A SEIR Epidemic Model with Infectious Population Measurement

Transcription

1 Procdings of h World Congrss on Enginring 2 Vol III WCE 2, July 6-8, 2, London, UK A SEIR Epidmic Modl wih Infcious Populaion Masurmn M D la Sn, S Alonso-Qusada A Ibas Absrac- his papr is dvod o h dsign of a vaccinaion sragy for a SEIR modl wih incompl knowldg abou h populaions h dsign is orind owards h masurmn us of h infcious populaion in h dsign of h vaccinaion rul wih h vnual incorporaion of an obsrvr o dal wih uncrain modl sa knowldg h obsrvr is no ncssarily paramrizd wih h xac known paramrs of h pidmic modl Kywords- Epidmic modls, SEIR pidmic modl, Obsrvr, Vaccinaion I INRODUCION Imporan conrol problms nowadays rlad o Lif Scincs ar h conrol of cological modls lik, for insanc, hos of populaion voluion (Bvron-Hol modl, Hassll modl, Rickr modl c) via h onlin adjusmn of h spcis nvironmn carrying capaciy, ha of h populaion growh or ha of h rgulad harvsing quoa as wll as h disas propagaion via vaccinaion conrol In a s of paprs, svral varians gnralizaions of h Bvron-Hol modl (sard im invarian, im-varying paramrizd, gnralizd modl or modifid gnralizd modl) hav bn invsigad a h lvls of sabiliy, cycl- oscillaory bhavior, prmannc conrol hrough h manipulaion of h carrying capaciy (s, for insanc, [-5]) h dsign of rlad conrol acions has bn provd o b imporan in hos paprs a h lvls, for insanc, of aquaculur xploiaion or plagu fighing On h ohr h, h liraur abou pidmic mahmaical modls is xhausiv in many books paprs A non-xhausiv lis of rfrncs is givn in his manuscrip, cf [6-4] (s also h rfrncs lisd hrin) h ss of modls includ h mos basic ons, [6-7] as follows: a) SI- modls whr no rmovd- by immuniy populaion is assumd In ohr words, only suscpibl infcd populaions ar assumd,b) SIR modls, which includ suscpibl plus infcd plus rmovd- by immuniy populaions, c)seir- modls whr h infcd populaions is spli ino Manuscrip rcivd January, 2 his work was suppord in par by h Spanish Minisry of Scinc Innovaion hrough Gran DPI also by Basqu Govrnmn by is suppor hrough Gran GIC74-I M D la Sn S Alonso-Qusada ar wih h Insiu of Rsarch Dvlopmn of Procsss, Campus of Lioa, Bizkaia, Apdo 644- Bilbao Spain (-mail: manuldlasn@hus) A Ibas is wih Escola cnica Suprior dç Enginyria, Univrsia Auonoma d Barclona, Crdanyola dl Valls, Barclona, Spain(-mail: AsirIbasu@abca ISBN: ISSN: (Prin; ISSN: (Onlin) wo ons (namly, h infcd which incuba h disas bu do no sill hav any disas sympoms h infcious or infciv which do hav h xrnal disas sympoms) hos modls hav also wo major varians, namly, h so-calld psudo-mass acion modls, whr h oal populaion is no akn ino accoun as a rlvan disas conagious facor h so-calld ru-mass acion modls, whr h oal populaion is mor ralisically considrd as an invrs facor of h disas ransmission ras) hr ar many varians of h abov modls, for insanc, including vaccinaion of diffrn kinds : consan [8], impulsiv [2], discr im c, incorporaing poin or disribud dlays [2-], oscillaory bhaviours [4-8] c On h ohr h, varians of such modls bcom considrably simplr for h illnss ransmission among plans [6-7] I is assumd ha SEIR modl is of h ru-mass acion yp II HE MODEL L S ( b h suscpibl populaion of infcion a im, E ( h infcd ( i hos which incuba h illnss bu do no sill hav any sympoms) a im, I ( ) is h infcious ( or infciv ) populaion a im, R ( is h rmovd by- immuniy ( or immun ) populaion a im Considr h ru-mass acion SEIR-yp pidmic modl: S I S S R N V N () S I E E N (2) I I E () R R I NV (4) subjc o iniial condiions S S, E E, I I R R undr h vaccinaion consrain V : R R In h abov SEIR modl, N is h oal populaion, is h ra of dahs from causs unrlad o h infcion,, aks ino accoun h numbr of dahs du o h infcion, is h ra of losing immuniy, is h ransmission consan ( wih h oal numbr of infcions pr uniy of im a im S I bing ), ar, rspcivly, h N avrag duraions of h lan infciv priods If hn nihr h naural incras of h populaion nor h loss of marnal los of immuniy of h nwborns is akn ino accoun If such a los of immuniy is considrd All h abov paramrs ar nonngaiv No h following: WCE 2

2 Procdings of h World Congrss on Enginring 2 Vol III WCE 2, July 6-8, 2, London, UK a) sinc, ohrwis, h avrag duraion of h lan priod is infiniy h infcious ar unrlad o h infcd from () b) sinc, ohrwis, h avrag duraion of h infcious priod is infiniy h whol immun populaion is no dynamically coupld o h infcious on c) If hn h moraliy by causs unrlad o h disas is no akn ino accoun If hn nihr h loss of marnal immuniy of h nwborns nor h naural incras of h populaion ar considrd If hn i is assumd ha hr is no moraliy dircly causd by h disas d) I is nonsns o vnually fix o zro h disas ransmission consan sinc his would dcoupl h infcd dynamics from h suscpibl on ) Som paricular modlling varians ( so- calld psudo mass-acion yp modls) fix o uniy h whol populaion in ()- (4) his modlling sragy dos no considr ha h disas ransmission of fw suscpibl infcd among larg populaion numbrs modras h disas voluion as h SEIR- modl ()-(4) ( so calld mass acion yp modls ) dos h mass acion modls ar basd of h mass acion principl from Chmical kinics following Guldbrg Waag (864) which rads as follows: For a homognous sysm, h ra of h chmical racion is proporional o h aciv masss of h racing subsancs, [5] h oal populaion is S( E( ; R By summing up boh sids of ()-(4), on gs: N ( (5) If hn h populaion is considrd o rmain consan hrough im; i ); R so ha: S ( E ( I ( R ( S () E () I () R () R h nx rsul sablishs ha if h infcion collapss afr a fini im hn h whol suscpibl plus immun populaions convrg asympoically o h whol populaion vn in h vn ha his on has no a fini limi Assrion Assum ha ) I ( ) ; hn, E( ) E ( ) ; S R ) as ) is uniformly boundd for all im if If furhrmor hn lim S R N N If hn, irrspciv of h iniial condiions, h ovrall populaion is consan if, h ovrall populaion divrgs if h ovrall populaion asympoically convrgs o zro if Proof: I follows ha E( ) E ( ) ; for som fini from () (2) so ha on gs from () (4): S R S R N S R S R N d N d as N d g N as ) so ha, if, hn S R ) as ) is uniformly boundd for all im sinc i is a coninuous funcion which is h uniqu im-diffrniabl soluion of an ordinary diffrnial quaion which canno possss fini scap ims S R ) ) as if ) is uniformly boundd for all im S R ) as if If, in addiion, hn N ) h ovrall populaion is consan if divrgs if convrgs o zro if irrspciv of h iniial condiions h following rsul xnd Assrion o h cas whn I ( vanishs asympoically Assrion 2 Assum ha ) I ( ) as If hn, E S R as Proof: h soluion of () saisfis I I E d E d ; Procd by conradicion by assuming ha h claim E as is fals hn, sinc E( is vrywhr coninuous in R bcaus i saisfis h diffrnial quaion (2), on has ha for any givn, hr xis som ral consans,,,, 2 2,, such ha E ;, 2 for h givn so ha: I 2 2 E d E d (,,, ) : 2 m 2 which conradics I as from Assrion so ha E as,sinc S R ) as, from Assrion I has bn provn in prvious paprs (s, for insanc, [6]) ha h vaccinaion conrol law has o ak valus in, for all im in ordr o nsur ha h SEIR modl ()-(4) is a posiiv dynamic sysm in h sns ha for any s of nonngaiv iniial condiions all h componns of h rajcory soluion of () (4 ) ar nonngaiv for all im his has o b accomplishd wih for cohrncy of h mahmaical problm wih h ral problm a h On h ohr h, i is assumd ha h whol s of paramrs paramrizing h SEIR modl () (4) is no known hn ISBN: ISSN: (Prin; ISSN: (Onlin) WCE 2

3 Procdings of h World Congrss on Enginring 2 Vol III WCE 2, July 6-8, 2, London, UK hy should b simad onlin o synhsiz h vaccinaion lawv : R, basd on hos simaions Assrions - 2 dica ha if h infcion collapss in som way hn h whol populaion of suscpibl plus immun asympoically convrg o h whol populaion vn if ha on has no a fini limi as im nds o infiniy his faur moivas fixing h adapiv conrol objciv as o synhsiz a vaccinaion law such ha h infcious populaion is asympoically rgulad o zro o achiv h sum of h suscpibl plus h immun asympoically rack h whol populaion as a rsul III SABILIY AND POSIIVIY RESULS h vaccinaion sragy has o b implmnd so ha h SEIR modl b posiiv in h usual sns ha non of h populaions, namly, suscpibl, infcd, infcious immun b ngaiv a any im his rquirmn follows dircly from h naur of h problm a h his scion invsigas condiions for posiiviy of h SEIR modl ()-(4) Firs, assum h consan populaion consrain (5) wih, implying dircly: S ( E ( I ( R ( S () E () I () R () ; R (6) is usd in (), ()-(4) o limina h infcd populaion E( lading o: I S S R S N V N (7) I I N S ( (8) R R I NV (9) for any givn ral consan / N sup I is possibl o rwri (7)-(9) in a compac form as a dynamic sysm of sa x ( S(,,, oupu y( S( whos inpu is approprialy rlad o h vaccinaion funcion as u V (, V ( his lads o: x A x N E u I + E x N 2 (a) N x N E u + + I A E E x N 2 (b) N x N E u A I + E x N y 2 N A x N V + I E x N y N V ( ) (c) 2 N (d) y x () whr i is h i-h uni Euclidan column vcor in R wih is i-h componn bing qual o on h ohr wo componns bing zro, ij having h i-h j-h componns bing on h rmaining on bing zro, so ha,,, A : A E A : (2) : E ; E :, E : () Simpl inspcion of ()-() yilds h following posiiviy rsul by aking ino accoun ha E ( if h rducd sysm (7)-(9) is posiiv by dirc calculaion of h soluion of (2) for, : horm Assum ha,,, a vaccinaion funcion PC R,, min ha V is usd hn, all h soluions of h SEIR modl ()-(4) saisfy S, E( ), ), ), N ; R if or if Furhrmor, ihr ( ),S, E( ), ), ) N as (i h oal populaion asympoically xinguishs) wih all h populaions bing uniformly boundd or all h parial oal populaions ar boundd h infcion dos no asympoically vanishs in h scond casd wih Proof: No ha marix A in (2) is a Mzlr undr h givn consrains hn, h dynamic sysm () is posiiv wih N ) N S( ) E( ) ) ) ; R if or if hn, maxs ( E( ) ; R Now, from (4), R ( R ; R his implis S, E(,,, N ; R sinc N ) S( ) E( ) ) ) ; R If hn N h proof rmains valid wih h changs N S( E( R S E(,,, N R,, ; h las par is provn by conradicion Assum ha I ( ) ; R wih hn N ; R if I so ha, S, E(, as wih (2) bing a posiiv sysm sinc R ( ) R hn I ( ) as ISBN: ISSN: (Prin; ISSN: (Onlin) WCE 2

4 Procdings of h World Congrss on Enginring 2 Vol III WCE 2, July 6-8, 2, London, UK (sinc, S, E(, as ) conradicing h assumpion ha I ( ; R Rmark h posiiviy propry is an ssnial ool o discuss h sabiliy of h SEIR- modl sinc all h parial populaions ar uppr-boundd by h oal populaion for all im I is also ssnial for appropria dscripion of h ral problm hrough h mahmaical modl Corollary Assum ha hn, h SEIR-modl ()-(4) is sabl Proof: N ; R from (5) so ha is monoon dcrasing hn i is uniformly boundd ) ; R Dfin indicaor binary funcions i : R, if N dfind as i : i ohrwis if N i ohrwis hus, Corollary xnds dircly as follows: Corollary 2 Assum ha limsup N i N i d hn, h SEIR-modl ()-(4) is sabl IVVACCINAION RULES An usful vaccinaion conrol funcion is on wih h goal of dcrasing approprialy h numbrs of suscpibl, infcd infcious whil including h nonlinar rm involving h produc S( of suscpibl infcious in () On has o cop wih wo major pracical problms, namly: h paramrs of h SEIR modl ()-(4) ar no usually known prcisly vn if h modl is considrd valid for a paricular sudy 2 h only populaions bing dircly masurabl wih a crain accuracy dgr for any im ar h oal populaion N ( h infcious on I ( For h rmaining populaions wha i can b said is ha S( E( a any im I could b calculad from h diffrnial sysm ()-(4) in h cas ha h modl paramrs h iniial condiions ar fully known Ohrwis, hy could b simad from paramr iniial condiion simad h gnral SEIR- modl ()-(4) may b compacd as h following dynamic sysm of sa x : S(, E(,, masurabl oupu y : I : bing h infcious populaion, i x A x ω ; x whr : ω,,, y S( (4) S( (5) : V (,,, V ( A : (6) Using h im-drivaiv opraor D : d / d in (4), i may b mor compacly rwrin as x D I A ω y x D I A ω subjc o x : S (), E(), ), ) (7) wih I bing h 4 4 idniy marix Sinc D I Aω N I ( D I A M ( D ) Adj D I A ω D D, ) ; R (8) hn Adj D I A ω ) D DI A 4 Adj D I A i ω i D DI A i D DI A 4 S I S I AdjDI A N V,,, N V i N N (9) wih DD I A D D D D M ( D ) : Sinc h calculaion of h adjoin marix involvs marix ransposiion hn h hird column of D I A has o b chckd in viw of (9) as follows;(a) h (,) ranspos, i h (,)- adjoin drminan of D I A is zro by inspcing (6); (b) h (2,) adjoin drminan of D I A is D D (c) h (4,) adjoin drminan is zro h (,) adjoin has no o b calculad sinc ω for all im hus, on gs from (9): S I M ( ) N I ( ; R (2) N ( D ) D D whr Sinc N I N I ( D ) N ( D ) I : M ( D ) M( D ) afr D D rmoving h sabl polynomial cancllaion D D, Eq 2 is quivaln o x M ( ) v I, I N S ( (2) N I whr v I c I c2 I is a ral funcion which aks ino accoun h conribuion o h soluion of (2) from nonzro iniial condiions of (2) which has bn ISBN: ISSN: (Prin; ISSN: (Onlin) WCE 2

5 Procdings of h World Congrss on Enginring 2 Vol III WCE 2, July 6-8, 2, London, UK N I ( N ( nglcd by h zro-pol cancllaion I M ( M ( which vanishs xponnially as wih c i I cii x() for i=,2 bing wo ral consans subjc o c I c2 I v I (i =,2) h vaccinaion conrol blow is nonngaiv for all im if i blongs o h inrval, for all im V ˆ N k Sˆ k Iˆ k Rˆ k Sˆ Iˆ 4 5 g N (22) h abov ha suprscrips on h various populaions dno hir simas hrough som availabl obsrvr in h cas whn h populaion amouns ar no prfcly known h following vaccinaion nonngaiv conrol combind of (9) (22) may b usd whn h posiiviy of h obsrvr h sauraion of h vaccinaion o uniy ar no imposd: V if V V k Ŝ k Rˆ k Ŝ Î g N, ohrwis 4 5 ˆ N (2) V : k Ŝ k2 Ê k Î k 4 Rˆ k 5 Ŝ Î g N ˆ N (24) Simulaions rsuls abou h abov vaccinaion laws ar in progrss sm o b promising I sms o b, in gnral, promising h ponial applicaion of som conrol hory chniqus, [7-2] o pidmic modls o dsign h vaccinaion ruls REFERENCES [] M D la Sn S Alonso-Qusada, A Conrol hory poin of viw on Bvron-Hol quaion in populaion dynamics som of is gnralizaions, Applid Mahmaics Compuaion, Vol 99, No 2, pp , 28 [2] M D la Sn S Alonso-Qusada, Conrol issus for h Bvron-Hol quaion in populaion in cology by locally monioring h nvironmn carrying capaciy: Non-adapiv adapiv cass, Applid Mahmaics Compuaion, Vol 25, No 7, pp , 29 [] M D la Sn S Alonso-Qusada, Modl-maching-basd conrol of h Bvron-Hol quaion in Ecology, Discr Dynamics in Naur Sociy, Aricl numbr 7952, 28 [4] M D la Sn, Abou h propris of a modifid gnralizd Bvron-Hol quaion in cology modls, Discr Dynamics in Naur Sociy, Aricl numbr 59295, 28 [5] M D la Sn, h gnralizd Bvron- Hol quaion h conrol of populaions, Applid Mahmaical Modlling, Vol 2, No, pp , 28 [6] Epidmic Modls: hir Srucur Rlaion o Daa, Publicaions of h Nwon Insiu, Cambridg Univrsiy Prss, Dnis Mollison Edior, 2 [7] M J Kling P Rohani, Modling Infcious Disass in Humans Animals, Princon Univrsiy Prss, Princon Oxford, 28 [8] M D la Sn S Alonso- Qusada, On vaccinaion conrols for a gnral SEIR- pidmic modl, 8 h Mdirranan Confrnc on Conrol Auomaion MED - Confrnc procdings, ar no , pp 22-28, 2, doi:9/med [9] M D la Sn S Alonso-Qusada, A simpl vaccinaion conrol sragy for h SEIR pidmic modl, 5 h IEEE Inrnaional Confrnc on managmn of Innovaion chnology, ICMI2, ar no , pp 7-44, doi:9/icmi [] N Orga, LC Barros E Massad, Fuzzy gradual ruls in pidmiology, Kybrns, Vol 2, Nos -4, pp , 2 [] H Khan, RN Mohapara, K Varajvlu S, J Liao, h xplici sris soluion of SIR SIS pidmic modls, Applid Mahmaics Compuaion, Vol 25, No 2, pp , 29 [2] X Y Song, Y Jiang HM Wi, Analysis of a sauraion incidnc SVEIRS pidmic modl wih puls wo im dlays, Applid Mahmaics Compuaion, Vol 24, No 2, pp 8-9, 29 [] L Zhang, JL Liu ZD ng, Dynamic bhaviour for a nonauonomous SIRS pidmic modl wih disribud dlays, Applid Mahmaics Compuaion, Vol 24, No 2, pp 624-6, 29 [4] B Mukhopadhyay R Baacharyya, Exisnc of pidmic wavs in a disas ransmission modl wih wohabia populaion, Inrnaional Journal of Sysms Scinc, Vol 8, No 9, pp , 27 [5] DJ Daly J Gani, Epidmic Modlling: An Inroducion, Cambridg Sudis in Mahmaical Biology: 5, Cambridg Univrsiy Prss, Cambridg, 999 [6] M D la Sn S Alonso- Qusada, On h propris of som pidmic modls, Inrnaional Journal of Mahmaical Modls Mhods in Applid Scincs, Vol4, No, pp 56-66, 2 [7] M D la Sn, A mhod for gnral dsign of posiiv ral funcions, IEEE ransacions on Circuis Sysms I- Fundamnal hory Applicaions, Vol 45, No 7, pp , 998 [8] M Dlasn, On som srucurs of sabilizing conrol laws for linar im-invarian sysms wih boundd poin dlays unmasurabl sas, Inrnaional Journal of Conrol, Vol 59, No 2, pp , 994 [9] N Luo, J Rodllar J Vhi M D la Sn, Composi smiaciv conrol of a class of sismically xcid srucurs, Journal of h Franklin Insiu- Enginring Applid Mahmaics, Vol 8, Issu 2-, pp , 2 [2] M D la Sn, On posiiviy of singular rgular linar im-dlay im-invarian sysms subjc o mulipl inrnal xrnal incommnsura poin dlays, Applid Mahmaics Compuaion, Vol 9, Issu, pp 82-4, 27 [2] M D la Sn, A mhod for gnral dsign of posiiv ral funcions, IEEE ransacions on Circuis Sysms I- Fundamnal hory Applicaions, Vol 45, Issu 7, pp , 998 ISBN: ISSN: (Prin; ISSN: (Onlin) WCE 2