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1 Avilbl hp://pvmu.du/pgs/398/sp Vol., No. (6) pp Applicions nd Applid Mhmics (AAM): An Inrnionl Journl ANALYSIS OF AN SIRS AGE-STRUCTURED EPIDEMIC MODEL WITH VACCINATION AND VERTICAL TRANSMISSION OF DISEASE Mohmmd El-Dom Cnr for Advncd Mhmicl Scincs (CAMS) Collg Hll, Room 46 Amricn Univrsiy of Biru P. O. Box: -36 Biru-Lbnon E-Mil: biomh4@yhoo.com Tlphon: (96) or x. 439 Fx: (96) Rcivd July, 5; rvisd Fbury 8, 6; ccpd April 8, 6 Absrc An SIRS g-srucurd pidmic modl for vriclly s wll s horizonlly rnsmid diss undr vccinion is invsigd whn h friliy, morliy nd rmovl rs dpnd on g nd h forc of infcion of proporion mixing ssumpion yp, nd vccinion wns ovr im. W prov h xisnc nd uniqunss of soluion o h modl quions, nd show h soluions of h modl quions dpnd coninuously on h iniil g-disribuions. Furhrmor, w drmin h sdy ss nd obin n xplicily compubl hrshold condiion, in rms of h dmogrphic nd pidmiologicl prmrs of h modl; w hn sudy h sbiliy of h sdy ss. W lso compr h bhvior of h modl wih h on wihou vricl rnsmission. Kywords: Vricl rnsmission,horizonl rnsmission,ag-srucur, Epidmic, Sbiliy,Proporion mixing MSC : 45K5; 45M; 35A5; 35B3; 35B35; 35B45; 35L4; 9D3; 9D5. Inroducion In his ppr, w sudy n SIRS g-srucurd pidmic modl, whr g is ssumd o b h chronologicl g i.. h im sinc birh. Th diss cuss so fw fliis h hy cn b nglcd, nd is horizonlly s wll s vriclly rnsmid. Horizonl rnsmission is h pssing of infcion hrough som dirc or indirc conc wih infcd individuls, for xmpl, mlri nd ubrculosis r horizonlly rnsmid disss. Vricl rnsmission is h pssing of infcion from prns o nwborn or unborn offspring, for xmpl, AIDS, Chgs nd Hpiis B r vriclly (s wll s horizonlly) rnsmid disss. Vricl rnsmission plys n imporn rol in minining som disss, for xmpl, s Busnbrg, l. (993), (988, p. 379), (98), (988, p. 8), (99) nd Busnbrg (986). In Fin (975), svrl xmpls of vriclly rnsmid disss r givn, nd in Busnbrg, l. (993), book is dvod for h sudy of h modls nd dynmics of vriclly rnsmid disss. Svrl rcn pprs hv dl wih g-dpndn vccinion modls, for xmpl, Hhco (983), (989), (997), (), Diz, l. (985), Diz (98), Kzmnn, l. (984), Schnzl (984), Andrson, l. (999), El-Dom (), (), (5), (6), M u llr (994), (998), Hdlr, l. (996), Knox (98), McLn (986), Couinho, l. (993), Lopz, l. (), Grnhlgh (988),(99), Csillo-Chvz, l. (998), Li, l. (4), nd Thim (). Much of his prvious work ssumd h h ol populion hs fixd siz. As on migh xpc 36

2 37 M. El-Dom hr r siuions, for xmpl, in dvloping counris, whr i is ncssry o considr ol populion sizs h vry wih im, morliy du o diss nd rducion of birh r du o infcion. Som of h ffcs of such ssumpions r considrd by McLn (986), El-Dom (), (4) nd Hdlr, l. (996). W no h svrl pprs hv dl wih SIR g-srucurd pidmic modls, bu wihou vccinion, for xmpl, Thim (99), Andrsn (995), Chipo, l. (995), Inb (99), Ch, l. (998), Grnhlgh (987) nd Csillo-Chvz, l. (989). Also, w no h som vccins wn ovr im giving ris o SIRS yp modls, for xmpl, s, Li, l. (4). In his ppr, w sudy n SIRS g-srucurd pidmic modl whn h diss is vriclly s wll s horizonlly rnsmid, nd hrfor our rsuls gnrliz hos in Li, l. (4), wr vricl rnsmission is no considrd. W prov h xisnc nd uniqunss of soluion o h modl quions nd show h soluions of h modl quions dpnd coninuously on h iniil g-disribuions. W drmin h sdy ss of h modl by proving hrshold horm nd obining n xplicily compubl hrshold R ν, in rms of h dmogrphic nd pidmiologicl prmrs of h modl, known s h rproducion numbr in h prsnc of vccinion srgy ν ( ), s in Hdlr, l. (996) or h n rplcmn rio, s in Thim (). And hnc, w show h R ν incrss wih q, which is h probbiliy of vriclly rnsmiing h diss (s scion for dfiniions) nd hrfor, incrss h liklihood h n ndmic will occur; lso R ν is usd o drmin criicl vccinion covrg which will rdic h diss wih minimum vccinion covrg. In ddiion, w show h h modl givs ris o coninuum of posiiv ndmic quilibriums in h cs of non-fril infcibls. This siuion dos no occur if hr is no vricl rnsmission, for xmpl, s, Li, l. (4). W lso sudy h sbiliy of h sdy ss. W sudy h sbiliy of h diss-fr quilibrium nd show h h diss-fr quilibrium is loclly sympoiclly sbl if R ν <, nd unsbl if R ν >. Also, w show h h diss-fr quilibrium is globlly sbl if R <, whr R is h bsic rproducion numbr, nd is inrprd s h vrg numbr of scondry infcions h occur whn n infciv is inroducd ino olly suscpibl populion. For ndmic quilibriums, w obin complicd chrcrisic quion, h llows us o prov locl sympoic sbiliy, in som spcil css. In ddiion, w driv gnrl formuls for h chrcrisic quions, in rms of ingrl quions. Alhough w do no obin xplici formuls for h soluions of hs ingrl quions, w will us hs ingrl quions o dduc h if h forc of infcion is sufficinly smll, hn n ndmic quilibrium is lwys loclly sympoiclly sbl. Th orgnizion of his ppr is s follows: in scion w dscrib h modl nd obin h modl quions; in scion 3 w rduc h modl quions o svrl subsysms nd prov h xisnc nd uniqunss of soluion s wll s h coninuous dpndnc on iniil gdisribuions; in scion 4 w drmin h sdy ss; in scion 5 w sudy h sbiliy of h sdy ss; in scion 6 w conclud our rsuls. Th Modl W considr n g-srucurd populion of vribl siz xposd o communicbl diss. Th diss is vriclly s wll s horizonlly rnsmid nd cuss so fw fliis h hy cn b nglcd. W ssum h following. s(,), i(,) nd r(,), rspcivly, dno h g-dnsiy for suscpibl, infciv nd immun individuls of g im. Thn

3 AAM: Inrn. J., Vol. No. (6) 38 sd id (, ) = (, ) = ol numbr of suscpibl individuls im of gs bwn nd, ol numbr of infciv individuls im of gs bwn nd. And similrly for r(,). W ssum h h ol populion consiss nirly of suscpibl, infciv nd immun individuls. L k (, ) dno h probbiliy h suscpibl individul of g is infcd by n infciv of g. W furhr ssum h, k (, )= k( ). k( ), which is know s h ``proporion mixing ssumpion'', s Diz, l. (985). Thrfor, h horizonl rnsmission of h diss for h suscpibl individuls occurs h following r: k ( ) s(, ) k ( ) i(, ) d, whr k ( ) nd k ( ) r boundd, nonngiv, coninuous funcions of. Th rm k ( ) k ( ) i(, ) d, is clld ``forc of infcion'' nd w l λ()= k( id )(,). And immun individuls r infcd du o wning of vccin ovr im h following r: ε k ( ) r(, ) ( ), λ whr ε is posiiv rl numbr [,]. Th friliy r β ( ) is nonngiv, coninuous funcion, wih compc suppor [, A ], ( A >). Th numbr of birhs of suscpibl individuls pr uni im is givn by β s(, )= ( )[ s (, ) ( qi ) (, ) r (, )] dq, [,], whr q is h probbiliy of vriclly rnsmiing h diss. Accordingly ll nwborns from suscpibl nd immun individuls r suscpibl, bu porion q of nwborns from infcd prns r infciv, i.., hy cquir h diss vi birh (vricl rnsmission) nd hrfor, β i(, ) = q ( ) i(, ) d, nd r(, ) =. Th dh r μ ( ) is h sm for suscpibl, infciv nd immun individuls, nd μ ( ) is nonngiv, coninuous funcion nd [, ) such h μ( )> μ > > nd μ( )> μ( ) > >. Th cur r γ ( ) is boundd, nonngiv, coninuous funcion of. Th vccinion r ν ( ) is boundd, nonngiv, coninuous funcion of. Th vccinion wns in immun individuls nd hy bcom suscpibl r δ ( ) which is boundd, nonngiv, coninuous funcion of. Th iniil g disribuions s (,)= s ( ), i (,)= i ( ), nd r (,)= r ( ) r coninuous, nonngiv nd ingrbl funcions of [, ). Ths ssumpions ld o h following sysm of nonlinr ingro-pril diffrnil quions wih non-locl boundry condiions, which dscribs h dynmics of h rnsmission of h diss.

4 39 M. El-Dom s (, ) s (, ) [ μ( ) ν( )] s(, )= k( ) s(, ) λ( ) δ( ) r(, ), >, >, i (, ) i (, ) [ μ( ) γ( )] i (, )= k( s ) (, ) λ( ) εk( r ) (, ) λ( ), >, >, r (, ) r (, ) [ μ( ) δ( )] r(, ) = ν ( s ) (, ) γ( i ) (, ) εk( r ) (, ) λ( ), >, >, s(, ) = β ( )[ s (, ) ( qi ) (, ) r (, )] d,, i(, ) = q β ( ) i(, ) d,, r(, ) =,, λ()= k ( id )(,),, s (,) = s( ), i (,) = i( ), r (,) = r( ),. () W no h problm (.) is n SIRS pidmic modl, h sm modl bu wih q =, i.., h cs of no vricl rnsmission, is dl wih in Li, l. (4), nd h sdy ss r drmind nd h sbiliy of h diss-fr quilibrium is sudid. In wh follows, w show h problm (.) hs uniqu soluion h xiss for ll im. Furhrmor, w show h soluions of problm (.) dpnd coninuously on h iniil gdisribuions. Also, w drmin h sdy ss nd sudy hir sbiliy. Rducion of h Modl, h Exisnc nd Uniqunss of Soluion nd Coninuous Dpndnc on Iniil Ag-Disribuions In his scion, w dvlop som prliminry forml nlysis of problm (.) nd show h problm (.) hs uniqu soluion h xiss for ll im. Furhrmor, w show h soluions of problm (.) dpnd coniuously on h iniil g-disribuions. W dfin p(, ) by p(, )= s (, ) i (, ) r (, ). Thn from (.), by dding h quions, w find h p(, ) sisfis h following McKndrick-Von Forsr quion: p (, ) p (, ) μ( ) p(, )=, >, >, p(, ) = B( ) = β ( ) p(, ) d,, () p (,)= p( )= s( ) i( ) r( ),. No h problm (3.) hs uniqu soluion h xiss for ll im, s Bllmn, l. (963), Fllr (94) nd Hoppnsd (975). Th uniqu soluion is givn by p ( ) π( )/ π( ), >, p (, )= (3) B( ) π ( ), <, whr π ( ) is givn by π μ( τ) ( )=,

5 AAM: Inrn. J., Vol. No. (6) 4 nd B( ) hs h following sympoic bhvior s : p B()=[ c θ ()], (4) whr p is h uniqu rl numbr which sisfis h following chrcrisic quion: p β( ) π( ) d =, (5) θ () is funcion such h θ ( ) s nd c is consn. Using (3.)-(3.), w obin h B( ) sisfis π ( ) B()= ( ) ( ) B( ) d ( ) p ( ) d. π ( ) β π β (6) Using (3.5) nd Gronwll s inquliy, w obin ( ( ) ) B ( ) ( ) p( ) [, ) β μ β, L (7) whr μ is givn by μ = inf μ( ). [, ) (8) From (3.) nd (3.6), w obin h following priori sim: ( ( ) ) pd (, ) p ( ) [, ) β μ. L (9) Also, from (.), s(,), i(,) nd r(,) sisfy h following sysms of quions: s (, ) s (, ) [ μ( ) ν( )] s(, )= k( ) s(, ) λ( ) δ( ) r(, ), >, >, s(, )= β ( )[ s (, ) ( qi ) (, ) r (, )] d,, () s (,) = s( ),, i (, ) i (, ) [ μ( ) γ( )] i(, )= k( ) s(, ) λ( ) εk( ) r(, ) λ( ), >, >, i(, ) = q β ( ) i(, ) d,, () i (,) = i ( ),, r (, )= p (, ) s (, ) i (, ). () By ingring problm (3.9) long chrcrisic lins = cons., w find h s (, ) sisfis

6 4 M. El-Dom μ( τ) ν( τ) δ( τ) k( τ) λ( τ) s( ) μ( τ) ν( τ) δ( τ) k( τ) λ( τ) δ( ) [ p(, ) i(, ) ] d, >, s (, )= (3) ντ ( ) δτ ( ) k( τλ ) ( [ B ( ) i(, ) ] τ) π ( ) ( ) ( ) ( ) k( ) ( ) d μτ ντ δτ τ λ τ τ δ( ) [ p(, ) i(, ) ] d, <, By ingring problm (3.) long chrcrisic lins = cons., nd using (3.), w find h i (, ) sisfis μ( τ) γ( τ) εk( τ) λ( τ) i ( ) μ( τ) γ( τ) εk( τ) λ( τ) k ( ) λ( ) μ( τ) ν( τ) δ( τ) k( τ) λ( τ) [( ε ){ s( ) μ( τ) b ν( τ) δ( τ) k ( τ) λ( τ) δ ( b) [ p( b, b) i( b, b) ] db} εp(, )] d, >, i (, )= (4) ε k( τ) λ( τ) i(, ) π ( ) μτ ( ) γτ ( ) εk( τ) λ( τ) k ( ) λ( ) μτ ( ) ντ ( ) δτ ( ) k( τ) λ( τ) [( ε ){[ B ( ) i(, ) ] μτ ( ) ντ ( ) δτ ( ) k( τ) λ( τ) b δ ( b) [ p(, b b) ib (, b) ] db} εp(, )] d, <, whr π ( ) is givn by π ( )= π( ). γ ( τ) (5) I is worh noing h if w cn sblish soluion for problm (3.3), hn soluion for problm (3.) is drmind, nd consqunly soluion for problm (.) is drmind by using quion (3.). To sblish h xisnc nd uniqunss of soluion o problm (.), w dfin h following s E o sisfy: E = { i(, ) : i(., ) L([, )); C[, ]), [, ), [, ], i(, ) = sup i(, ) }, L ] [,

7 AAM: Inrn. J., Vol. No. (6) 4 whr C[, ] dnos h Bnch spc of coninuous funcions in [, ] nd L ([, )) dnos h spc of quivln clsss of Lbsgu ingrbl funcions. W no h E is Bnch spc. In ordr o fcili our fuur clculions, w nd h following lmm: Lmm Suppos h xy,, hn y x y x. x Proof: L f ( x) =, hn us h mn vlu horm o sblish h rquird rsul. Also, for h sm purpos w no h by suibl chngs of vribls nd rvrsing of h ordr of ingrion, w obin h ( ) ( ) k( ) ( ) d μ τ γ τ ε τ λ τ τ k ( ) λ( ) δ( b) ( ) ( ) ( ) ( ) ( ) b μ τ ν τ δ τ k τ λ τ p( b, b) dbd d ( ) ( ) k( ) ( ) d μτ γτ ε τ λ τ τ k ( ) λ( ) δ( b) ( ) ( ) ( ) k( ) ( ) d b μτ ντ δτ τ λ τ τ p(, b b) dbd d μτ ( ) γτ ( ) εk( τ) λτ ( ) = k ( b ) λ( ) δ( ) b b ( ) ( ) ( ) ( ) ( ) k b d p(, b) d ddb. μτ ντ δτ τ λτ τ In h nx horm, w prov h xisnc nd uniqunss of soluion o problm (.) vi fixdpoin horm. Thorm Problm (.) hs uniqu soluion h xiss for ll im. Proof: Dfin h s Q by Q { i Ei i M} sisfis h following: (6) = (, ), (, ), (, ), whr M is consn which ( ( ) M > p( ) L β. μ ) (7) W no h Q is closd s in. Now, for fixd iniil g-disribuions s ( ), i ( ), r ( ) nd p ( ), dfin h mpping T : Q E E by E

8 43 M. El-Dom μ( τ) γ( τ) εk( τ) λ( τ) i ( ) μ( τ) γ( τ) εk( τ) λ( τ) k ( ) λ( ) μ( τ) ν( τ) δ( τ) k( τ) λ( τ) [( ε ){ s( ) μ( τ b ) ν( τ) δ( τ) k ( τ) λ( τ) δ ( b) [ p ( bb, ) i ( bb, )] db} εp (, )] d, >, Ti(, )= (8) ε k( τλ ) ( τ) i(, ) π ( ) μτ ( ) γτ ( ) εk( τ) λ( τ) k ( ) λ( ) μτ ( ) ντ ( ) δτ ( ) k( τ) λ( τ) [( ε ){[ B ( ) i(, ) ] μτ ( ) ντ ( ) δτ ( ) k( τ) λ( τ) b δ ( b) [ pb (, b) ib (, b) ] db} εp(, )] d. <, W noic h, by h priori sim (3.8), w obin h following sim for i (, ): ( ( ) ) id (, ) p ( ). L β μ (9) Accordingly, w s h, T mps Q ino Q. Now, w look for fixd poin of his mpping o provid xisnc nd uniqunss of soluion for problm (.). To his nd, w l i (, ) nd i (, ) b lmns of Q, hn using (3.6)-(3.8), (3.5), nd Lmm (3.), w obin h following: Ti(., ) Ti (.,) K( M, ) i(., ) i (., ) d, L L () whr KM (, ) is consn which dpnds on M nd. Thrfor, Ti(., ) Ti (., ) K( M, ) i(., ) i (., ). () And hus, by inducion, for ch posiiv ingr n, w obin [ (, )] n n n KM T i(., ) T i(., ) i(., ) i (., ). () n! N Inquliy (3.) implis h hr xiss posiiv ingr N such T is sric conrcion on Q. Thus T hs uniqu fixd poin in Q. Sinc is rbirry, i follows h problm (.) hs uniqu soluion h xiss for ll im. This compls h proof of h horm. In h nx horm, w show h soluions of problm (.) dpnd coninuously on h iniil gdisribuions, hrfor, problm (.) is wll posd.

9 AAM: Inrn. J., Vol. No. (6) 44 Thorm 3: L p(, ) nd p (, ) b wo soluions of problm (.) corrsponding o iniil gdisribuions p( ), s( ), i( ), r( ) nd p( ), s( ), i( ), r( ), rspcivly. Also, suppos h p(, ) = B( ) nd p(, ) = B( ), nd l i (, ) nd i (, ) b h corrsponding soluions of problm (3.). Thn h following propris hold: ( β ( ) μ ) B ( ) B( ) β ( ) p( ) p ( ), ( β ( ) (., ) (., ) ( ) ( ), L L p p p p i(., ) i (., ) L L μ ) K ( M, ) [ s ( ) s ( ) i ( ) i ( ) C p ( ) p ( ) ], (4) L L L (3) whr C is consn h dpnds on h prmrs of h modl nd. Proof. No h (3.) nd (3.3) follow dircly from (3.6) nd (3.8), rspcivly, by linriy. To obin (3.4), firs w us (3.3) nd (3.5), nd hn (3.9) o obin h following: i(., ) i (., ) [ s ( ) s ( ) i ( ) i ( ) C p ( ) p ( ) ] L L L KM (, ) i(., ) i(., ) d. L Now, h forgoing inquliy yilds (3.4) by h id of Gronwll s inquliy. This compls h proof of h horm. W no h (3.)-(3.4), show h soluions of problm (.) dpnd coninuously on h iniil g-disribuions, nd hrfor, problm (.) is wll posd. L Th Sdy Ss In his scion, w look h sdy s soluion of problm (.), undr h ssumpion h h ol populion hs lrdy rchd is sdy s disribuion p ( ) = cπ ( ), i.., w ssum h (3.4) is sisfid wih p =, s, for xmpl, Busnbrg, l. (988, p. 379). W considr h following rnsformions, clld h g-profils of suscpibl nd infciv, rspcivly: s (, ) i (, ) u (, )=, v (, )=. p ( ) p ( ) Thn wih hs rnsformions, (3.9)-(3.) sisfy h following sysms of ingro-pril diffrnil quions: u (, ) u (, ) ν ( u ) (, ) = k( ) u(, ) λ( ) δ( )[ u(, ) v(, )], >, >, (5) u(,)= v(,),, u (,) = u( ),,

10 45 M. El-Dom v (, ) v (, ) γ ( v ) (, ) = k( ) u(, ) λ( ) εk( ) λ( )[ u(, ) v(, )], >, >, v(,)= q β() π()(,) v d,, v (,)= v ( ),. (6) A sdy s u ( ), v ( ), nd λ mus sisfy h following quions: du ( ) d u () = v (), [ ν( ) k( ) λ δ( )] u ( )= δ( )[ v ( )], >, (7) dv ( ) [ γ( ) εk( ) λ ] v ( ) = λ k( )[ ε ( ε) u ( )], >, d v ()= q β( ) π( ) v ( ) d, (8) λ = c k ( ) π( ) v ( ) d. (9) Aniciping our fuur nds, w dfin hrshold prmr R ν, nd is givn by whr ( ) f γτ ( ) R = c k ( ) π( ) k ( ) D ( ) dd ν γτ ( ) f cq β( ) π( ) k ( ) D ( ) dd k ( ) π ( ) d, q β( ) π ( ) d D nd π ( ) r dfind s f (3) [ ντ ( ) δτ ( )] d Df ( )= ( ε ) ν ( ) τd, (3) π π ( )= ( ). γ ( τ) (3)(33) Hr, w no h h hrshold prmr R ν, known s h rproducion numbr in h prsnc of h vccinion srgy ν ( ), s in Hdlr, l. (996) or h n rplcmn rio, s in Thim (). And if w s ν ( ) =, in h formul for R ν, hn w obin R, usully clld h bsic rproducion numbr, nd is inrprd s h xpcd numbr of scondry css producd, in lifim, by n infciv, in olly suscpibl populion. Also, from quions (4.6)-(4.7), w cn s h R ν dcrss wih ν ( ), nd hnc Rν < R. In h following rsul, w drmin spcil cs of h sdy s soluion of problm (.), nd, in priculr, w look wh is clld diss-fr quilibrium, nd w show h i is possibl h λ = nd posiiv ndmic quilibrium xiss, in fc, coninuum of posiiv ndmic

11 AAM: Inrn. J., Vol. No. (6) 46 quilibriums. This bhvior is solly crd by vricl rnsmission nd is no prsn whn q =. Thorm 4 Suppos h h following condiions hold: (i) k ( ) is idniclly zro, (ii) q =, nd (iii) h suppor of γ ( ) lis o h righ of h suppor of β ( ). Thn, problm (.) givs ris o coninuum of ndmic quilibriums of h form: [ ντ ( ) δτ ( )] [ ( ) ( )] d ντ δτ τ u ( )=( v ()) δ ( ) v ( ) d, (34) whr h rl numbr v () (,], γ ( τ) v ( )= v (), (35) is rbirry. If ny on of h condiions in () dos no hold hn, if λ =, hn h sdy s of problm (.) is h diss-fr quilibrium: [ ντ ( ) δτ ( )] [ ( ) ( )] d ντ δτ τ v u δ d ( )=, ( )= ( ). (36) Suppos h h following condiions hold: (i) k ( ) is no idniclly zro,(ii) q =, nd (iii) h suppors of γ ( ) nd k ( ) li o h righ of h suppor of β ( ). Thn problm (.) givs ris o coninuum of ndmic quilibriums of h form: [ ντ ( ) δτ ( ) λ k [ ντ ( ) δτ ( ) λk( τ] u ( )=( v ()) δ ( ) v ( ) d, (37) [ γτ ( ) ελk [ γτ ( ) ελk v ( )= v () λ k ( ) F( ) d, (38) whr h rl numbr v () (,], is rbirry, nd F( ) is dfind s F (39) ( )= ε ( ε) u ( ). Proof: To prov (), w no h if w solv h sysm of h ordinry diffrnil quions (4.3)- (4.4), hn w obin (4.)-(4.3). Thn, if w s λ = in (4.3) nd usd h rsuling quion nd quion (4.4) o find v (), w find h v () is undrmind by (ii) nd (iii). Now, using quion (4.5), w obin h following quion for λ nd v () : λ = () ( ) π( ) [ γτ ( ) ελ k cv k d [ γτ ( ) ελk cλ k π k F d ( ) ( ) ( ) ( ) d. Thrfor, if w s λ =, hn w s h v () is undrmind by (i). And ccordingly, for rbirry fixd v () (,], compls h proof of (). (4) w obin n ndmic quilibrium givn by (4.9)-(4.). This To prov (), w no h i is sy o s h v () =, ihr from (4.5) nd λ =, if k ( ) is

12 47 M. El-Dom no idniclly zro or if ny of h ohr condiions in () is no sisfid, hn q β( ) π ( ) d < nd hrfor, using (4.4), w s h v () =. Now, if w us quions (4.)-(4.3), w obin (4.). This compls h proof of (). To prov (3), w no h using (4.4) nd (4.3), i is sy o s h ssumpions (i)-(iii). Thrfor, for rbirry fixd v () (,], v () is undrmind, by w us quion (4.5) o drmin ls on λ >. To his nd, w rwri quion (4.5) in h following form: [ γτ ( ) ελk λ c k ( ) π( ) k( ) F( ) d d π [ γτ ( ) ελ k = cv () k ( ) ( ) d. (4) W cn sily s h h righ-hnd sid of (4.6) is dcrsing funcion of λ, wih vlu grr hn zro whn λ =, sinc k ( ) is no idniclly zro, nd nds o zro if λ. On h ohr hnd, h lf-hnd sid of (4.6) hs vlu qul o zro whn λ = nd pprochs, whn λ. Thrfor, quion (4.6) hs soluion λ >, nd his vlu of λ givs ris o n ndmic quilibrium vi quions (4.)-(43). Hr, w no h h sysm of ODEs (4.3)-(4.4) hs uniqu soluion for fixd λ nd known v (). Also, w cn sily s h u ( ) nd known v () : whr f λ sisfis h following ingrl quion, which hs uniqu soluion for fixd ( ; ; v ()) nd u ( ) = f( ; λ ; v ()) K( s, ; λ ) u ( s) ds, λ (4) Ksλ (, ; ) r funcions h dpnd on h prmrs of h modl only, for fixd λ nd known v (), nd hy r dfind s follows: ( ; ; ()) = ( ()) ( ) ν( ) δ( ) λ k ( τ) [ ν( τ) δ( τ) λ k f λ v v δ d [ ντ ( ) δτ ( ) λk γτ ( ) εk( τ) v () δ ( ) d ελ [ ντ ( ) δτ ( ) λk δ ( ) k ( s) (43) s γτ ( ) εk( τ) ds d, ν( τ) δ( τ) λ k( τ) γ( τ) ελ k( τ) d τ s (, ; λ )= ( ελ ) () δ ( ). s Ks k s d (44) This compls h proof of (3) nd hrfor, h proof of h horm is compl. In h nx rsul, w prov h xisnc of n ndmic quilibrium whn >, R ν howvr, his

13 AAM: Inrn. J., Vol. No. (6) 48 ndmic quilibrium my no b uniqu du o possibl lck of monooniciy, w no h his is lso h cs for svrl g-srucurd pidmic modls, for xmpl, s, Csillo-Chvz, l. (998), Ch, l. (998) nd El-Dom (6). Thorm 5 Suppos h q, nd R ν >, hn λ = nd λ > r possibl sdy ss for problm (.). Proof: No h if q, hn w cn us (4.4) nd (4.3) o obin h following: [ γτ ( ) ελk q ( ) ( ) k ( ) F( ) d ελ k( τ) q β( ) π ( ) d λ β π d v () =. (45) Now, w cn us (4.5) nd (4.) o obin h ihr λ = quion: [ γτ ( ) ελk = c k ( ) π( ) k ( ) F( ) dd or λ sisfis h following [ γτ ( ) ελk ελ k( τ) cq β( ) π( ) k( ) F( ) dd k( ) π( ) d (46), ελ k( τ) q β( ) π ( ) d whr F( ) is dfind by quion (4.4). Noicing h, if q, i.., q < hn from quion (4.4), w s h u () > nd hrfor, v ( )< [, ). Accordingly, from quion (4.5), w s h, λ < c k( ) π( ) d. Now, using his vlu for λ in quion (4.) nd h fc h v ( )< [, ), nd quion (4.3), w cn dduc h h righ-hnd sid of quion (4.) is lss hn on his vlu of λ. Also, i is sy o s h h righ-hnd sid of (4.) is qul o R ν >, whn λ =. Thrfor, quion (4.) hs ls on soluion λ >. This compls h proof of h horm. Hr, w no h from horm (4.), n ndmic quilibrium would xiss if R ν >, nd from quion (4.6) h ffc of vricl rnsmission vi is prmr q which is h probbiliy of vriclly rnsmiing h diss, is sn R ν incrss wih q [,), nd hrfor incrss h liklihood h n ndmic will occur. So, in ordr o prvn n oubrk nd conrol h sprd of h diss, w nd o rduc R ν o vlu lss hn on. If ν ( ) is consn hn hr xiss uniqu vlu for ν ( ) which rducs R ν o on, bu if ν ( ) is g-dpndn nd no consn hn ν ( ) cn b chosn ccording o som consrin h rducs h cos of vccinion or in gnrl o obin wh is clld n opiml vccinion srgy, for xmpl, s M u llr (994), (998), Hdlr, l. (996), Csillo-Chvz, l. (998) nd Li, l. (4).

14 49 M. El-Dom If q =, nd ohr condiions hold, for xmpl, s Thorm (4.), hn problm (.) givs ris o coninuum of ndmic sdy ss, nd his siuion dos no occur whn hr is no vricl rnsmission. Also, if q =, nd γ ( ) is idniclly zro, hn problm (.) hs h sdy s s h ol populion consising of infciv only. Also, i is sy o s h his sdy s is cully, by uniqunss, h soluion for problm (.), in his spcil cs. Th ffcs of crin vccinion srgis for h rdicion of imporn communicbl disss such s msls, rubll, prussis nd ubrculosis r dl wih in svrl pprs, for xmpl, s Hhco (983), (), (997), (989), Diz (98), Knox (98), McLn (986), Kzmnn, l. (984), Schnzl (984), Couinho, l. (993), Grnhlgh (99) nd Andrson, l (999). Sbiliy of h Sdy Ss In his scion, w sudy h sbiliy of h sdy ss of problm (.), nd, in priculr, w sudy h sbiliy of h diss-fr quilibrium, nd h ndmic quilibriums. Thorm 6 Th diss-fr quilibrium, givn by quion (4.), is loclly sympoiclly sbl if R ν <, nd unsbl if R ν >. Proof. Srighforwrd linrizion of quions (4.)-(4.) round h diss-fr quilibrium yilds h following chrcrisic quion: [ γτ ( ) ξ] = c k ( ) π( ) k ( ) F( ) dd [ γτ ( ) ξ] β π π (47) ξ cq ( ) ( ) k ( ) F( ) d d k ( ) ( ) d, ξ q β( ) π ( ) d whr F( ) is givn by quion (4.4) nd u ( ) in h dfiniion of F( ) is dfind s in quion (4.), nd ξ is complx numbr. W no h, whn ξ is rl, hn h righ-hnd sid of quion (5.) is dcrsing funcion of ξ, nd pprochs zro s ξ, nd quls R ν whn ξ =. Thrfor, quion (5.) hs soluion ξ > if R ν >. Accordingly, h rivil quilibrium is unsbl if R ν >. And if R ν <, hn i is clr from (4.6) h h only possibl soluions of quion (5.) mus sisfy ξ <. Th locl sympoic sbiliy of h diss-fr quilibrium is compld by obsrving h h rl roo of quion (5.) hs h dominn rl pr, nd his is obind by considring bsolu vlus. This compls h proof of h horm. In h nx rsul, w show h h diss-fr quilibrium is globlly sbl whn h R ν < R. Thorm 7 Th diss-fr quilibrium is globlly sbl whn R <. R <. W no Proof. By using quion (3.3), w find h i (, ) sisfis

15 AAM: Inrn. J., Vol. No. (6) 5 μ( τ) γ( τ) εk( τ) λ( τ) i ( ) cπ( ) k ( ) ( ) k( ) ( ) d γ τ ε τ λ τ τ λ ( ) F (, ) d, >, i (, )= ε k( τ) λ( τ) i(, ) π( ) cπ( ) k ( ) γτ ( ) εk( τ) λ( τ) λ( ) F (, ) d, <, (48) whr F(, ) is dfind s F (, )= ε ( ε ) u (, ). (49) From problm (.), β i(, ) = q ( ) i(, ) d, hn using (5.), w obin h following: ε k( τ) λ( τ) (, )= { β( ) π ( ) (, ). i q i d γτ ( ) εk( τλ ) ( τ) c β( ) π( ) k ( ) λ( ) F(, ) dd γ( τ) εk( τ) λ( τ) c β( ) π( ) k ( ) λ( ) F(, ) d d ( ) ( ) ( ) ( ) μ τ γ τ εk τ λ τ mod.5 cm. β( ) π( ) i ( ) d}. (5) Also, from problm (.), λ()= k( id )(,), hn using (5.), nd chnging h ordr of ingrion svrl ims nd mking ppropri chngs of vribls yilds λ ε k( τλ ) ( τ) () = k ( ) π( )(, i ) d. γτ ( ) εk( τλ ) ( τ) c k ( ) k ( ) π( ) λ( ) F (, ) dd (5) ( ) ( ) k( ) ( ) d μ τ γ τ ε τ λ τ τ cm k i d mod.5. ( )( ) ( ) No h by Assumpions -5 of scion nd h domind convrgnc horm, w obin ( ) ( ) ( ) ( ) μ τ γ τ εk τ λ τ ( ) ( ),. k i d s Also, by similr rsoning s bov, w obin ( ) ( ) ( ) ( ) μ τ γ τ εk τ λ τ β ( i ) ( ) d, s..

16 5 M. El-Dom And γ( τ) εk( τ) λ( τ) c β( ) π( ) k ( ) λ( ) F(, ) dd, s. limsup Now, w l i = i(, ) nd λ = λ( ), hn from quions (5.4)-(5.5) nd Fou s Lmm, w obin h following: limsup i qi β( ) π ( ) d cqλ β( ) π( ) k ( ) dd, γτ ( ) λ i k ( ) π ( ) d cλ k ( ) π( ) k ( ) dd. Thrfor, λ λ R < λ, sinc R <, which givs λ =. Accordingly, h diss-fr quilibrium is globlly sbl, if R <. This compls h proof of h horm. In ordr o sudy h sbiliy of n ndmic quilibrium, w linriz h sysm of quions (4.)- (4.) by considring prurbions w (, ) nd η (, ) dfind by w u u η (, ) = (, ) ( ), (, ) = v (, ) v( ). Accordingly, w obin h following sysms of ingro-pril diffrnil quions: w (, ) w (, ) ( ) ( ) k( ) w(, ) ν δ λ = δ( ) η(, ) k( ) u ( ) ψ, >, >, w(, ) = η (, ),, w (,)= w( )= u( ) u( ),, (5) η(, ) η(, ) ( ) k( ) (, )=( ) k( ) w(, ) γ ελ η ε λ k( ) ( ε) u ( ) ε( v ) ψ, >, >, η(, ) = q β( ) π( ) η(, ) d,, η(,)= η( )= v( ) v ( ),, whr ψ ( ) is givn by ψ ()= c k ( ) π( ) η(,) d. (53) (54) Now, w ssum h w (, ) = ξ f( ), η(, ) = ξ g ( ),

17 AAM: Inrn. J., Vol. No. (6) 5 whr ξ is complx numbr. Accordingly, w obin h following sysms of ODEs: f ( ) ξ ν( ) λ k( ) δ( ) f( )= δ( ) g( ) k( ) u ( ) ψ, f() = g(), (55) g ( ) ξ γ( ) ελ k( ) g( )=( ε) λ k( ) f( ) k( ) ( ε) u ( ) ε( v ) ψ, g()= q β( ) π( ) g( ) d. whr ψ is dfind s ψ = c k( ) π ( gd. ) ( ) (56) Using (5.9)-(5.), w obin h following chrcrisic quion, in h cs ψ : [ ξ γ( τ) ελ k = c k ( ) π( ) k ( ) F ( ) dd [ ξ γ( τ) ελ k ξ ελ k( τ) cq β( ) π( ) k ( ) F( ) dd k ( ) π( ) d, ξ ελ k( τ) q β( ) π( ) d (57) whr F ( ) is dfind s ( ελ ) f( ) F ( )= ( ε) u ( ) ε( v ( ). ψ (58) W no h if w s λ = in (5.), w obin (5.). In h following rsul, w sblish h locl sympoic sbiliy of n ndmic quilibrium, in h spcil cs ε =. Thorm 8: Suppos h sympoiclly sbl. R ν >, q, nd ε =, hn n ndmic quilibrium is loclly Proof: W no h if ψ =, hn g ( )= from quion (5.), sinc g () = bcus ε =. Thrfor, i follows from quion (5.9) h f( )=, nd hnc, sbiliy follows in his cs. If ψ, w k ξ in quion (5.) o b rl, nd hn i is clr from h chrcrisic quion (4.) h ξ <, sinc v ( ) sisfis quion (4.3), nd R ν >, hrfor h locl sympoic sbiliy of n ndmic quilibrium follows from h fc h h rl roo of h chrcrisic quion hs h dominn rl pr. This compls h proof of h horm. In h nx rsul, w will prov h locl sympoic sbiliy of n ndmic quilibrium, whn q = nd δ ( ) is idniclly zro. This rsul will llow us o dduc h sbiliy of h ndmic

18 53 M. El-Dom quilibrium of h SIR g-srucurd pidmic modl, no h h ndmic quilibrium is uniqu in his spcil cs bcus of h monooniciy of h righ-hnd sid of quion (4.). Thorm 9: Suppos h R ν >, q = nd δ ( ) is idniclly zro. Thn h ndmic quilibrium is loclly sympoiclly sbl. Proof: No h if ψ = q =, nd δ ( ), hn i is sy o s h f( ) = from (5.9), sinc f() = g() =. Accordingly, i follows h g ( ) =. chrcrisic quion (5.) ks h following form: [ ξ γ( τ) ελ τ π And hnc, if k d = c k ( ) ( ) k ( ) F ( ) dd, ψ, hn h (59) whr F ( ) is givn by quion (5.). Now, (5.3) cn b rwrin in h following form: = c k ( ) π( ) k ( ) [ ξ γ( τ) ελ k F k s u s ds v d d whr F( ) is givn by (4.4). [ ξ ν( τ) λ k s ( ) λ ( ε) ( ) ( ) ε ( ), (6) Thrfor, if w ssum h ξ is rl in (5.4), hn from quion (4.), w conclud h ξ <, nd ccordingly, h locl sympoic sbiliy follows from h fc h h rl roo of h chrcrisic quion hs h dominn rl pr. This compls h proof of h horm. No h if δ ( )= q = ε =, hn w obin h SIR g-srucurd pidmic modl, nd horm (5.4) shows h h ndmic quilibrium of h SIR g-srucurd pidmic modl is loclly sympoiclly sbl. In h nx rsul, w prov h locl sympoic sbiliy of n ndmic quilibrium in h cs q, nd δ ( ). Thorm : Suppos h h following hold: (i) R ν >, (ii) δ ( ), nd ντ ( ) ( ελ ) k( τ) γτ ( ) (iii) ( ελ ) k ( ) d<. Thn n ndmic quilibrium is loclly sympoiclly sbl. Proof: If ψ =, hn from quions (5.9)-(5.), w obin h following: cg() k ( ) π ( ) ξ ελ k( τ) γ( τ) ντ ( ) ( ελ ) k( τ) γτ ( ) [ ( ελ ) k ( ) d] d =. From quion (5.5) nd ssumpion (iii), w dduc h g() =, f () =, nd from ssumpion (ii) nd quion (5.9), w obin h nd quion (5.), w obin h g ( )=. (6) nd hn i follows h f( )=. From f( )=

19 AAM: Inrn. J., Vol. No. (6) 54 g() Now, if ψ, hn using quions (5.9)-(5.) nd solving for, w obin h following ψ chrcrisic quion: ξ ελ k( τ) q β( ) π ( ) d [ ξ γ( τ) ελ k π { c k ( ) ( ) k ( ) F( ) d d [ ξ γ( τ) ελ k ξ ελ k( τ) cq β( ) π( ) k( ) F( ) dd k( ) π( ) d } ξ ελ k( τ) q β( ) π( ) d ξ ελ k( τ) c q β( ) π ( ) d [ ξ γ( τ) ελ k k k v d d { ε ( ) π( ) ( ) ( ) λ ( ε) s [ ξ γ( τ) ελ k [ ξ ν( τ) λ k k () π() k ( ) k () s u () s dsdd} [ ξ γ( τ) ελ k [ ξ ν( τ) λ k qλ ( ε) β( ) π( ) k ( ) dd{ [ ξ γ( τ) ελ k k ( ) π( ) k( ) F( ) εv ( ) dd cλ ( ε) s [ ξ γ( τ) ελ k [ ξ ν( τ) λ k k () π() k ( ) k () s u () s dsdd}= εqc [ ( ) ( )] ( ) β π π ( ) ( ) ( ) ( ) c ξ γ τ ελ k τ ξ ελ k τ ( ) ( ) k ( ) v ( ) d d( k ( ) ( ) d) [ ξ γ( τ) ελ k [ ξ ν( τ) λ k qcλ ε k π k dd ( [ ξ γ( τ) ελ k β ( ) π ( ) k ( )[ F( ) ε v ( )] d d) qc λ ( ε ) [ ξ γ( τ) ελ k [ ( ) k( )] d ξ ν τ λ τ τ s β( ) π( ) k ( )( k ( s) u ( s) d ) ( s d d

20 55 M. El-Dom [ ξ γ( τ) ελ k [ ν( τ) ( ε) λ k ( τ) γ k( ) π( )[ λ ( ε) k( ) d] d). (6) Now, suppos h ξ is rl, hn from h chrcrisic quion (5.6) nd quion (4.), nd h fc h R ν >, w obin h ξ <, sinc for ξ, h righ-hnd sid of (5.6) is nonposiiv whil h lf-hnd sid is posiiv. Thrfor, n ndmic quilibrium is loclly sympoiclly sbl, sinc h rl roo hs h dominn rl pr. This compls h proof of h horm. W no h if, in ddiion o h ssumpions of Thorm (5.5), w ssum h ε = in h modl, hn w obin h SIR g-srucurd pidmic modl sudid in Ch, l. (998) nd hrfor, Thorm (5.5) sblishs h locl sympoic sbiliy for h ndmic quilibrium of h modl. Also, w no h if δ ( ), nd ε =, hn condiion (iii) of Thorm (5.5) gurns h uniqunss of h ndmic quilibrium, nd his follows from h monooniciy of h righ-hnd sid of quion (4.). Now, w look h cs in which δ ( ) is no idniclly zro bu q =, for xmpl, s Li, l. (4). In his cs, w firs us h sysm of ODEs (5.9)-(5.) o obin h if ψ =, hn f( )= g( )=, nd h is bcus f() = g() = nd h uniqunss of soluion of h sysm of ODEs. Also, if w ssum h ψ, hn using h sm sysm of quions, w cn dduc ( ) h G ( )= g sisfis h following Volrr ingrl quion: ψ whr Ks (, ) nd d ( ) r givn by G ( )= KsGsds (, ) ( ) d ( ), (63) Ks (, ) = ( ) ( ) ( ) s s [ ξ γ( τ) ελ k [ ξ ν( τ) δ( τ) ( ε) λ k( τ) γ s λ εδs k d, (64) d ( )= [ ξ γ( τ) ελ k [ ( ) ( ) k( )] d ξ ν τ δ τ λ τ τ s λ ( ε) k( )( ( k( su ) ( sdsd ) ) (65) [ ξ γ( τ) ελ k k ( )[ F( ) εv ( )] d. Also, in his cs h chrcrisic quion is givn by h following quion: = c G( ) k ( ) π ( ) d. (66) W no h h Volrr ingrl quion (5.7) hs uniqu soluion, nd his fc cn lso b sn from h sysm of ODEs (5.9)-(5.).

21 AAM: Inrn. J., Vol. No. (6) 56 In h nx rsul, w show h if h forc of infcion is sufficinly smll, hn n ndmic quilibrium is lwys loclly sympoiclly sbl. Thorm : Suppos h R ν >, nd q=. If λ is sufficinly smll, hn n ndmic quilibrium is loclly sympoiclly sbl. g ( ) [ v (, ) v( )] Proof: W no h G ( )= = ξ, nd so for R ξ nd ψ, G ( ) is ψ ψ boundd nd is ingrl from o is lso boundd, bcus w r ssuming h h ol populion hs lrdy rchd is sdy s. Now, sinc ε >, w cn us quion (4.) o show h h following ingrl is boundd: [ γτ ( ) ελk π c k ( ) ( ) k ( ) d d. Using quions (5.7) nd (5.), w obin h = c Ksk (, ) ( ) π( Gsdsd ) ( ) c k( ) π( dd ) ( ). Thrfor, from quions (5.8)-(5.9), w obin h h righ-hnd sid of quion (5.) pprochs [ γτ ( ) ελk c k ( ) π( ) k ( )[ F( ) εv ( )] dd s. λ And hrfor, in viw of quions (4.) nd (4.3), h righ-hnd sid of quion (5.) is sricly lss hn on for ny ξ wih R ξ, nd λ sufficinly smll. Accordingly, n ndmic quilibrium is loclly sympoiclly sbl if λ is sufficinly smll. This compls h proof of h horm. Similrly, w obin h following ingrl quion in h cs δ ( ) is no idniclly zro, q, nd R ξ : (67) (68) (69) G ( )= K(, s) G ( s) ds d( ) q q [ ξ γ( τ) ελ k β π ε H( ) q { ( ) ( ) k ( ) F ( ) v ( ) d d Δ (7) [ ξ γτ ( ) ελk [ ξντ ( ) δτ ( ) λk s λ ( ε) β( ) π( ) k( ) δ sgq() s k() su() s dsd d}, whr Ks (, ) nd d ( ) r s bfor, r givn by quions (5.8)-(5.9), nd H ( ) nd Δ r dfind s follows H ( )= [ ξ γ( τ) ελ k k [ ξ γ( τ) ελ k [ ξ ν( τ) δ( τ) λ k λ ( ε) ( ) d, (7)

22 57 M. El-Dom [ ξ γ( τ) ελ k β π Δ = q ( ) ( ) d (7) [ ξ γ( τ) ελ k [ ξ ν( τ) δ( τ) λ k qλ ( ε) β( ) π( ) k ( ) dd. Also, in his cs, h chrcrisic quion is givn by = c G ( ) k ( ) π ( ) d. q (73) W no h if w pu q = in quion (5.4), hn w obin h Volrr ingrl quion (5.7). Also, w no h in h cs q, δ ( ) no idniclly zro, nd R ξ, hn if w ssum h ψ =, hn from quions (5.9)-(5.), w obin h following wo formuls for g(): [ ξ γ( τ) ελ k λ ( ε) ( ) π( ) ( ) ( ) [ ξ ελ k k ( ) π ( ) d k k f d d g() =, (74) q ( ) ( ) ( ) k( ) f( ) β( ) π ( ) [ ξ γ( τ) ελ k λ ε β π dd g() =. [ ξ ελ k q d (75) And hrfor, if w ssum h β ( ) nd k ( ) r consns in quions (5.8)-(5.9), w find h g() =, nd ccordingly, f() =, nd hnc, f() = g() =, by h uniqunss of soluion for h sysm of ODEs (5.9)-(5.). In h nx rsul, w show h n ndmic quilibrium of problm (.), is loclly sympoiclly sbl, if h forc of infcion is sufficinly smll. Thorm Suppos h R ν >, nd k ( ) nd β ( ) r consns indpndn of g. If λ is sufficinly smll, hn n ndmic quilibrium is loclly sympoiclly sbl. Proof: Th proof of his horm is similr o h proof of Thorm (5.6), nd hrfor, w omi h proof. Conclusion W sudid n g-srucurd SIRS pidmic modl, whn h diss is vriclly s wll s horizonlly rnsmid nd h forc of infcion of proporion mixing ssumpion yp, suscpibl individuls r vccind wih vccin h wns ovr im, nd hrfor immun individuls r suscpibl wih som rsisnc o h diss. Th morliy nd friliy rs r g-dpndn. W provd h xisnc nd uniqunss of soluion o h modl quions nd showd h soluions of h modl quions dpnd coninuously on h iniil g-disribuions, nd hrfor,

23 AAM: Inrn. J., Vol. No. (6) 58 h wll posdnss of h problm is provd. Furhrmor, w drmind h sdy ss of h modl nd xmind hir sbiliy, whn q, by drmining n xplicily compubl hrshold prmr R ν, in rms of h dmogrphic nd pidmiologicl prmrs of h modl, known s h rproducion numbr in h prsnc of vccinion srgy ν ( ), s in Hdlr, l. (996) or h n rplcmn rio, s in Thim (). And hnc, w showd h R ν incrss wih q, which is h probbiliy of vriclly rnsmiing h diss, nd hrfor, incrss h liklihood h n ndmic will occur; lso R ν dcrss wih ν ( ), nd is usd o drmin criicl vccinion covrg which will rdic h diss wih minimum vccinion covrg. If R ν, hn h only sdy s of problm (.) is h diss-fr quilibrium, nd is loclly sympoiclly sbl if R ν <, nd globlly sbl if R <. If R ν >, hn diss-fr quilibrium nd n ndmic quilibrium r possibl sdy ss, h diss-fr quilibrium is unsbl. Th qusion of uniqunss of n ndmic quilibrium is n opn problm, nd his is lso h cs for svrl g-srucurd pidmic modls, for xmpl, s Csillo-Chvz, l. (998), Ch, l. (998), nd El-Dom (6). Concrning h sbiliy of n ndmic quilibrium, w obind complicd chrcrisic quion h llowd o prov locl sympoic sbiliy, in som spcil css, which r: (i) h cs ε =, (ii) h cs q =, nd δ, (iii) h cs δ, nd ντ ( ) ( ελ ) k( τ) γτ ( ) ( ελ ) k ( ) d<. And w drivd gnrl formuls for h chrcrisic quions, in rms of ingrl quions. Alhough xplici formuls for h soluions of hs ingrl quions r no obind, w r bl o us hm o dduc h if h forc of infcion is sufficinly smll, hn n ndmic quilibrium is lwys loclly sympoiclly sbl. If q =, nd ohr condiions hold, for xmpl, s Thorm (4.), hn problm (.) givs ris o coninuum of ndmic sdy ss, nd his siuion dos no occur whn hr is no vricl rnsmission. Also, if q =, nd γ ( ) is idniclly zro, hn problm (.) hs h ol populion consising of infciv individuls only s h sdy s. Acknowldgmns Th uhor wro his ppr whil h ws visiing h Insiu of Mhmics of h Univrsiy of Posdm, Posdm, Grmny, nd h would lik o hnk h Grmn Acdmic Exchng Srvic, DAAD: Duschr Akdmischr Aususch Dins.V., for suppor nd Prof. Dr. N. Trkhnov nd Prof. Dr. B.-W. Schulz for n inviion nd hosbiliy during his sy in h Insiu of Mhmics. This work is compld whil h uhor is n Arb Rgionl Fllow h Cnr for Advncd Mhmicl Scincs (CAMS), Amricn Univrsiy of Biru, Biru, Lbnon, h is suppord by grn from h Arb Fund for Economic nd Socil Dvlopmn, nd h would lik o hnk h Dircor of CAMS, Prof. Dr. Wfic Sbr, for n inviion nd hospiliy during his sy in CAMS. H would lso lik o hnk Profssor Mimmo Innlli nd Dr. Xu-Zhi Li for snding rfrncs, nd wo nonymous rfrs for hlpful commns nd vlubl suggsions on h mnuscrip.

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