Abstract. 1. Introduction. 1.1 Notation. 1.2 Parameters

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1 1 Mdels, Predici, ad Esimai f Oubreaks f Ifecius Disease Peer J. Csa James P. Duyak Mjdeh Mhashemi {pjcsa@mire.rg, jduyak@mire.rg, mjdeh@mire.rg} he MIRE Crprai 202 Burlig Rad Bedfrd, MA Absrac Cveial SEIR (Suscepible Expsed Ifecius Recvered) mdels have bee uilized by umerus researchers sudy ad predic disease ubreak. By cmbiig he predicive aure f such mahemaical mdels alg wih he measured ccurreces f disease, a mre rbus esimae f disease prgressi ca be made. he Kalma filer is he mehd desiged icrprae mdel predici ad measureme crreci. Csequely, we prduce a SEIR mdel which gvers he shr erm behaviur f a epidemic ubreak. he mahemaical srucure fr a assciaed Kalma filer is develped ad esimaes f a simulaed ubreak are prvided 1. Irduci Mahemaical mdels have bee used sudy he ubreak f a umber f ifecius diseases [1, 2, 6]. I paricular, differece ad differeial equais are he mehdlgies i which such mdels are wrie [4, 5, 6]. May research hspials ad/r public healh deparmes are maiaiig a daabase f emergecy rm visis by paies wih caegrized cmplais. he cmbiai f a mahemaical mdel f a ubreak wih daily measuremes becks he applicai f a Kalma filer prvide a pimal esimae f he umber f ifecis. his paper will prvide he mahemaical ifrasrucure required impleme a Kalma filer simulaed emergecy rm daa. he prgram f his discussi will be prvide a geeral mdel, discuss mdel simplificai, ad demsrae he efficacy f he filer simulaed daa. I his firs seci, we esablish cmm ai ad a geeral mdel fr he ubreak f a specific (bu ukw) ifecius disease hrugh a geeral ppulai. 1.1 Nai S = S = umber f peple i he ppulai suscepible he disease a ime E = E = umber f peple i he ppulai expsed/ifeced by he disease a ime I = I = umber f peple i he ppulai wh are ifecius a ime R = R = umber f peple i he ppulai wh have recvered frm he disease a ime here are a umber f parameers which will eed be eiher mdeled r esimaed frm he daa. I is assumed ha hese parameers are ime ivaria hugh mre sphisicaed effrs ad ifrmai culd prduce ime varyig mdels. A descripi f hese parameers is lised belw. 1.2 Parameers β = prbabiliy f disease rasmissi v = rae f sercversi (i.e., frm expsed ifecius) µ I = deah rae f ifecius due he disease α = recvery delay rae ρ(i) = βi = cversi rae frm suscepible expsed/ifeced (als called he frce f ifeci) I figure 1 belw, a schemaic diagram expresses he graphical represeai f he spread f a ifecius disease hrugh a ppulai. Implici i his figure is he assumpi ha everye i he ppulai is suscepible he disease. he firs bxes illusrae he migrai f he ppulai f suscepibles S hse expsed ad ifeced E. he rae a which he suscepibles are ifeced is prprial he umber f cacs c wih he ifecius ppulai I imes he prbabiliy f disease rasmissi per cac β imes he prpri f he ppulai which is

2 2 ifecius: ρ(i) = βi. Sice he ifeced leave he ppulai f suscepibles a egaive sig is aached his quaiy. Csequely, ds/ = 0 ρ(i)s βis. I a similar maer, he disease dyamics f equai (1.1) are frmed. Suscepibles S ρ(i) Expsed ν Ifecius I E µ I Deah by disease 1 µ I α Recvered R Figure 1.1. Disease dyamics 1.3 Disease Dyamics ds = ρ( IS ) βis = ρ( I) S ve βis ve = ve µ II (1 µ I α) I ve( ) (1 α) I( ) dr = (1 µ I α) I ( ) (1.1) his full mdel expressed i (1.1) peraes uder he simplifyig assumpi f a sufficiely shr ime scale such ha sigifica ppulai eers he suscepible ppulai ad ha he parameers β, ν, µ I, ad α d vary wih respec ime. he effrs behid his wrk are prese a mdel fr a shr ime scale wihi he epidemic cycle (i.e., he rder f 2 3 weeks). Csequely, a series f simplifyig assumpis ca be made which are lised belw. Assumpis (i) Shr ime scale: [, + ] where he chage i ime is less ha hree weeks. (ii) N immigrai r emigrai frm he subppulais (iii) Isufficie ime fr R (recvereds) reur he ppulai f suscepibles (iv) Fr [, + ], S = S( ) = S. ds Frm (iv), = 0 ad S = S (csa). Se ρ(i) = βsi ρ I, where ρ βs, s ha he secd ad hird equais f he disease dyamics becme = ρi ve = ve (1 α) I (1.2)

3 3 Observe ha he furh equai f he disease dyamics is cmpleely decupled frm he middle w equais. Csequely, he ppulai f recvereds ca be cmpued as R = R ( ) + (1 µ α) I( τ) dτ. (1.3) I By seig X = [E,I], he reduced se f disease dyamics ca be wrie i he vecr marix frm dx = AX (1.4) ν ρ where A = ν 1 α. he measuremes f his sysem are a pri f he umber f ifecius which repr emergecy rms a day day basis. Mre precisely, le be he prbabiliy ha a member f he ifecius ppulai appears i a reprig emergecy rm. he, he measuremes are m = I. (1.5) he measured quaiy, I, raher ha he mdeled ppulai f ifecius peple I, is wha emergecy deparmes repred. hus, make he fllwig chage f variables (1.6) rasfrm he prblem a dimesial framewrk. I a I I$ E a E E (1.6) $ Sice, = ad =, he muliplyig (1.2) by ad simplifyig yields he dimesiless disease dyamics = ρ I $ ve $ = ve (1 α) I$ ad he assciaed measuremes (1.7) m = I $. (1.8) Nw wih X = EI, $ ad A as abve, he disease dyamics ca be wrie as d X = AX (1.9) where X is he sae vecr. As equai (1.9) illusrais, he disease dyamics are liear. Mrever, here are regular ime measuremes (1.8). Mder crl hery was develped arud his very sceari: he eed slve liear differeial equais i assciai wih regularly sampled (i ime) measuremes. A pimal esimae f he mdel prediced/ measureme crreced sae f a disease ubreak ca be baied via he Kalma filer. he discussi is hereafer, framed i he Kalma filer cex. 2. he Kalma Filer Sice he mahemaical mdels f he disease dyamics (1.9) ad measuremes (1.8) are iherely imperfec, ise i he frm f zer mea Gaussia radm prcesses are added ehace hese mdellig deficiecies. hus, he sae dyamics, add a vecr w ~ N(0,Q) called he sae r sysem errr. he marix Q is called he sae r sysem ise cvariace. Similarly, cmpesae fr he variabiliy i he measuremes, a vecr v ~ N(0,V) called he measureme errr is added (1.8). he marix V is called he measureme ise cvariace. he defiiis belw help develp he Kalma filer (see, e.g., Csa [3]). Sae Vecr: E X = I$ Sae Dyamics: d X = AX Sysem Mdel: dx = AX + w Cv w Q Sysem Nise Cvariace: [ ] Measureme: m = I $ HX Measureme mdel: m = HX + v, Measureme Jacbia: H = [0,1] Measureme Nise Cvariace: Cv v V [ ]

4 4 rasii marix: Φ (, ) = exp ( A [ ] ) ν ρ where A = ν 1 α Measureme Esemble ime : M = m ( ), m ( ), L, m ( ) Sae Predici: p M { } 1 2 = {} = he empy se X (, M ) =Φ(, ) X (, M ) k k 1 k k 1 p k 1 k 1 Xp( 1, M) =Φ( 1, ) X p(, M) Φ( 1, ) X( ) Sae Jacbia: F = A Cvariace dyamics (Ricai Equai): dp = PF + FP + Q Cvariace predici: P ( ) =Φ(, ) P ( ) Φ (, ) + Φ(, s) Q( s) Φ (, s) ds Kalma gais marix: K( ) = P( ) H I ( ) Ifrmai marix: 1 ( ) [ ( ) I = HP H + V( )] Sae crreci: X (, M ) = K( )[ m( ) m ( )] c p Prediced measureme: mp( ) = I $ p (, M 1) Sae esimae: X ˆ (, ) p (, ) (, ) M = X M 1 + X c M Cvariace updae (Jseph frm): P ( ) = [ I K ( ) HP ] ( )[ I K ( ) H] + x x K ( ) V ( ) K ( ) S = 1000, E = 10, I = 1, R = 2, ν = 0.4, β = 0.5, α = 0.3, ad µ I = 0.1. he sysem ise cvariace Q was seleced as a 10% variai f he iiial sae cvariace P ( ) = ( X( ) µ )( X( ) µ ) ad ( ) 1 µ = E + I. Fially, he measureme 2 ise cvariace V was seleced as he variace i he daa. he filer was ru ver he simulaed daa (3.1) esablish a baselie esimae f he umber f expsed/ifeced ad ifecius reprig a emergecy deparme. he resuls are depiced i Figure 3.1 belw. A e sadard deviai eighbrhd, based he esimaed cvariace marices P was cmpued fr he ifecius class; see p pri f Figure 3.2. he a simulaed e week (i.e., seve day) ubreak was irduced i he ppulai a a radm seed ime i he frm f (3.2). Figure 3.1. Baselie Kalma filer esimaes frm simulaed daa 3. Simulai A mahemaical mdel ha simulaes he uderlyig dyamics f he hspial daily visis ha are iflueza relaed was develped i he frm f equai (3.1) D = 2cs(2 π/365) + 8+ w. (3.1) Here = 0, 1, 2,, 5x365 is measured i sigle days ver five years, ad w N(0,2) is rmally disribued ise. We assumed he fllwig se f iiial cdiis ad parameers: Figure 3.2. Oe sadard deviai eighbrhd f he ifecius class

5 5 0 fr < fubreak (, ) = 2( ) fr + 6 (3.2) 0 fr > + 6 ha is, fubreak (, ) was added he simulaed daa D i (3.1). If he Kalma filer esimae f he ifecius class I $ ( k, Mk), reflecig he ifluece f he measuremes D + fubreak (, ) hrugh ime k, exceeded he e sadard deviai eighbrhd esablished fr he baselie case wihi e days f he sar f he ubreak (i.e., fr k [, + 10] ), he a rue psiive fr ubreak deeci was recrded. Oherwise, a false egaive was recrded. isure a sufficie umber f measuremes were prcessed by he Kalma filer, he rage f he radm ubreak ime was resriced: [50,1800] days. Oe husad radm ubreaks were esed ad he umber f rue psiives (p) ad false egaives (F ) were recrded. Fr his es, 100% f he ubreaks were discvered wihi he requisie ime perid (10 days). I paricular, 2.9% f he ubreaks were deeced day 2, 17.3% were deeced day 3, 59.8% were deeced day 4, 19.9% were deeced day 5, ad 0.1% were deeced day 6 f he ubreak. [3] P. J. Csa, Bridgig Mid ad Mdel, S. hmas echlgy Press, S. Paul, MN, 1994 [4] G. Fulfrd, P. Frreser, ad A. Jes, Mdellig wih Differeial ad Differece Equais, Cambridge Uiversiy Press, 1997 [5] S. Gupa, R. M. Aders, ad R. M.,May, Mahemaical Mdels ad he Desig f Public Healh Plicy: HIV ad Aiviral herapy, SIAM Review, Vlume 35, Number 1, March 1993, pp [6] J. D. Murray, Mahemaical Bilgy, Secd, Crreced Edii, Spriger Verlag, Berli, 1993 Summary A se f mahemaical mdels gverig he ubreak f a ifecius disease have bee deailed. Simulaed daa have bee geeraed. he assciaed Kalma filer has bee develped ad esed agais he simulaed daa wih psiive resuls. Aalysis ccerig he variai f he mdel parameers ad heir effec up he Kalma filer esimaes ad he applicai f his mehd real recrded emergecy deparme daa will be he fcus f fuure wrk Refereces [1] R. M. Aders ad R. M., May, Ifecius Diseases f Humas, Oxfrd Sciece Publicais, 1992 [2] N. G. Becker ad L. R. Eger, A rasmissi Mdel f HIV, Mahemaical Biscieces, Vlume 119, 1994, pp