A Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate

Transcription

1 A Condiion for abiliy in an I Ag rucurd Disas Modl wih Dcrasing urvival a A.K. upriana, Edy owono Dparmn of Mahmaics, Univrsias Padjadjaran, km Bandung-umng 45363, Indonsia fax: , mail: asupria@yahoo.com.au; aksupriana@bdg.cnrin.n.id Financial and Indusrial Mahmaics Group ITB, Bandung 43, Indonsia Absrac In his papr w prsn an I modl for disas ransmission wih an assumpion ha individuals in h undr-laying dmographic populaion xprinc a monoonically dcrasing survival ra. W show ha h rsuls in an analogous I modl for disas ransmission ar h spcial cas of h I modl in his papr. W found ha hr is a hrshold for h disas ransmission drmining h xisnc and h absnc of h ndmic quilibrium. W invsiga h sabiliy of his quilibrium via a Gronwall-lik inqualiy horm. Unlik in h I modl, h hrshold for h xisnc is no quivaln o h hrshold for h sabiliy of h quilibrium. W provid an addiional condiion which consisnly gnralizs h rsuls in h I modl.. Kywor : Disas Modling, I Modl, Thrshold Numbr, abiliy of an Equilibrium Poin I. Inroducion Ag srucur is among h imporan facors affcing h dynamics of a populaion in rlaion o h sprad of conagious disass. To sudy h ffc of ag srucur in h dynamics of conagious disass, a las hr ar wo approachs, firs by dvloping a populaion modl wih coninuous ag [,] and scond by dvloping a populaion wih ag groups [3]. A modl of I disas ransmission is sudid in [] and a modl I disas ransmission is sudid in [] by assuming coninuous ag. An I modl only fis o disass ha caus an infciv individual rmains infciv for lif. To incras ralism, in his papr w prsn a modl for an I disas ransmission by assuming coninuous ag. r w assum individuals in h undr-laying populaion xprinc a monoonically dcrasing survival ra as hir ag gos by. W also assum ha hr is a dnsiy-dpndn bu ag-indpndn birh ra. W show ha hr is an ndmic hrshold, blow which h disas will sop, and abov which h disas will say ndmic. Th rsuls in h I disas modl in [] gnraliz ino h I modl. II. Th Mahmaical Modl Th modl discussd hr is h gnralizaion of h modl in [] o includ an comparmn as an amp o incras h ralism of h modl. Throughou h papr w us h following noaions: N = Toal numbr of individuals in h populaion = Th numbr of suscpibl individuals in h populaion I = Th numbr of infciv individuals in h populaion = Th numbr of rcovr or immun individual in h populaion B = Th rcruimn ra or h birh ra β = Th ransmission probabiliy of h disas W assum ha h populaion N is dividd ino hr comparmns,, I, and, such ha N = I. To includ ag srucur, suppos ha hr xiss Q ( a ), a funcion of ag dscribing h fracion of human populaion who survivs o h ag of a or mor, such ha, Q = and Q ( a ) is a non-ngaiv and monoonically dcrasing for a. If i is assumd ha lif xpcancy is fini, hn Q a = L< and aq. a < Furhr, l also assum ha N,, I, and dnos, rspcivly, h numbrs of N,, I, and who surviv a im. Thn w hav

2 N N B Q ( a) =. inc h pr capia ra of infcion in h populaion a im is β I, hn h numbr of suscpibl a im is givn by a β I s B Q =. 3 If h ra of rcovry is γ hn h numbr of infciv a im is givn by βi γ a a I = I B Q βi a = I B. Q Furhrmor, considring ha = N I hn w hav = N BQ( a) βi a BQ( a) βi a I B Q βi a = B. Q I is clar ha N =, =, I =, and =. 6 nc, quaions (3), (4), (5), and (6) consiu an I ag srucurd disas modl. III. Th Exisnc of a Thrshold Numbr In his scion w will show ha hr is a hrshold numbr for h modl discussd abov. L us considr h following i sysm of quaions which has h sam bhavior wih h sysm (3) o (6) whnvr : N B Q a I a β =, 7 = B Q, 8 βi a I = B Q 9 βi a = BQ( a) Equaions (7) show ha h valu of N 4 5 is consan, hnc h quaions for h agsrucurd I modl rduc o hr quaions, (8) o. Th Equilibrium of h sysm is givn by (, I, ) wih I saisfying I a I a γ = B Q a. I is asy o s ha (, I, ) = ( N,,) is h disas-fr quilibrium. To find a non-rivial quilibrium (an ndmic quilibrium), w could obsrv h following. Ia βb Q( a) =. β I Th L of () is a funcion of I, say Ia a γ f ( I) B Q( a) β I This funcion is monoonically dcrass wih f( I ) I Ia B Q a I βi B aq f( I ) I Ia B Q I βi = I Thrfor, a uniqu non-rivial valu of if and only if occurs B aq( a) >. 6 inc h L of (6) drmins h occurrnc of h non-rivial valu of I, hn i will b rfrd as a hrshold numbr of h modl. nc, an ndmic quilibrium (, I, ) ( N,,) occurs if and only if >. IV. Th abiliy of h Equilibria To invsiga h sabiliy of h quilibria w us h mhod in [] and us h lmma hrin.

3 LEMMA 4.. (BAUE, ). L f b a boundd non-ngaiv funcion which saisfis an sima of h form f f f( a) ( a), whr f is a non-ngaiv funcion wih f = and a is a non-ngaiv funcion wih a <. Thn f. = POOF. []. I is also showd in [] ha h lmma is sill ru if h inqualiy in h lmma is rplacd by f f sup f s ( a). 7 a s Th following lmma is h xnsion of Braur s lmma. LEMMA 4.. L f j, j =, b boundd nonngaiv funcions saisfying f f sup f s ( a), a s f f sup f s ( a), a s whr f j is non-ngaiv wih f j = and j ( a) is non-ngaiv wih j ( a) <. Thn li m f ( ), j j = =,. POOF. f sup{ f, f } sup sup{ f, f }sup{ ( a), ( a)} a s f sup{ f, f } sup sup{ f, f }sup{ ( a), ( a)} a s and hnc, sup{ f, f } sup{ f, f } sup a s sup{ f, f}sup{ ( a), ( a)} From Lmma 4. w conclud ha sup{ f, f} =, and his is suffic o show ha f j =, j =,. 4.. Th sabiliy of h disas-fr quilibrium. W invsiga h sabiliy of h disas-fr quilibrium for h cas of <. Considr h following inqualiis. βi a a β I nc w hav, I = I And aβ sup I 8 a s βi a B Q ( a)( ) I B Q ( a)( aβ sup I ) 9 a s = βi a B Q β a s B Q ( a)( a sup I ) Morovr, sinc I = and B Q a aβ = < hn using Lmma 4. w conclud ha I ( ) =. Nx, l us s h xprssion a γ B which, Q a aβ if = B, can b wrin in h Q a aβ form. nc, if < hn B Q a aβ < (Appndix ). Furhrmor, sinc = hn using Lmma 4. w conclud ha =. Consqunly, = ( N I) = N This shows ha h disas-fr quilibrium ( I, I ) = ( N, ) is globally sabl. V 4.. Th sabiliy of h ndmic quilibrium. Th ndmic quilibrium (, I, ) appars only if. L us s h > prurbaions of I and, rspcivly, by u and v. Dfin I = I u and subsiu his quaniy ino quaion (4) o obain h following calculaions.

4 I u = I β[ I u] a B Q u = I I βi βu a a B Q Ia = B Q a I u Ia β a B Q a Ia u = B Q ( a)( ) I Ia B Q( a)( ) u Ia β a BQ( a) Ia u = BQ( a)( ) I u Ia β a B Q Ia B Q ( a)( ) I nc, w hav Ia β a s B Q ( a) a sup u Ia ( ) u B Q I sup us BQ ( a ) β a Ia a s By dfining f = u, I a a γ β a = B Q a, and I a a γ = f B Q ( a)( ) I, hn w hav f f sup f s ( a). a s f = W s ha ha and i can b shown ( a) is non-ngaiv wih ( a) < (s Appndix ). Thn by Lmma 4. w hav f =, mans ha I = I. Nx, dfin = v and subsiu hs quaniis ino quaion (5) o obain h following calculaions: v = β[ I u] a B Q ( a)( ) v = βi βu a a B Q = B Q a Ia Ia β u s a B Q Ia = ( ) Ia BQ a u Ia β a B Q v B Q Ia v = BQ( a)( ) I u s a β a B Q Ia BQ( a)( ) Ia BQ ( a ) βa sup a s u nc, w hav Ia ( ) Ia sup a s β v B Q I us BQ a By dfining f = u, f = v Ia a = BQ βa, and ( I a ) a γ f = BQ a I, hn w hav f f sup a s f s ( a). W can show ha in Appndix 3 ha f = and ( a) is non-ngaiv wih ( a) <. Thn by Lmma 4. w hav f =, mans ha =. Finally, sinc N is a consan, =, and I ( ) I = hn (, I, ) is globally sabl.

5 V. Concluding marks In his papr w hav discussd an agsrucurd I disas modl wih a dcrasing survival ra. W found a hrshold numbr for h xisnc and uniqunss of an ndmic quilibrium, ha is, B aq( a). As is h cas of h I disas modl discuss in [], an ndmic quilibrium appars if > and disappars if <. In h I disas modl, h hrshold for h xisnc of h quilibrium is also h hrshold for h sabiliy of h quilibrium. owvr, in our cas in which hr is a rcovr comparmn, hr is an addiional condiion for h quilibrium o b sabl. r w found ha hr is a sabl ndmic quilibrium if > and B Q ( a ) a< is a sabl disas-fr quilibrium if, and hr < and B Q (. W noic ha a ) a< his condiion is consisn wih ha in [] if h rcovry ra γ =, sinc in his cas is quivaln o. nc, w conclud ha h I modl in [] is naurally nsd in h I modl discussd in his papr. VI. frncs [] F. Braur. A modl for an I disas in an ag-srucurd populaion. Discr and Coninuous Dynamical ysms ris B. (), []. Busnbrg, M. Ianlli, and.. Thim. Global bhavior of an ag-srucurd pidmic modl, IAM J. Mah. Anal. (99), [3].W. hco. An ag-srucurd modl for prussis ransmission. Mah. Biosc. 45 (997), Appndix If < hn ( a) = BQ( a) βa <. I is sraighforward from h dfiniion ha a = B and Q a aβ γ = BQ( a) aβ. Appndix W claim ha f j = and j ( a) is non-ngaiv wih j ( a) <. I is clar ha, if X, Y {, V} wih X Y hn f = j XIYa X X X BQ( a)( ) I =. To winss ha, l us procd for j a < ( a) as follows. Dfin ax IVa a g( x) = Bβ Q( a) β x which is a dcrasing funcion of x. W s ha IVa aγ a g( IV) Bβ = Q( a). βiv inc I V is an quilibrium valu, hn w hav g( I V ) =. Furhrmor, β ( a). g = B aq ( a) = g IVa Considring ha is dcrasing funcion hn ( a) = g < g( I V ) =. Appndix 3 If < hn Ia a γ a = B Q βa is lss han on. Using h rsul in Appndix w hav Ia B Q βa BQ( a) βa <.