Hybrid model analysis on host factor of epidemics involving fuzzy Markovian chain

Transcription

1 Theoreical Mahemaics & Applicaios, vol., o.2, 20, 7-27 ISSN: (pri), (olie) Sciepress Ld, 20 Hybrid model aalysis o hos facor of epidemics ivolvig fuzzy Markovia chai B. Aruraj Absrac I his sudy, a hybrid model aalysis o hos facor of epidemics ivolvig fuzzy markovia chai is preseed. This hybrid model cosiss of sochasic approach ivolvig markovia chai, fuzzy exper sysem, simulaio approach ad fuzzy se heory i order o obai he sysem respose i a more realisic maer. A sochasic model of he wo sae markovia-chais is applied o cerai epidemics by age group. A age specific member rae ad fuzzy markovia risk rae are defied ad used as ovel measure idices o prioriize age groups. A fuzzy exper sysem is used i medical diagosis for various classificaios. For simulaig he process, we have ake he sympom daa for he diseases from he experieced medical expers. The compuaio of max-mi composiio is assumed o describe he sae of he paies (age wise) i erms of diagosis as a fuzzy se characerized by is membership value. The aalysis of age specific diagosis of cerai diseases is esseial o provide he daa for he plaig of healh services ad o seig up of prioriies amog hose services. The combiaio of sochasic approach ad he fuzzy logic eables our compuaioal simulaio o much more faihfully porray he pheomea uder sudy. Mahemaics Subjec Classificaio: 0E72, 94D05 Deparme of Mahemaics, A.V.V.M.Sri Pushpam College (Auoomous), Poodi 6 50, Thajavur (DT), Tamil Nadu, Idia. Aricle Ifo: eceived : March, 20. evised : May 9, 20 Published olie : Jue 25, 20

2 8 Hybrid model aalysis o hos facor of epidemics Keywords: Fuzzy Se, Compleme of Fuzzy Se, Fuzzy elaio, Max-Mi, Composiio of Fuzzy Se, Fuzzy esricio, elaioal Assigme Equaio, Possibiliy Disribuio Fucio, Fuzzy Markov Chai, Age specific member rae, Epidemics Iroducio Mahemaical modellig aalysis is used exesively i he field of epidemiology of diseases. May mahemaical models are used usig pure iiial ad boudary value codiios i solvig he problems. Mahemaical modellig aalysis, i spie of is advaages is o well suied o hadle he uceraiies associaed wih he problem due o lack of clear cu ierpreaios of he parameers. Thus here is a eed o improve he exisig mahemaical model of he problem i order o brig is behaviour as close as possible o he acual behaviour of he sysem. I requires he eed o sudy he effec of sysem respose i he presece of uceraiies of he problem. The review of lieraure reveals ha he deermiisic approaches ad sochasic approaches are used o model he respose of he problem. Fuzzy se heory is used o measure uceraiies. Exper kowledge i medicie is applied usig he echique of fuzzy logic. CADIAG 2 for ieral medicie, EMEGE for ches pai aalysis, EXPET for ophhalmology, MYCIN foe disease diagosis ad reame was developed. The above sudies were cofied o he sudies of he effecs of uceraiy aloe. A he same ime, he realisic effecs of sysem respose were o cosidered. There is o exisig lieraure, which icorporaes he sysem respose ad fuzzy respose i he aalysis of huma healh problem. This requires he redefiiio ad exesio of he mahemaical modellig. This is a hybrid model aalysis which combies he mahemaical models ad fuzzy se heory. I his paper, we cosider he hos facor of epidemics for aalysis. The causaive facors of disease are classified as age, hos ad evirome. The ieracio of hese facors is required o iiiae he disease process i ma. The hos facors iclude age, sex, ehiciy ad may biological facors. Here, were cosidered he hos facor o age specificaio. Age is srogly relaed o disease ha ay oher sigle hos facor. Cerai diseases are more freque ha cerai

3 B. Aruraj 9 age groups. If he aack rae of a epidemic is uiform i all he age groups i implies ha all he age groups are equally suscepible ad here was o preimmuiy. May chroic ad degeeraive diseases show a progressive icrease i prevalece wih advacig age. This may reflec a persise ad cumulaive exposure o causaive age or risk facors. I some chroic diseases he child prevalece are less i umber or disappearace represes ha he diseases are goig o pass from he exisece ad vice-versa o be i high umber. Epidemic refers o he uusual occurrece of disease clearly i excess of expeced occurrece of a commuiy. Periodical flucuaios are he commoes characerisics of may epidemics. The seasoal variaios of disease occurrece may be relaed o eviromeal codiios such as emperaure, humidiy, raifall, overcrowdig ad life cycle of vecors which direcly or idirecly favour he rasmissio. I his sudy, we choose cerai diseases ha are saisfyig he above codiios. The aalysis of age specific diagosis is esseial o provide he daa for he plaig of healh services ad o seig up of prioriies amog hose services. The applicaio based o he fuzzy Markovia chai ad he rasmissio membership fucio of membership i he 2 x 2 marix is calculaed. I addiio a procedure is proposed for he diagosis, which is based o he max-mi composiio of fuzzy relaio. Moreover, a calculaio of a idex for medical diagosis sysem is made. The objecive of he hybrid model aalysis is o predic he behaviour of he problem accuraely. Medical aalysis have caused huma suppor, cos effecive ad caused diagosis accuracy o icrease. Neverheless, he epidemiological surveillace is expesive, ime cosumig ad require higher admiisraive procedures. I his sudy, we developed a rule based fuzzy sysem ha uses exper kowledge ad oher daa ha mimic a real oucome. The aricle is orgaised as follows. I secio 2, we briefly oulie he defiiio of possibiliy disribuio fucio followed by fuzzy exper sysem desigig i secio. I secio 4, we prese he applicaio of hos facors of epidemics, umerical simulaio ad he resul will follow i secio 5. A he ed, some coclusios are highlighed i his paper.

4 20 Hybrid model aalysis o hos facor of epidemics 2 Basic Defiiios We use he sochasic approach o modellig o hos facor of epidemics o faciliae age specific diagosis aalysis. Usig his mehod, we esimae he possibiliy of oucomes based o available daa. Here, we desiged he defiiio of possibiliy disribuio fucio. Based o he defiiio of possibiliy disribuio fucio iroduced, he fuzzy Markov chai was framed. Defiiio 2. (Possibiliy Disribuio Fucio) Le F be a fuzzy se i a uiverse of discourse U, which is characerized by is membership fucio µ ( u), which is ierpreed as he compaibiliy of u U wih he cocep labeled F F. Le X be a variable akig values i U ad F ac as a fuzzy resricio, ( X ), associaed wih X. The he proposiio X is F, which raslaes io ( X ) = F associaes a fuzzy eve, π, wih X which is posulaed o be equal o ( X ). The possibiliy disribuio fucio, π ( u ), characerizig he fuzzy eve π is defied o be umerically equal o he membership fucio µ ( u) of F, ha is, π = ˆ µ F, ha is, π = ˆ ( u ). The symbol =ˆ will always sad for deoes or is defied o be. π F Defiiio 2.2 (Fiie - Fuzzy Ses) A (fiie) fuzzy se (a fuzzy eve, a fuzzy relaio or a fuzzy resricio, ) π, o S, is characerized by possibiliy disribuio fucio π, π : S [0, ], ha akes o a fiie umber of possible fuzzy ses will be deoed by π = {, = 0,, 2,... }. The se of all fuzzy se o S is deoed by F ( S ). π Le { π, = 0,, 2,... } be a sequece of radom fuzzy ses (or a discree fuzzy parameer sochasic process). Le he possible oucomes of π be i ( i = 0,, 2,... ), where he umber of oucomes may be fiie (say, m) or deumerable. The possible values of π cosiue a se S = {0,, 2,... }, ad ha he process has he fuzzy sae space S. Uless oherwise saed, by fuzzy sae

5 B. Aruraj 2 space of a fuzzy Markov Chai, we shall imply discree fuzzy sae space (havig a fiie or a couably ifiie umber of elemes); i could be F ( S ) = { 0,, 2,... }. Defiiio 2. (Fuzzy Markov Chai) The fuzzy sochasic process { π, = 0,, 2,... } is called a fuzzy Markov Chai, if, for i, j, i, i,..., 0 i F ( S ). π { x + = j x = i, x = i,..., x = π { x = π i j + = i, x 0 = i0 } = j x = i } ( say) wheever he firs member is defied. Fuzzy Exper Sysem Desigig Exper sysem i medicie cosiss of medical exper ad fuzzy sysem. We apply he fuzzy exper sysem i medical diagosis. Fuzzy exper sysems ad various iellige echiques help for classificaio. The followig classificaios are relaed o medical diagosis: sympoms ad fidigs, he medical kowledge, disease ad diagosis. We use he followig symbols o skech he above classificaio. S = ( s,..., sm ) : se of sympoms. D = ( d,..., d ) : se of diseases. T = (,..., q ) : se of age groups. All s i, d j, l are fuzzy ses characerized by heir respecive membership values. The membership values of hese relaioships [ T, S ] ad [ S, D ] covered io fuzzy relaio [, 2 ] are meioed i wo aspecs. (i) Occurrece of s i i case d j (ifeced- I, I 2 ), (ii) No-occurrece of s i i case d j (uifeced- U, U 2 ).

6 22 Hybrid model aalysis o hos facor of epidemics This leads o he defiiio of fuzzy relaio. Assume wo fuzzy saes such as occurrece ( x = ) ad o-occurrece ( x = 0 ). The rasiio of possibiliy disribuio fucios of fuzzy Markov process i a 2 2 marix is cosidered: Π = π π 00 0 π π 0 All elemes i marix Π are o-egaive umbers, ad π =, i = 0,. j = 0 For occurred value o re-occurred, we ge π = 0 0, π = for π ( π. π ( ) give by π ) π { x x } π ( x ) π ( x ). 0 ) = Thus he -sep rasiio process ca be wrie as i j ( 00 = = + 00 ( 00 + k = ( ) π ) = π ( k) = π ( x ) π ( x ). From he origial daa of he proposed age disribuio, mulisep rasiio process is calculaed. Here is he sae of age group, ad we ge he ew agespecific member rae (ASM) bewee he age group o + is give by 00 π ( x ) ( ( ) ( )) ( ) π x π x + ( ASM) = 00π ( x ) 0 ( ) = π. π ( x ) The membership fucios for hese (ifeced- I, I 2 ad uifeced- U, U 2 ) wo fuzzy saes are defied o be µ ( x) = max{mi ( µ ( x), µ ( x) )}, x X X The s i, d j I s S I I2 µ ( y) = max{mi ( µ ( y), ( y) )}, y Y U µ s S U U 2 occurrece relaioships are acquired empirically from medical expers usig he membership values. Oher relaioships such as age\sympom, sympom\disease are also defied as fuzzy ses. Possibiliy ierpreaios of relaios (max-mi) are used. Give a age-group\disease relaioships yield fuzzy diagosic idicaios ha are basis for esablishig occurrece ad ooccurrece diagosis. Fially, wo differe relaios are calculaed by meas of fuzzy relaio (fuzzy composiio). I = I I = {( x, max{mi ( µ ( x), µ ( x) )} x X } 2 X s S I I2 U = U U 2 = {( y, max{mi ( µ ( y), µ ( y) )} y Y Y } s S U U 2 Y

7 B. Aruraj 2 To calculae he age specific member rae for he above fuzzy ses, we use he followig formulas. 00 µ ( I ) = I ( x ) ( µ µ I I ( x ( x ) µ ) I ( x + )) ( U ) 00 µ = U ( y ) ( µ U µ U ( y ( y ) µ ) U ( y + )) 4 Applicaio o Hos Facor of Epidemics Applyig he fuzzy exper sysem o hos facor of epidemics is compoud of exper ad fuzzy sysem. The microbial ages ha cause epidemics are umerous ad iclude viruses, parasies ad ohers. The daa for he developed sysem were ake from he lieraure ad wih he help of medical expers. We cosider he followig se of epidemics (commo disease) D as follows: d - Lepospirosis d - Malaria 2 d - Measles d - Iflueza 4 d 5 - abies d 6 - Filaria d 7 - Degue The experieced medical expers have ake he sympom daa for he diseases age wise. We cosider he followig se S of commo sympoms: s - Sore hroa s 2 - Lymph edema s - Head ache s 4 - uig ose s 5 - Fever s 6 - Cough s 7 - Chills We cosider he followig T of commo age group T = ( 0,,..., 7 ). The membership values of hese relaioships [ T, S ] ad [ S, D ] are covered io fuzzy relaios ad wo marices have bee formed. The wo marices are

8 24 Hybrid model aalysis o hos facor of epidemics Age, sympom cofirmaio relaioship-marix, ] [ Sympom, disease cofirmaio relaioship-marix, ] The above marices cosis of simulaio daa obaied from exper are abulaed (Table ad Table 2). The he compuaio of max-mi composiio = is assumed o describe he sae of paie (age-wise) i erms of 2 diagosis as a fuzzy subse of µ [ 2 T D characerized by is membership fucio (, d ) T (, d ) = max{mi( µ (, s ), µ ( s, d ))}, s S 2 The possibiliy disribuio fucio of he compleme of fuzzy se, π (, d) is deoed by π (, d) = π (, d ). The compleme of fuzzy se is abulaed i Table. From, he fuzzy Markov risk rae (FM) is se up as a ew measure of ifecio risk rae ad abulaed i Table 4. I varies as he produc of proporio of becomig ifecio a a give age i wo age/ime ( 2) seps, π (, d). Is calculaed formula is U I FM ( T, D ) = 00 π (, d) π U 4 ( 2) I (, d) where ( 2) π (, d) ( π (, d) π ( +, d) ) π (, d), T, d D. U I = Thus, he fuzzy Markov chai model ca give precise ad reliable iformaio for differe age-specific prevalece ad is relaed facors. D. Table : ] Sympom Age (T) S S 2 S S 4 S 5 S 6 S 7 Bor ( 0 ) ( ) ( 2 ) ( ) ( 4 ) ( 5 ) ( 6 ) Above 70 ( 7 ) [

9 B. Aruraj 25 Table 2: ] [ 2 Sympom Disease d d 2 d d 4 d 5 d 6 d 7 S S S S S S S Table : The compleme of fuzzy se d d 2 d d 4 d 5 d 6 d Table 4: Ifecio risk rae FM ( T, D) d d 2 d d 4 d 5 d 6 d

10 26 Hybrid model aalysis o hos facor of epidemics 5 esuls ad Discussios Table 4 shows ha here is wide variaio as he produc of proporio of becomig epidemics a a give age i wo age/ime seps. From he Table 4, i is see ha he epidemics affec all ages. The diseases d, d ad d 4 are mosly commoly prese i childre ha aduls. The disease d 5 ad d 6 shows very high prevalece up o 4 age groups. I is see ha he diseases of Lepospirosis, Measles ad Iflueza are rakig high amog childre. I is recommeded ha hese age groups should be prioriized durig he oubreak of epidemics. Furher, i should be ake io accou ha he differe FM be he resul of differe healh codiios ad differe exposure a differe age groups. Eve hough all epidemics affec all age groups geerally, we sugges o use FM o arge age groups o be prioriized. 6 Coclusio I his sudy, a hybrid model aalysis o hos facor of epidemics ivolvig fuzzy Markov-chai is proposed. The model is cosruced usig he codiioal possibiliy disribuio fucio, Markov-chai ad fuzzy exper sysem. A agespecific member rae ad Markov risk rae are defied ad used as ovel measure idices o prioriize age groups o ideify ad decide he age group ha are argeed for moivaio. I is cocluded ha due o uceraiy of he pheomea. Combiaio of sochasic model ad fuzzy se heory eables our compuaioal simulaio o much more faihfully porray he pheomea uder sudy. efereces [] H.J.Zimmerma, Fuzzy se heory ad is applicaios, 2 d revised ediio, Kluwer Academic publishers is published by Allied Publishers Limied 996.

11 B. Aruraj 27 [2] J.E. Park ad K. Park, Tex Book of preveive ad social medicie, ih ed, M/s Baarasidas Bhao publishers, Jabalpur, Idia, 200. [] S. Jayakumar ad B. Aruraj, O Fuzzy Modellig for Compuig A Idex for Medical Diagosis Sysem, Bio-sciece esearch Bullei, 25(2), (2009), [4] S. Jayakumar ad B. Aruraj, Evaluaig he Medical Diagosis Sysem wih a Emphasis o elaed Eviromeal Facors Usig Fuzzy Modellig, Biosciece esearch Bullei, 26(), (200),