Fros, Bayesia, Prior, Poserior, Likelihood.. NTRODUCTON Avia ifluece, SARS, HV/ADS, degue, malaria, measles are some ifecsious diseases which sill become impora issues wih high prevalece ad cause epidemic. Oe of he effor is o overcome hose diseases i.e. sudy he way how he spread diseases happes, herefore i ca be coduced is preve ad reame. Accordig o Hehcoe (), oe of mahemaical model, which describes he spread of diseases i.e. SR (Suscepible-feced-Recovery) model. Suscepible (S) is a idividual has a poeial o be ifeced he disease. feced () is a idividual which is ifeced. Recovery (R) is a idividual which is ifeced he diseases he becomig recovered. Rae of spread i he SR Reed- Fros model is defied as probabiliy coac bewee suspeced ad ifeced. SR Reed-Fros model assume he spread of diseases happeed i closed ad

2 homogeeous populaio. The rae of spread depeds o coac probabily (p). The paer of predicio diseases spread, whe maximum ifecio ad he process of diseases ifecio eded ca be kow based o parameer value which is applied i he model. The esimaio of parameer ca use Bayesia mehod. Esimaio parameer i he Bayesia mehod used daa sample ad prior iformaio. Prior iformaio ca be used o improve predicio qualiy or parameer predicio.. SR REED-FROST MODEL To formulae he model, we mus deermie he variables. Noaio is period, measured i icubaio ime. The variables are S is umber of suscepible i he populaio, is umber of ifeced i he populaio, ad R is umber of recovery i he populaio. Accordig o Term (7) ad Fie (4), here are some assumpios of he SR Reed-Fros model. ifecio is spread direcly from ifeced o suscepible oly by adequae coac, he ifeced will become recovery,. each idividual has a equal coac probabiliy, 3. he populaio is homogeeous ad closed, 4. ifecios occur idepede. The umber of ifeced a ime will become recovery i he ime +. Accordig o Term (7), if he q probabiliy a idividual o appear adequae coac he each suscepible has a probabiliy q because ifecios occur idepedely. he ex period some of suscepibles have adequae coac wih ifeced will become ifeced. The ohers suscepible will remai i suscepible group. This eve be biomial disribuio, so he suspecible has probabiliy of avoidig ifecio give by S + S + P( S+ S, ) = ( q ) q. S+ f he ifecious occur durig m period he he umber of ifeced each period will give as a pidemic vecor (,,.., m ), Term (7). The ifecious each

3 period occur idepedely, so he joi probabiliy become ifeced for m period is P(,,..., m ) = P( m m, S m )... P(, S ) P( ) () wih P( ) =. Accordig Fie(4), Reed-Fros model followig he chai of biomial, so he umber of suscepible i he ex period is expeced value for radom variable S +, ad give by [ S + ] = Sˆ + Sq. () E = The ew ifeced i he ex period is some suscepible who have adequae coac wih ifeced.he umber of ifeced i he ex period give by (3) + = S S + Based o assumpio ha probabiliy ifeced will become recovery is oe, so he umber of recovery i he ex period is give by R + + = R. (4) From equaios (.), (.3),ad (.4), Reed-Fros epidemic model is obaied S = S q + + = S S + R = R + + where S >, >, R =, q. (5) 3. PROR DSTRBUTON The spread of diseases paer i he model of SR Reed-Fros epidemic is a case which Beroulli disribued, Fie(977). The deermiaio of prior disribuio is based o cojugae prior. Because of he spread of diseases is Beroulli disribued, i is discree disribuio, while q is parameer which is disribued o coiue (q rus all dos i he ad ierval). i eeds cojugae from Beroulli disribuio i he coiue form. is bea disribuio, Soejoei ad Soebaar(988). f he daa sample follows Beroulli process ad is likelihood fucio are biomial disribuio, herefore he parameer esimaio uses Bayesia mehod eeds bea disribuio, cojugae from Beroulli 3

4 disribuio. Based o he iformaio, prior disribuio from SR Reed-Fros epidemic model is bea disribuio. Radom variable of q is disribued bea wih α ad β, oaed q ~ Bea( β ) i.e. where p ( q, β ) = q B( β ) α ( q) β Γ( α) + Γ( β ) B ( β ) =, cosa. α >, β >, ad q. Γ( α + β ) α Expeced value of bea disribuio is E [q] = α + β 4. LKELHOOD FUNCTON FOR HOUSEHOLD OF SZE 3 Le a household of size 3, he he possible pahs i epidemic vecor are (), (, ), (,, ) ad (, ). Usig he equaio (.) wih p = -q, we ge probabiliy each vecor are give o Table.. Vecor () (,) (,,) (,) Table. Probabiliy each vecor probabiliy q pq p q p f we had a larger populaio, le observed umber of household each vecor oaed by m,,o,p wih m observed umber of chais of form (), observed umber of chais of form (,), o observed umber of chais of form (,,), ad p observed umber of chais of form (,). Based o probabiliy each vecor, if we have household i populaio he he expeced umber of household i he populaio give by E(i) = * probabiliy of vecor i occurig wih i is epidemic vecor, he he likelihood fucio ca be cosruced by L )] m o p = E[()] E[(,)] E[(,,)] E[(, (6) 4

5 Le j be he umber of households where a epidemic of fial size j occurs where j=,,, ad le (,,) be expressed as. Nohig ha = (,,) + (, ) so = (,,) (, ) ad so hece = (, ), usig Equaio (3.), his gives + + he full codiio likelihood fucio for q be ( q;,,, ) = ( L q q ) 5. POSTEROR DSTRBUTON Accordig o Soejoei ad Soebaar (988), poserior disribuio is equivale wih muliplicaio likelihood fucio ad prior disribuio, so from prior disribuio as kow ad likelihood fucio obai, poserior disribuio for q,,, ) give by ( p q,,, ) L q;,,, ) p( ) ( ( q B( β ) α β ( q ( q) ) x q ( q) α + + β B( β ) q ( q) ( α, + + β ) Bea wih parameer ( α ) a ad b = + + ) = ( β Accordig o Bai ad Egelhard (99), esimaio of q cosiue expeced values from poserior disribuio, give by cosider,,, kow. E (q) = qˆ = a = (7) a + b ( α ) ( α + β ) 5

6 6. EXAMPLE Refer o ivesigaio io measles i Rhode slad, O Neill ad Robers (999). There are oly usig he daa o household of size 3. All members of he populaio are aged bewee 7 mohs ad years, he daa have a populaio of size = 334 household. The resul shows i Tabel.. Table.. Number household for each vecor Type of chai Expeced umber Household umber ( ) (, ) q 34 pq 5 (,, ) p q 39 (, ) p 36 Based o Tabel., = 34, = 5, = = 75, = 36. Usig equaio (7) esimaio parameer of q obaied qˆ =. 5, so pˆ = Based o qˆ spread of measeles i Rhode islad occur wih probabiliy coac pˆ = Wih equaios (5), SR Reed-Fros epidemic model is obaied S+ = (.5) S, + = S+ S, ad R = R +. Each household i his ivesigaio of size 3, so umber of idividu i he populaio is. Predicio spread of measeles usig SR Reed Fros model wih = shows i Tabel.3. + Tabel.3. Number spread of diseases Periode S R

7 Tabel 4.3 shows ha spread predicio measeles every period. Maximum ifecio happes i he secod periode, while fial epidemic eds afer he fourh period. Compariso umber of ifeced each period for differe ca be see i he Figure.. Number of fec feced wih o= feced wih o= feced wih o= Period Figure.. The spread of ifeced 7. PROBABLTY VALUE FOR DFFERENT Spread of diseases i SR Reed-Fros model deped o firs ifeced ad probabiliy value. Probabiliy coac obaied based o equaio 6. Used he daa ivesigaio io measeles i Rhode islad wih umber of husehold =334, ca be made simulaio for umber household wih vecor (,,), oaed, are difere. Compariso probabiliy value of coac show i abel 4. Table.4. Probabiliy value for differe qˆ pˆ

8 Table 4 show ha more ad more spread of diseases wih vekor (,,) show he smaller he probabiliy coac bewe ifeced wih suspecible. 8. CONCLUTON. The esimaio of SR Reed-Fros model uses Bayesia mehod wih household of size 3 is obaied is equaio i.e. qˆ = wih,,, is kow. ( α ) ( α + β ). Based o observaio umeric wih householf of size 3, spread of diseases deped o firs ifeced ad probabiliy coac i populaio. larges probabiliy coac occur if he umber of household ha epidemic smalles. REFERENCES Bai, L. J. ad M. Egelhard, roducio o probabiliy ad mahemaical Saisics, ed., Duxbury Press, Califoria, 99. Fie, P. E. M., A commeary o he Reed-Fros epidemic model. America joural of Epidemiology 6 (977), 87-. Hehcoe, H. W., The mahemaics of ifecious diseases, 4 (), Philip O Neil, P. ad Gareh Robers. M, Bayesia ferece of fo Parially observed Sochasic Epidemics, J.R. Sais. Soc A (999), 6. par. pp-9. Soejoei, Z. da Soebaar, feresi bayesia, Uiversias Terbuka, Karuika, Jakara, 988. Term, H., Usig probabilisic models o ifer ifecio raes i viral oubreaks. hp : //mahge.sas.ox.ac.uk/bioiformaics/ifecio_raes.pdf (7). 8