emic Mod

Glob al Dyn amics of Two Epidemic Model s

hy

©

ZhenWang

A thesissubmitte d tot.hc SchoolofGra d uateStu d ios in pnrt.ial Fulfihucuto ft.hc roq u irc mc nts fort.hcdogrccof

Masu-rof Science

Dcp.ut.mcutofMolhcnuu.ics['1Statis tics MemorialUniversityof Newfoundland

Augnst,20 /2

St,Jo hn's Nowfouud lnnd

Abs t ract

Mat.h ctnut.icalstu d ios of inlccti ou sdiseaseinvol vedelaydillcrcntia!oq nn tio us whichare morearcu rut.c inrep rese nti nglIIodel~withge~tat. i o ut.imcs,incuhat.ion period s,or iutraccllulardcla vs,and periodic equati on» whichacc ountfor impa ctof seasonal ,ordiurnal env iro n m e nts.ThepurpO~()of thisthcsi»istoiuvcsti gutct.he globaldyn.uui csofatimc-d clnycddenguetransmi ssionmod elandaperi odic with in- host virus mod el .

\Vebeginwith nuu.hcm at.icalprclimiuuricsfor thist.hcsi ».Weprovide somerunt.he- ruati cnldefinitions andtheor emsrclat.cd tothe theoryofcoopc rn t.ivcdelaydillcrcntiul cq uu t.iou,un iform persi sten ce andcoexiste ncestutcs,chaintrunsiti vc sets,and basi«

rcpro d uc t.ion numbers.

InChap t er2,wepresent atimo-d cl a vcdden guetransiui ssionmodel wit.hage struc turefor t.hc vector population. \Ve firstintroducet.lrobusicrcp ro duct.io n uumbc r, undshowt.hu t t.hcdiseas epe rs is ts when R o

>

I. It.is aboshow n t.hatt.hcdisoaso will dieou tifR o

<

I,provid edthattheinvasionintensityi~not.strong.Wcfurther establishasetofsufficientcond itio nsfortho cxistc ncoandgloba latt rac tiv ityof the ende m ic equilibri umbytilemethodof Iluctuution s.Nu me ricalsimula t.ious arc perform ed t.oillustra te ouruu alyt.icrosu lts.

Chaptc rSisde vot edto theinvestigati onof t.hocllcc tsof peri odicdrugtrcat.mc u t onstand urd within- host.virusmodel.Wefirs t introducethebasicreproducti onrati o forthemodel,then show thatt.hcinfecti onfreeeq u ilib ri u m is globallyasym pto t.icnll,' st.ah lc and thediseas e eventually disappearsif Ro

<

I,wh ilethere ex istsat. I('a"t.one posi ti veper iodicstat.oand t.hcdiseasepersi s ts whenR o>I. \Ve alsoconside r

op ti m izat ion probl emshyshift ingthephase ofthesedrug eflieaeyfu nctio ns. Itturns out that shifting thephase cancert ain lyafrectthe st.nbilitvof theinfecti onfre(~st.<'ady state.Anumericalstudyisperformedtoillust ra te omana lyt icresults.

At last,wcsiuu uuuizctheresul tsin thist.hosis ,and alsopoin t ou tsome problems forfutureinvesti gati oninChap tc rG.

iii

A ck 110Wled g ern en ts

First aw l for em ost ,I wouldliketorecordmydeep andsinceregra tit ude 10Pro- fcssorXiaoqi au gZhaofo rhis superv isio n,ad vice.andgu ida ucc froIIIthe curlvst age of this resea rch aswell as givingme cxtraord inarvcxpc ricurcs throu gh out thework.

Above alland themostneeded.heprov idedmeunllinching cnco urugc mc utandsup- portillvario us ways.Histrulyscientistintuitionand wideknowlcd g«havenuulchim asa coustunt.oasisofidr-asandpassionsinmnr.hcuin t.irsrcsoarch,whichexcept iollafly inspireandenrichIlly gro wthas a studentanda researcher.Myt.hnn ksalso goesto 1111'S.Zhao .IIPr kind helpmndcIllylifeillSt..lohu' smuch1I1on~cnjoyahlc.

IgratefullyacknowledgePro fessorYua n Yua nfo rtc~a ch i llgmeIunct.ionnldiller- cnti ulcqua t.iouandgivillgII}(~frequent cncourugc mc nt..Iwarmlyt.lumkProfessors Xla rcoMcrkli,1'01 11BairdandHOllaldHaynesforteachillgmefuuct.ionulalialysis.

top ology, andnumcric nl soluti onor dilfcrcntiuleq ua t io n,Professor Chu nhua Ou1'01' teach ingmemod ernperturbati on theor y.IIIaddit.ion ,Iam alsoindebt edtoProfessor .JieXiaoforgivi ng me cous t.antenco urage me ntandiuva lua h lcsugges t.ions.

Itis apleasu ret.opaytribute alsotothe NSE HCofCana du,lIHTAC S of Cana da.

the Schoo lofGradua te St ud ies forprovidingfin an cialsu ppor ts.Iwouldliketot.hnu k the Department or Mat.lrcm ati csand Statisticsforprovidin gmetea ch ing assistalit fellowship andconvenient faciliti es.Thnnks alsogoest.oallsta fr member s at. the dcpnrt.mcntfortheirindisp cnsahlchelpdealingwithtravel funds,nrh uin ist.rnt.ionnnd bureaucra t icmattersduri ngmystayin MllN.

Iwouldliketot.akot.his opportunityt.oexpressIllysinceregrat.i t u(k toYuxi.uu;

Zha llg ,Ha ifellgSong ,Rui1'('lIgfor t.hcir help andsup port.1I10stim portant.Iwould

iv

liketothank Che nZha ngfor his care,suppo rt and underst anding.Tha nk s goesto all myfriend sinSt. John's.Mv time her e wasmndc enjoya b le in largepartdueto lIlanyfriend s.

Last,Illy spec ialgra t.it.udc isduetomybro th er, andtuost.import.ant.ly,Illypar ent s.

to who m I.u n for everindebtfortheirlove,suppo rt and undcrs uuuling.

Contents

Abst r ac t

Ack no w led gem ents

List of Figures

1 Prolim iuar ic s

1.1 Coopera l ive d day differclIl ialcqllat.iolls. 1.2 Uuilo rm pcrsiston rc andcoexistencestntos. 1.3 Chai n t.ransit ivc scts

1.4 Bas icropro d uctionuumbcr.

2 A Time- Delay ed Den gu eTransm issio nModel 2.1 lut.rod uct.iou.

2.2 Thcmodcl. 2.:\ Thrcsholddvnum ics 2.4 Glohalut.tr.u-t.ivity 2.5 Nu mericalsim u la tions

vi

iv

ix

17

2:\

:\!i

·1·1

3 A Within-HostVirusModclwithPeri odi cMultidrugThcr ap y 47

3. [ Int ro d uction. 47

:\.2 The glohal dyna m ics. 50

3.:\ Casestu d ies. (l.[

3.3.1 Drug efficacies of t.hohan g-hangtype (il

3.:\.2 Anactualplunmarokincti cmod el. 71

3..1Discussion . 80

4 Su m ma ryan d FuturoWork 82

,1. [ Research sununary 82

4.2 Fu turewor k.. 8,1

Bibli o gr aphy 86

vii

List of Figures

2.1 Long-term bclunnoroj thepopulation ofeach. classuhcn.R.o

<

Iand the in vasionissm all. ...

2.2 E' glo/JIl.lly asymptotically atlraciiuc whenRo

>

Iandcon di tions

(I1J)and (l/2)hold. 45

2.3 Persisten ceo]injectedniosquiio sand ltunutn in dividu als.. ¥;

:3.1 Hosiereproduc ti onratioRo·I'S.e.lficac yJOl'in- phase tuul out-oi-phase G5

3.2 BusicrcproductiotiniiioTc«us.phasedUfe'l'l:ncel/J.. 3.:3 Busic.reproducti on.ratioRoasa [unctio no] (ilicw:ycand duration/1

GG

with1/)

=

O. Thehorizontal surjacc correspondsto R o

=

1. 72 :I A BasicreproductionratioRoasa [unction oje.lJil.'acycand duration/1

with1/)=0.5.Thehorizontal surj ace1.'00TI.'SP011ds toRo

=

1. 7:1 :3.5 Plasma G

"

(1'I:d solidline),intracellular G" [blue solidline)conccuini- tunis ,ande.ll icacy111'ofriionuu ir(green dashed line).Secthetcrijor ptinu nc lcrualucs..

viii

75

3.G ln tracel luiar concentrationsoj the native (C,.)(blue),m.onophosl'l101"Y- latt»! (C,'l')(rul). linddiphosp l101"yloted(C"pp)[qrccn] [oruis.linde.lJi- elleytttrr ojtenoJovlr'DF[hlack},Seetheter![or pammr:le"l" nllllws. . 77 3.7 The inuuccilularomccutraiion of diph ospho"l"ylated[orniso]lenojfJlli"l

DF,C"pp andits jitiitu;curve.

3.8 Tlicbusi crcproduciiouratioRs,liS.IJ!lOsetli.lTe"l"ene e t!J.

ix

7!J 80

Chapter 1

Preliminaries

III t.hisrhap tc r.wepresentSOllie dclin iti o usam!knownthoornm s wh ich willbeused ill theres t.of thisthesis .'I'hcyarcinvolvedillcoope ra tivedelaydillcreut.ia lcquat.ions, ulliforlllp cr si slclIee alldco exist.ellce states,chaillt.ra llsiti ve set salldhasicreproduc- tiouuurubcr.

1.1 Coope rat ive delay differ ent ial equations

Let V he aBanachspacewithall orderCOile V+wit.hnoncmptyinterior1111.Y+.For YEV,we wr it e:1'

:s:

yif,lj- :1'EV+ ,:1'<,Ijify-:1'EiIII V+\{O},and:1'

«

,Ijif y-:rEIIIIYt-.

Fordelaydiflorcutinlequation,letI'denotestheIIH1XilllUIIldelayappcarillgill the cqunt.ion,thenthe spaceC:=C( [-l',OJ,IR")isanutu ra lchoiceofsta t espace.Define C+:={¢JEC:cjJ(O)

2:

0,- rT

:s:

0

:s:

O}.Thenotation

<,

:S:,

«

willheused for theorde r relations011C gene ra te d hyC+.IIIparticular,II~

:s:

~JillCifam !oulvif I/J(S):S:4J(s)ho ldsillIR"for everysE[-1',0].

Cons ide rsystem

d.:~~t) =

f(l.:rtl, (1.1)

whe ref:JRxD-+JRIIiscont.iuuous andDeCis ope n.It.is alsoassume dtha tf isLipsclritzinits secondargume ntoneachcom pactsubsetofJRxDsot.hntiniti al valueproblemassocia te dwith(1.1)hasunique solutio ns.

The or em1.1.1.([2D,TIIEOBE ~I5.2 .1])As.mllll;thniif</iEf)satisfies<Ii::::

o,

<lii(O)=0[or sottu:i tuul IE

K

then/;(1.r/J)::::

o.

Ifr/JEf)satisfies</i::::0tu u!

10EJR.then:1'(1,/0,/p)::::0[oralii::::10initsnuirin uil internalofeii stencc.

Definition1.1.1.fissaid10bequusinunuiiotu:ij]«: anyr/J:::;

4'

unllir/Ji(O)=//J,(O) [orsomei.wcluiucfi(/p):::;fi(</!).

Theorem1.1.2.([2!J.TI IEOB E~I5.1.1])Let

I.» n

-+JR"lu: continuous.Lipscliii:

01/each.colIIJHu:1subsetofn. andOSS U1II CthaieitherfOJ'!Jsatisfies thequusim on oi ntn:

condition.Assllmc also tliat f(l, r/J) :::;!J(I.<!i)for all (I,r/J)E

n.

If (10.

1).

(lO.</i)E

n

satisflJr/J:::;</i,the n

:r(l./o. r/J. J):::;.r(l ./o.</i,!J)

holds]oralii::::10[orwhich bothon;defined.

Next. we cons ide r t.hc goncrnlnonauto noiu ou slinoa rsystcm

d::~~t)

=1,(1):1'"

where1,:JR-+L(C, JR")is conti nuous awlL(C, JR" )ist.hc spaceofbound edlinear mapslrouiCtoJRII. LetLi(l)</idenotet.hci-t.hcom po nentof[,(I)/p.Thou1,(1) sa tis fiest.hequasimonotoncconditio nifandon ly ifthefollowing condit.ion hold s :

(K)If1>2':0and1>i(O)=0,then1;(1)1>2':

o.

Theorem1.1.3.([2!J,LE~I ~IA5.1 .2])(l\)holtls ifiuulonlyifthere e:ristsll;(t)EIR [ori::=;i::=;nalld llOsitive !Jol'd llletJsul'Csl/i)t) forl::=;i,)::=;nsuclilhai

1i(l)1>=11;(1)1>;(0)

+ t t

,/» (O)tlOI/i)(I,(i)

)=I,-rj

(1.2)

IIntil /ii(l ){O}=O.Moreover,if(II)holds.thenthercprcscnlniio ii(1.2)isuniqueanti ai(l)anti J/i)(I )mecontinuausjunciionsojt.

Dcfin iti on1.1.2. MalrixJ1 = (ai))"X"issuidto beirreducibleiff ol'c /lel'YIlO111:111»Iy

»I'01'el'subset.1of the setN

=

{I,2, .. ,n},thereisaniE1anti)E.J

=

N\1such lh.altu,

# 0.

Wethellintrodncethefollowing collditi on:

(I)Themnt.rixA(1,)(I)delilied by

isirre ducib le ,where'\ECisthe clc mcutwithi-thc-om ponen tIan dtheot.hor

com pone nt 0 for all0E[-r,O).

Dcfinit ion 1.1.3. System (1.1)issaidtohe coopcmlii»:iff) ordercon'IJe:I: and tlf(,p)sulisji csthecondition(J()forracl:~~ED.

Ifsystem(1.1) iscoopera tive,then thederiva t ive

,(/(1»

cunbe rcpros cu lcda,;in (1.2)where!I;

=

^!Ii(<I»and'/i)

=

>1;) ('/»arecontinuousfunct.ionsof4~ED.

Dcfinition1.1.4 .Systcm(1.1)issau! to becoopctuiim:awlirreducibleifit iscoOJl- crat.iucandllicjollounnq conditionshold:

(1)For111111~;ED.df(ljJ)satisfies(I);

(2)For^{CIIC 17J}j[orwhichI'j

>

O.llutrccri sisisucl: Iluil.[or IIIIIjJED.

fOl'1I1Is11lIlIlE

>

O.

Topresentson icresultsabo ut.thest.ahi litvof allequilib ri u mofsystem(1.1).11'(' ns su me

f

is conti nuo us ly dilluront.iabl c andcoo pera tive illadom ainD.Su ppose[I is alleq u ilib ri u mor(1.1),thatisii E IR"is such thai.f(v)=O.Then thelinear var ia t iollalsys (c lIlcor rcs po lld illgtofiis

y'{l )=LYr, L=r(j'{v).

Definition1.1.5.ThestabilitytuodulusofLisdefinedas 8(L)=max{9\ "\:Ddt::.(,\)=O}

whrTc9\"\ denot estherealpart.oj X.

(1.3)

Suppose syste m(1.1) is cooperati veand irred ucib le.Thenwocau dcfincu roopor- at ive andirre d uc ib lesys tem of ordinary differont.inl cqua t.ioushyignorin gallYdcluvs whic happearill(1.1).Thislead s to t.hc followi ngsystem

;1:'

=

F(;r). F(J:)

= ft :/:) .

where-den o tethe incl usio nIR"->Chy./:-> :i'i(())

==

:I'i .for allIJE[-r,O), i=1,...11.Observethat(1.'1)husthe same cquilibrinas (1. 1).

The o rem1.1.4.([20,CO HA LLA HY5.5.2])8(L)

<

0(8(L)

>

0) iflind onlyif s(D F(1I))<0(8(DF(v))

>

0).

Nextweintroduce some not ati onsuboutmatrices.Formatrices :\and13,0:::;

rI,0

<

A menusthatA is cutry-wiscuonncgntivo,pos iti ve,respect.ivelv.A:::;13 mean sthat ():::;!3-A.Aisquasi-positi vemeans allofitsoff-diugonu lentrie s arc nonn cgativo.Theexpo ne nti alofasq uare matrixAisexpresseda~exp[A].LetpiA) he thcspectralrad iusof thematrixA.Thefollowingstan dn rdresults(sec,e.l-\..

[aj)

willheusedlater.

Theorem1. 1.5.Thefllllllwilly siu lcuunitslire valid:

(l)If Ais quu si -posiiiucIIl1dA:::;13 but A

#

U,then

0 :::;

exp[IA]:::;cxp[l!3]but exp[IA]

#

exp[l!3 ],'VI

>

O.

(2)ifA

>

0atu}1320luis1111zeronnuIIrZITIICOIU/II.II,tlicnAU

>

0undllA

>

O.

(:1)if0<A:::;U lntlA

#

U,then.p(A)<p(!3).

1.2 Uniform persi stence and coe x is ten ce st a tes

Suppose.\ isametri cspacewithmetri cd.Let

f :

X--->Xhea conti nuous map andXo CXanopenset.DefineDXo:=X\Xo,aIHIMa:={:rEDXo,f"( .r)E DXo, 'V1I20}.

Definiti on1.2.1.A boundedsci.Aissaid IIIutii uct a bounded.,,:1IJillXif

lim~lIp { d(J"(:r) ,A)}=O.

1l-00J'EIJ

A subset A

c

Xissaid IIIbeallatimciorif Aisiun unnp h)compact1I11dinuariont.

(f(A)=A).tuulAullrucl»sonic open.nciqhborhood.IIfiiscl].IIyllllmiult.nu.t.orfill'

X--->Xisall uttrucioriluitallractseVIT Ypoint.illX.FI!1"atununnpli;innariuut

sclM.llu: setII'-'(M )

=

{:I'EX:lim,,_ ood(f "(:I'),M)

=

O}iscalled ilu:slublcsri oJM.

Dcfinition1.2.2. Aconti nuou snuui

f :

X--->Xissaid¹⁰bcpointdissipal^n»:iJ there is a boinulcdscl.130inXsuchtliat/30nllruclscuclipoint illX.

'I'hcorem 1.2.1.([4 2 ,TIIEOH E~I1.3.1])IJ J:X--->Xiscotnpnctandpoint.

diss ipalinc.then.thereisaconnectedylobalalirnciorA tluiialtro ct» cucliboun ded sut.

inX.

Theorem1.2 .2.([42,'I'll EOIU%11.3.1ANDIlE ~IAIlI,1.:1.1])AssUIIWiluil.

(CI)f(Xo)CXoiuul

f

hasa ylo/w/ al lrac torA:

(C2)Therecrisisafinite:sequenceM ={MI...,J'''-}o]disj oint,compucl, urul isolatedimiari untsets in0,\0suchtliai

(b) Nosubset.o]A1JOI1I1Sa(;yd(:illOXo:

(c)M;isisolatrdinX:

(d)lV'(M;)

n

Xo=0 JOl'eacli1

< i-:

i:

Then.

f

isunijormu;persisten t.nntlircspcc!10(Xo,DXo)illilu:SCIIS Ciluil.tlWI'I;erisls 117//1>0such Ow/limillf,,_ ood(f " (:I'),DXo)~"[orall:1'EXo.

Hecalltha t a familyofmnpp ings<I>(I ),I~DOilXis calledacout.iuuous-ti mc scuriflowprovid cdthntxl'{U)=1,<I>(I):l'is collti llllollsjoillt.lyill(I,:r ),Hlld<I>(l)o<[>(s )=

<1>(I+s)fora!!I,s~D.

De finiti on1.2.3 .A continuous[unction/I:X--->[0. ) iscalleda qcu cralizcd distance[unction[or<fl{l)ijilhasllu:P1'O/)(:I'lylhal ll«II{l);,,)

>

0[ort

>

0ijeither /1(;")=0awl;"EXn01'ij/1(.,,)

> o.

Theo re m 1.2.3.([31.THEO H E ~I:1])Letpbea qcu ctulizcddisltu tct:[unction

Io:

Ihe!lill en snnij[o l/l(I>(I).Assu nicthut.

(1'/)<1>{I)hasa!llo[m[a/lm elor;

(1'2) There erisis ajiniu:sn/ ,wneeM={Ml,M2••• ,M,,}ojlm.irwisedisj oinl.1'0111- pactandisolui cdinooriouiselsin DXnunll! ilu:[olltnuitu;propert i es

b) No sU[JSI:lo]M[onusacuclcinDXn: c) M,isiso[aledi1l X :

d) IV' (M;)n/l-I(O,oo)=0 [orIll/ I:::::i:::::k.when:1I"'(Mi)istlu:stublc.,,:1 ojM;.

ThentherecrisisII

>

0suchIhall imiIlft_ oo/l«II{l);,,)

2:

II[orall:"EXn.

AssumeX isa dosedsubset of Banachspac eE,and Iha t,XnisaCOIl\'('Xand rcla t.ivclyopensu bset.illX.Then DXnis rclativclv dosedillX. Weha vethe followingresult..

'I'hco rcm1.2.4,([42 ,THEOHEM1.:1.6])LetS X--->Xbeaconii nu ous11Iafluuth S(Xn)cXn.Assuuf(:iluii

(1)S:X--->Xispoinidissinatiu«;

(2)S compucl:

(,'1) S unifonnlypersisten tunllirespect.to(Xo,DXo);

Thentherecrisis(Jglobalalti nriorAoforSinXothatallrucisstrong lybo un dedsets inXo^andShas a roeristctu»:state:roEAo.

Letw

>

O.Afamily of mappiug»'1>(1): X--->X,

t ::::

0,is calledallw-periodie scmi llowOilXif ithm;thefollowingpropert ies:

(1) '1>(0)=I,where1isthoidentityruaponX;

(2) '1'(1

+

w)='1>(1)0'I>(w),Vt::::0;

(3)'1>(I).riscont.iuuousill(I,:r)E[0,00)xX.

Themappin g'I>(w)is calledthePoiuc.ux'map (period map ) associatedwit.hthis pcri odi c scmillow.

Theorem1.2.5. ([42,TIIEOIWM3.1. 1])Lel'I>(1)beaui-ncri odic scniijloiuonX with,[>(t)Xo

c

Xo,Vt::::O.AssumethatS='[>(w)satisfiesthe followingcondit ion s:

1) SispointdissipoiiocinX;

2) Sis compact:

TlicuuuijormpersistenceofS unllirespectto(Xo, D.':o)im pliesthat of'I>(I):\"--->X.

1.3 Chain transitive sets

Let.'1>(1),t.::::0,he a «out.inuous-timcscmillow011ametricspace Xwith met ricd.

Wc say eEX isalleqllilihrillmofcI>(t)ifc!>(t )e = efor all

t ::::

O.

Defini tion1.3.1.LetA

c

Xbeanonempty,inuariant.srl. [or'1>(/ ).IFc say IIis intcrna lls;elwintransiiim:

iJ

[orUIl Ya ,bEllundUIlYE

> (),

10

>

O.thereisU

I

:Si :S

III- I.sucli thatd('I>(li):ri, .ri+d

<

E[orall;

:Si :S

111- I.

Theor em 1.3.1.([,(2 .LE~I ~IA1.2.1'])Let'[>(/ ):X->X,I:::::O.beacont in uous timesenujlom.Thentlu:otnequ[ulplia}lim itset o]allYprcctnnpnctposii.in«(ncqatirc]

orbitisinternalhjcluu titransitive.

Theorem1.3.2.([42,TIIEO IlE ~11.2.2ANDnE ~I Al U(1.:1.2])ASSIIII Wihaicarl:

equilibriumof'[' (1)isauisolatedinnari ttnl.set.thaithen:isno r,yclic chainoj cqui- Iibria.U1/( /thaiclleryprcnnnpm'!orbitC01IIlr:7yestoS01l/Ccquililninuio](['(1).Tlu-u allY iulcrnullucluiititransiiincset isunequilibriumoj (1'(1).

1.4 B a si c rep ro d uct io n number

Bas ic rcpro d uct ionuumbcrof aninfect iousdiseaseisa fuudmncutu landimportaut conceptint.hcst.udv ofdisease control,It. isdefined as the expecte dnumberof secondary infecti ous aris ingfroma sing leiudividua l duringhis orhercuti reinfecti ou s period. inapopulationofsusccp t ihlc. Usually.thebasicrcprod uct.iou uumborscrves asat hresho ld paralllet er in t he senset.ha t.t.hediseascd iesoutif t hebasierepro dlletioll 1IIIIIIbcrislessthuu unity.andthediseasepersistsill thepopulnt.iouif it is grea lcr thnu uuity.Thus,illorde rtocontrolthcdiscus c,\\'eneed to reduceR.o(0beI,'ss thanI.The explicit forlllulaof R_IIwas givenill

[:1,1 1

forala rgeclassofaut.ouomous cOlllpa rtlllcnt alepidcllliclllodeis. Illt hisscct.ion,\\'cprcscn t thctheor.vof basie

rep rodu ctionratiosforcom partmentalepide m icmod elsill periodiccuvirom uon ts,

whichwasdev elopedill

[3G].

\Veconsiderahete rogeneouspopula ti on whosein d ivid ualscaubogrou ped into1/

hOIlIOg(~II('OUScompurt. mc nts.Lct:»=(:rl,".•:r,,)'I".witheach :0::0. bothe)st ate ofind ivid ualsilleach«orupart.mcnt..Assu methat t,]IC coru part.iuent.sca ll bcrlividcd

into twotypes : infectedcompart. mcnts,labeledhyi= 1,.. .,11/,nnduuinfoctcd

«oru purtmcnts,labolcdhyi=III

+

I. .. ,u,Deno teX.,tohetheset ofall dis('aSl'- Iroc st.atcs :

X.,:={:r:O::0:Xi=0,Vi=J....III}.

Letfi(t ,:r)hetheiupu t rateof newlyinfectedind ividu ulsilltheithcom part.ruout, vt(l.,:r)hethein p utratc of' ind ivid uu lsbyothermeans(Iorux.u u plo,births.inu ui- gra tiolls ).aJl(IV-(t ,:r)hytherate of"t rallsferof illdividualsollt of«ompnrtu rcn trffor exam ple,deaths,recovery andcmigrntious ) .Th us,thediseasetnmsmission mod e l is gove rnedhya ll aut{1I10l 1l0Ilsord illnrydifrcrelltialsy st.elll:

o/!f

=fi(t,:r)-Vi(l.,:r):=fi(t.:r),i=I, (1.5)

whereVi=Vi--Vr We assu methat.themod el(1.5)nd mit.s adiscnsc-frcoperiodic solnt.ion:r°(t )=(0,...,0,:r~~'+1(I.),...,:r~~ (I. ) fwit.h:r~) (I.):0::0,

11I +

1::;i::;n.forall t.Ld

f =

(/1," "/")'1',and define

F(t):=

( Dfi(~ , :rO (l.) ))

,\1(1.):=(DVi(t, :I:O(l.))) .

D.f) l~i•.i~1II lJl'j l~i,jSHl

i\/(t):=

(D{;(~ ;r()(t))) ^.

"J 1II+ISi.j Sll

10

It.then followsthat.

J)Jf(I,:r0(t))=(F(t)

0 ).

J)JV(t ,.r0(t))=

( \1(/) 0 )

o

0 ./(t) -M(I)

whe re./(1)isan(/I-111)x/Iuiatrix.Dcuo to«(>,,(1)hethemonod roiu ymatrix of t.ho linea rw-per iodicsystem

;t

=:1(1).::.\Vemakethefollowiugassum p tions :

(A I)ForeachI::;

i ::;

/I,thefuncti onfj(l, :r),Vt(l ,:r),andVj-(l,;/:)arenonn egati ve andcoutinuo usonIR xIR~andcout iuuouxlydifferent ia lwit h respectto .».

(A2)Thcrcis arcaluutnhcru.

>

(Isuchthat fo r eachI::;

i ::;

/I,thefunctionfj(l, :r), vt(l,:r),andVj-(l,:r )arew-p(:rio d ieinI.

(AJ)If:rj

=

0,t.hcnVj-

=

O.Inp.uticulnr,if;/:EX."thenVj-

= ()

fori

=

I.

(A4) fj =0ifi

>

^/11.

(A5)If.»EX."thenfi(:")

=

Vt(:r)

=

0 fori

=

I.

(Mi)p(I>,IJ(w))

<

I,wherefI(I>.IJ(w))isthe spectra lradiusof«(>,I[(w).

(A7)fI("-dw))

<

J.

LetY(I,8).1

>

8,het.hccvolutionoperatorof theliuca rco-pcriod icsvstcm

~ = -\l(I).'I'

Thatis.foreach8E IR,theII/xlI/lIIat.rixY(I,8)sa t isli('s

~}'(1 ,8)

^=-V(I)Y( I,s),VI::;8,Y(8,8)=I,

whore1isthe111xII/idcnt.itvmatrix .Set.C",het.hoorderedBanachspa ce ofallw- periodicfllncti onsfrolnIRtoIR"'e,!nippedwiththelllaxinnnnnorlll and thepositil'c

II

cone C::=h~ECw:¢(t)2:0,VI2:O}.Thenwo candefinea linearopcrator I.:C'" ->Cwby

(L¢)(I)

= L ""

^{}' (t.1-}a)F(t-a)¢(t -a)t!a,VIE

IR.

¢E

C~

^..

Wc ca ll Lthcncxtinfedionopcrator,allll dcfi nct hcspeclralradiusofLasthcbasic reproductionratio

Ro:=p(L)

fortheperi odicep ide m icmode!(I .S).

LetIV(I,s,A),12:s,sEIR,bethe cvolut.ionoperatorof thefollowinglinear systom

~ =( - \f(t) + ±F(t))IlI,

^{IE IR. (I.G)}

Thefollow ingtheo rem isusefultonumcricall v comp utethebasicrepro d uctio n ratioRo.

ThCOI'Clll1.4.1.([:lG,TIIEORE~I2.1])Let (A J}-(A7)hold. ThcIItheJollowillg st atements af"{:»alid:

(1)IJ\V(w.0,A)has a posiiincsolutionAo.then'\0isIIlIciqcnualsu:oj L. utul111:/11'(' Ro>O.

(2)IJRo

>

O.llien.'\0=RoisIlu:ItlliIJIU:solutiono]p(lI' (w,O,,\))=I.

(.'I)

n;

=0

iJ

III/dmt.!.t/

iJ

p(IV(w.O, ,\))

<

Ijill'all,\

>

O.

Thefollowin gresultshowsthatRoisathn:sholdpa ranlel(:rfort heloca! st ability ofadiseas e- freeperiod icsolu tio n:r°(t).

12

Theorem1.4.2.([3G,TIIEOIlEI-I2.2])IlssIl1llCIhal(A I)-(A7)hold. '1'1/('11tlu:

[olloutnu;slnl.cnum ls areoalu]:

[l]Ro

=

Iijuiulonly i,lp('I'F_I'(w))

=

I.

(2)

u; >

IiJ awl olllyiJp«(>F-I'(W))

>

I.

(3)Ro

<

I

iJ

andoulst

iJ

p«(>F-dw))

<

I.

Tliu»,:,,0(1) isaSY1lljilo!.imllysiub l«iJRo

<

I.atulunstableiJRo

>

I.

Fina lly,wegivc auumcricn l ulgori t.ln ufort.hc eoruputn t.iouof Ro(~('C['\'1]).

Let<1>(1,.\ ),I:::::O,bethe sta ud urd Iuudmucut.almnt.rix solu tionof (I .G)with

«>(0,.\)=I.Fo r anygiven.\

>

0,wecan uumc ri cnll y eom p utcallcigcnvn lur-sof

<!>(W,.\)by Mut.lnb,or Map l«,andhen ce,the spec tralradius,p«I'(w..\)),of«>(w.A).

LetJ(.\):=p«!>(w,.\)).Sinr:cF(I )isnouucgat.iv c and-\1(1)is «oo pcru ti vc,it fol- low sthatJ(,\)iscontin uousuud uou inc n-as iugiu.\E(0.(0).Further.lilll,\_ oof(,\)= p«ll_dw))

<

I.l3y thefollowi ngfUIIl"ste ps,wecannumcri cnll v calcu la teRo.

(I)Choosetwoposi ti venumbe rs110

<

bosucht.h.uf(II0

>

I

>

/(bo)).Ifthorois nosll Cha tlot.honThco rcm Ld.H iii] implies t.hat.Ro=0.

(2 )Definetwoseq ue ncesII"andb;byind uctionsif/(~):::::I.define11,,+I=

~,audb"+1

=

b«:otherwise,definea,,+1

=

II",andb,,+1

=

~.It follow s thatII"::;

i.:

11,,+1 :::::a" ,b" ,::;b" ,amif(II,,):::::I:::::f(b,,)for allII.

(:3)By~tep(2),weha ve[1111+1,bll+dC[1I11,bll]unrlbll-all=+,(110-(10)'Thus liulll_ex.,all=lilllll_oo('II='\ 0

>

O.Sincef(II,,):::::1:::::f(bll)fo rallII,weha ve /('\0):::::I:::::/(.\0),and hon coJ(.\o)

=

1.Conseq ue ntly,weha vetc«

=

'\0,

13

I

{I"-Ro

I::;

li;-II"=f.;(bo-110).Givenallerro rtolerancee,W(~callchoose allN

> ()

surht.luuiv(bo-(10)::;f.Thus,wehaveRo~liNorRo~{IN .

14

Chapter 2

A Time-Delayed Dengue Transmission Model

2.1 Introduction

Dengu efeveristhe1II0St.COIII IIIOIIviral discaso sp rca d tolnunnnsh.vmosqu itos,nne]

hasbecome alliutc rna tio nalpublichealthconcern.Dengueis caused hyagroupof fournntigcnicnllydis tin ctIlavivirussoro types:DEN-I. DEN-2,DEN-:l, and DEN-,I:

audisprimarily trans mitte d hyAed esmosquit os.part.iculnrlvA,acq.vpt.imosquit os.

Den gueisfoundill t.ropical andsubtropicalregionsaround theworld.prcdoruiuatclv ill urban and peri -urb an areas.Theincidenceof den guehasgrowll dr.uu ut.ir-nllv aroulld theworldillrecellt.decades.It.i selldelllicilll uoret.halll lOcoull tr iesill Africa,theAmericas,theEast ernMcditcrrancau,Sout h-ca stAsiaandt.hc\\'cst erJI Pacific,It.infects50 (0100 millionpeople worldwidea year,lcadiru;to 50million hospitaliza tions, nndapproxima tely12,500 to 25.000 deathsayear[G,7,27,:\7].

15

Thehuma n is thomninan rp lifviug host of t.hcvirus,alt houg hstudioshav eshow n thatillsome par t. oft.hcworldruon kcvsmay becomeillfl'('(l'daIIIIperh ap s serveas a sourceofvirusforuuinfcc tod mosquitos[6].Hu man mayget.infoetcdhya hite fromtheinfect edmosqui tos,and A.neq v pt.i mosqui t osmay acquiretheviruswlu-u theyIced011allin fectiousindi vidual.Much hav ebeendoneill ten nsof modeliug and alia lysi s ofdiseasetransmi ss ionwit.hstructu redvectorpop ula ti ou. Wall gandZhao [3G]prop osedauoul ocnl and time-delayedrea ction-diffusi onmodelofdellglll~fever, andcst ublishcdathresholddyn umi csilltenliSoft.hebnsicreprod uctionIUIIIII)(~rRo.

Lou andZhao

[221

presentedamalariatransmi ss ioumodelwit.hst.ruct.u rcdvex-tor populut.iou,nndalsoes tab lisheda th resho ld typ eresu lt,wh ichst.ates tha t.when Ro

<

1and thediseas einvasion is1I0t.strong ,t.hodiseasewill dieout.;when Ro

>

I, thediseasewillperxist..

III this cha pter,weincorpo ra tet.hcst.agl~st.ruct.ureof mosq uit os(see,e.g..

1 22]),

since thedevel opmen t.sta gesofmosq u itoshave aprofoundim pnctOilt.hctransruis- sion ofdisease:first,theimma ture mosquitosdo1I0t.flyandbitehuman. so t.h('ydo 1I0t.pa rt.icipateill thein fectioncycle:second,ma turemosquitos arequitedille rcn t fromim maturemosqui tosfrom biologicalandepide miologicalpcrspcctives.IIIview ofrcalis t.ic collsiderat.ioll,wetaket.hesediffel"l'lItstag l'sillto a cl'Ou llt..\\'ea lsoillclude t.hct.imodelay todescri betheincubatiouper iods of1I10Sll11it osandtheluu nanpop- ulnt.ionsvwh ich isimportau tbecausethereareincubut.ionreaIist.icallvaudt.hctimc periodis1I0t,small.IIIfact.,from t.hccxpressionofZq,inscct.iouB,we callsecthose del aysred ucet.hc valuesof Ro.The refo re, t.hcucgloct of t.hodel ays ovcrcs t.imntcd thcinfcx.tiourisk.

Thepurposeof t.h ischa p te risto studythe gloh a l dyn.unirsofat.imc-del uvcd

IG

dcnv;uc t rau smi ssionm odcl.lnsection 2,wcprcscntthcmodel syst cmallllpro \'eits wcllposod ncss. Insectio n3,wefirstintroducethe basi«reprodu cti onnumberRo, and then showthatt.hodiseaseisuniformlypersist ent when Ro

>

Ibyappealinv;to thetheo rydevelop edin[5, 31).Unde rcer ta incond itions, we also oh t a in thenonl oeal stabilityof' thediscns o-Iroc eq uilibri umwhenR o

<

I.Inscct.ion},we ohta ina set ofsullicicntco ndit ions forthe endemicequilibriu m tohe v;lohallyntt.ractl vchythe methodofH uct.unt.ious. Insection5.wepcrformnumcn cal sim ulnt.ious toillustrate ouranalyti cresults.

2.2 The model

In this sectio n,follo wingthc~ideasin[35]'wepresentanngc-st.ruc t.urcden gu emodel witht.imcdcluyforthecrossinfecti onbetweenmosquit osand humuu iudividu nls .We divid e themosquitopo pula ti onintotwosuhc lassos:aq ua tic population andwinged pop ulation.Wingedfemal eA. aegyp t.imosqu it oeslayeggsin unattended water.Eggs maydevelopintolar vaefrom twodaysupt.ooncweek.ThcInrvnc spend uptot.hrcc daystopas »throu ghlourinstal'Stoente rthepupalstav;e.Thepupadevelopsinlo anad u lt.afteruhout twodays.Theimmaturemosquitosliveinaq ua tichubit at.sand matu remosquitosdisp ersetosearch forfood.Let.Adenot ethedensityof aq ua t ic popul ut.ion ofmosquitos ,IVhetheden sit y ofwinged popul ati on of mosquitos,and T,\hethelen gth ofimmatu re stageofmosquitos .Followin gthe modeltoformula te astage -s tru ctured po pul ati onin Aielloand Freedm an[2),we su p pose thedyn amics

17

of mosq uito sisdescribed hy

d~~;,t)

l3((IV(I ))IV(I )-oA(I)-('-aT"I3(IV (I-T,d)IV(I-T,d,

d l; ;?)

c-aTAI3(IV( t_T,d) IV( 1-T,d-II",IV (I),

where I3is t.heper capitah irthrutc ofn du ltmosquito s,0is t.hcpcrcapitndeathruto ofnqu ati cmosquitos,awlII",isthcdcal.hrateofa d ultmosquitos.Follow ing

[:IGj,

we assume thatthofu ncti onofIJ(IV )IVisthelogist.ic grow t.hrate:

For thedyna mics of hum an populat.ion,we nssuuu:t.luuthedcn sitvNof the huuuuipopulnt.iouoboys

dN

dt

=II-IlhN.

whereIIis a cous tn ut recruitmentrat.cuudIII.isthedeath rate,

To cons iderden gu etransmi ssionbetw een mosquitos awl lunu an individ ua ls.welet lVI,IV"and11'2denotethedun si tv ofsusccp t.iblc,expose d, and infccl.iousmosquit os ofwingedpopu lation,respect ively ; and dividethehumaupopu lnt.ioninl«: fourCOIll- p.utmcnts:suscep ti ble(5),exposed(E),infecti ous(I)aw lreco vered(H),Let.T", ),etheiw:n ),atioupe riodofdeugne vil'llswit hin lllosqnitos aud Thhethl,illcn hat.ion periodofdengu e viruswith in hosts,Following Chowcllctnl.[8]' we supposethat t.hcinfccl.ion rat es ofxuscc pt.iblc mosqui tos andsuscep t ib le hunumindiv idu als arc describedby

resp ectivel y,whereIiisthemoa n rateof'mosqu itohit.o»per1Il0sqn ito ,]I istheproba- bility tha t a bit ebyasusccpt.ililomosqu itoto an infectioushost willcauseinfoct.ious.

18

qistheprob uh ilit y tha tahitchy aninfecti ousmosquito10a suscoptib lr-host will causeinfect ion tothehost, andN=S

+

E

+

I

+

I?is thetot aldousit v oflru nuui popul a tion.Sinceaninfcctiousmosq uito uuiyhavelower fecundity thnna xusrup tihlc mosquito,weletaE [0. I]den otetherelativefecun di tyof aninf(-el('d mosqu itotoa susceptib lemosquito.Spccilicallv,the iufect.iousmosquitohast.hcsameroprod uct.iou rateas a suscepti b lemosquito ifa=I,and havelower repro ductio nrate ifa

<

I.

Thenwr:havethefollowi ngmod el:

dA(I)

---;i/

dW1(l) -dt -

r[1-

1\;~t) I+ II'.,(I)

^-aA(I)-re-ur

^"[1- W(l/~

^T·Il]+W.,(1-T...), I'I,-ur"[I-

W(/l~

T,tl ]+W.,(I_T..tl-/J",1II1(l )-/J",

j~~;)

1111(1).

111,.(1) dl l'Al ) -dl - dS(I)

---;i/

£(1) dl(l )

----;i/

dl?(l )

---;i/

^{,/(1)-/11.11(1),}

(2.1) (2.2)

(2.:1) (2..1) (2.5)

(2.G) (2.7)

ln],,is therecovery rateofi nfected lunnanindividuals,e.; andc:" are(heinfection- ind uced deathrates ofin fect ed mosqu itosand luuunu ind ivid uals,rcspcctivclv

Notethattheequa tio n foraq unt.ic populat.ion of mosquito»isdecollpledfroiuthe ot he rcq uatio us.Itthen suffices to considersystem(2.1)-(2.7)which isaniutogro-

I!J

diflcrc u t.ial eq ua t ionsystem.Diffcrcnt.int ing (2.2)an d(2.5)gin's

(2.8 )

d~?}

^/I,,1V2(/}

~r~I/}

^{-I/,,£ (I}-}

^/j"c- "

^{"T"1V2(1-T,,}}

~~II = ~;,})"

^{(2. 9)}

Thesystemcons is t.iugof(2.1).(2. 8 ),(2. 3),(2A),(2. 9 ),(2. G) uu d(2.7) isall ordinnry diflcreutialsystem withtimedel ay s. For sim p licitv,wewillrefertothis syste mas"t.hcmodelsystem"ill therest of thischa pter.

LetT=Illa x{ TA,T""Td,and ddille C:=C([-T,O],~7 }.For</>=('/JI. 4)2.. .,'/J7)E C,define

II

4)11=

'L:=I II

'/Ji1100'where

II

qJi 1100=llI<IXOE[_T.O!

1

'/Ji(lI}

I.

ThellCis aBnnuchspa ce.DelineC+={</>EC:~6i( lI }~0,VI::;i::;7,11E[-T,O)}.Then C+isanorm nl COlleofCwit.hnon cmp t.vinter iorillC.For a cont.in uo us func t ion

/I:

[-T,

a¢}--->~7wit.h a¢

>

0,wedefine/IIECforea chI~0hy/l1(1I}=/1(1

+

II}.

\I{)E[-T,O].

IIIviewof(2. 2) and(2.5),we choose theinitial da l afo rthemodelsvstotnill,1:\.

whichisdefined as

forsm a llr5E(0,

" '~I''' ) '

^Thefoll owing resu ltshowsthat.t.homod ol syste m iswell- posedill,Y,I.and the soluti oll semiflow adm it s agloh al att ract or oll,1'oI•

T'hcorc m2.2.1. Forlilly

'/J

E,1'01.themodel syslem luis11uniquenonncqat.iucsoluliou

hasacotnpuc t.qlobalallrtu.lor.

20

where

G,((jJ) 1'(,-uT,\[1-Li;,

~:.(-T,d l+(1"(_T,d +

cPA-T,d

+

a(11,:,(-T,I))

-II..,~'J,

^(0)-

/J .., Lru,(~;(O)

<P'(0),

G~(~'J) /3"'Lt,(~;(0)

^{1', ((l)-11..,1'AO)-}/J",('-II",r",

Ltl(;~:)T"')

<p,(- T",).

G:M)

(J",('-I,,,,r"' Lr(,il(;~:)T,,,) ¢' (-T"')

^-(II..,+E..,)¢:,(D).

G,((/J) 11-1',,1',,(D)-

/J" Ltl(:~;(D)

^(/JAD),

G,,(1J)

/J" Ltll(~;(O)

^¢:I(D)-I',,¢,,(O)-/-J"e-III,TI,

Lt'(;~~T")

^1':,(-T,,).

Gli(¢) (J"c-II"r"

Ltl(;~~T")

^{1'A-T,,)-(II"}

⁺

^E"

⁺

^,,)1'li(D),

G7(¢) ,.~'Jli (O)-11,,1J7(D).

Notetha t,y.isdosedillC.andfor all¢ E,l,\ ,G(¢)isconti nuousand Lipschitzill 4'illeachcom pnctset.ill~x,'I.'.,13y[IG,Theorem2,3]'it thenfollowst.hal,forallY

¢ E.Y•.t.hcrcisanuuiqucsolutionoft. hcmodelsystem through(O, 1J)Oilitsmaximal interva l[0,a,,)of'existe nce.

SinceGi(¢)2Dwhenever

l'

E.1:'.withq'i(D)=D,Thoo rcrul.l.!im pliest.hat.t.ho solutionsof themodel systemarenonncgut.iveforalltE [D,a,,) .Notethatthetotal hostpopulat.iousat isfies

1"01'SYStCIlI¥If=H-(II"

+

E,,)y(l ).t.ho equilibrium""~EI'is globu llvaSylllptotically

stable. For allYD

<

8

<

II"/~E'"

¥ifIll;.

=H-(II"+E/, ),)

>

D,So ify(D)28,t.hon 21

.'1(1)

2:

<5.for nnv1

2:

O.From(2.8). wr:~el

Byintcgratingonbothsidesfrom0 toI. weobtaiu

c"""\V,,(t)-IV,(O)

=

l

("''''' fl",

l~r~;) IVJ(.~)ds -l

c'""(.'-T" )(J,,,

1~~: ^{-=-:: ::)}

^1V\(s-T",)ds

=

l

('I'""'(J",

1~~;)

^{1V1(s )ds-}

.L~T'"

^d'''''(i",

l~(t!)

^IVJ(s)ds

=[ '

('I'''''/i'''l~((';) IVI(S)dS- t' ("''''''fi", ~(('~!) lVI (S)dS

.I-rll,

. I-TII '

Similarlv,ifE(O)=.I~TI'C/I"'(i"1\'2~'(~(.')d8is satisfied,then

Thisimpliesthat1/1E,1',\.VIE[0,0",).

Notethat

Forsyste m

(2. 10)

thecq ui lih ri u mN'=

It

isgloballyaSYlllptot ically st.ablc.Bythocom pa risonpri u- ciplc,itfollowsthat

(2.11)

HegardilIgthetotalvect or populat.ion,wehave dlV(t)

-dl - IT-UT'[I-

W(tI~

^T,Il]+Wa

U-

T,Il -II",W(I)-

f",W~(t)

rc-aT'[J-

W(tI~

^T..Il

I

^+W(1-T,Il-ll",IF(t)

For system

'!If

=rc-^aT,I1;: -11",y(t),theequilibri umr"~I;·,~:.' I\isglohallyasyi u p to t- icallystab le.Bythecom pa rison principle,it followsthat

IT-UT,\I\~

Iimsu p W (t)::;-'-.- .

'- 00

"II",

(2.12)

By (2.11)awl(2.12), it followsthat(T" =00,allthesoluti onsexistglohally, andarculti matelybounded.Moreover,whenN(t)

>

max{ /!;,'·'·-I;'~:."\ }andW(I)

>

max{/!;'7'r~I;'~:.'l\},wehave

dJ~?) <

0,

ill;;?) <

0

wh ich impliesthatallsolutio nsarcuuiformlybounded.Th()rdore,tho soluti on scmillow<I>(t)=/I,(-):;t'J--->;t'Jispointdissipntivc,By [IG, Theorem:I.<i.l],'I>(t) is compact for anyt.

>

T.Thus, [17,Theorem3..1.8)impliesthat<1>(1)has acompnct gloha l a t tractorilI ;t'J.

2.3 Threshold dynamics

o

IIIthisscct.io u,we esta blis h thethresholddynamicsfor themodelsystem ill(('I'1IISof tho bnsiorop roductio n uumbcr.

\Vedefinethe"d iseased classes" asthemosquitoand lnu nnnpopul a t.iousthatan) cit.he rexposed or infectious,i.c, IV".W2,EamiJ.To get. thediseas e-fre e equilibrium,

23

lct.t.ingIV"

=

W~

=

E

= / =

0,wethengetI?

=

0and

,.r-"T"[I _

1V1(I/:T..d]+IV,(1_ T..I) -/1",11',( /).

11- /1,,5 (1).

(2. 1:1) (2. 1,1)

Etl=(0 . O.O.N' .O.O.0)andE,=(IV'.O.0,N'.O.0,0)arctwodis('asplroceq u ilib ria.

whereIF'=I\(" ;.~::,':,~""').By[4:3.Pro positiou-l.I].forsys tem(2.13).the equilibrium IF'is globallyaSylllp toticallystable if thefollowi ng condit ion is sat.isficd

Linearizin gthe1II0d el sysl.om at thediseaseIreoeq u ilibri u mE1•we ob t ai n t.lu: fol- lowing system(he rewcoulywritedown thccq unt.ionsfort.hcdiscascdclass('s):

dll',.(I) 11" IF'

-dl - (-J"'"'"fF1(1)-/1",11",,(1)-/-J,,,C-""'T"'"'"fF1(I-T",),

dll'AI} 11"

-dl - jj",c-I''''T'''"'"fF1(I-T",) - (/1",

+

£",)II'~(I ),

d/~~1} /-J"IV~(I)

-/1,,£(1) -f-J"c-""T"IFAI-T,,).

d~l~l}

^/1"c-I",T"IVAI-T,,)-(/1"

+£"

+1)1(1).

Followin gtheideain

[381,

weintroducethebasicrcpro d uction uumberforthe 1II0d ei syste m.Donotcr j,:1'2,:1':\and:1'.,bet.hcnumberofeach diseasedclassat time1=0.

and:,.,(1), ,"A/).:1':1(1)and:,.,,(1)betheremainingpopulnt. ions of cachclassattimcI.

respecti vely, thenwe obtain

24

o "" (;:;:'~I;;'+1) J

/1",('-/, ,.·r"'1\' ·

"" (1" , + £,,+1)

o o

Thetot nl nu mbcrofnewlyinfect edineachdisl'as('dclass is

Siucc

[

: :] =[:: 0 0

~:':~::::

^{;::] [:}

^{: J}

TI 0 I"I~€, 0 0 II

11 ()

d ;I::':..

T/r 0

1 1

We callseethatthe2x2nuu.rix:

[ ^{:: ::}

l\1_n=

o _I' "~~£'''

() '1;~:~:';'11~'1

isthenextinfect.ionopera tor.Asusual.wedefinethe spectra l radius oft.hcmatri x

Mn asthehas ic rl'p rodlltl.ioll llulllhcrRn forthctuodclsystom.Itthcn followst.luu R /J",fJj,e-II,,,.r"'+I,,,r"l1V'

"n

=

N'(/I".

+

C",)('II,

+

CI,

+ , )'

011I' firstresultshowsthat.thediseaseisuniformlyporsistontifRn

>

I.

Theor em 2.3.1.Let(Ill) hold.IfR.n

>

1,thenthereisanII>0suclitlmirnu, soluti onut],rll)of the model "ystemunlli 'IlE,yo,,/>:1(0)

i'

0and,1'li(O)

i'

0"a1i"jic s

25

Proo].Defille

Clearly,wehave

Defille

Claim L.Tlwn:cnist.s11,)1>0,such1lIlIlfm'lIny'pEXo,lim sllp '_ N

II

<I'(I),!,-Eo 112':

SinceII",</'(~-"T,\,we callchooseco>

°

and,)\>

°

sullicicut.lysmall,surhthat

11\

+ II~

^{': lI;l}

⁺

^11,1

<

co,

VI

(111,

II~,

^11;"

^{lid -}

(N ',0,0,0)

1 < ,)"

(2.lfJ)

11", + !j",co < ,.e-"T'\ ( I - ~ ).

(2. IG)

For allYr/JEXoI,since4),,(0)

i'

0,and9(i((l)

i'

0,itfollowsfrom Theorem1.1.1.we get

W~ (I)>0.l(t)>0,"11>0. (2.17) Nextweshowtha tthere existsa102':0,suchtha t1I',(lo,r/J)>0.tor all^41,EXo.

Otherwise,thereexists

4 '

EXo,suchthat1I'\{t,r/I)=0, for all12':0.From(2.2).

we get11',.(1):=0forall12':T""t.hcn from(2.1),wegotII'Al):=0,for all12':T""a cont.rndict.ionwit h(2.17).Theil,"yTheol'Cm!.!.I,1I',{t)>0,forall 12':10,

Suppose,hycont.rnd ict.ion,thatlill\SUPt-oo

II

<I'(I)/j,-Eo11<8,forsome/j'EXo.

Thus.

II

<I'(I)/j,-Eo11<,),holdsfor all lar geI.

2G

Thenwc rnnchooselar~enumberII>10•suchthatforalii2:II.therehold s that

Consid erthcucxt.luu-a rand1Il0l1 0 t OllCt.imc-dolnvcdsystem

Let.Aohetheprincipal eigenvalueofthecorres pondi ngcigcuvalucproblemof cqua t.ion(2.18).Sinceforany¢JEC([-T,OJ,IR+)with¢J(O)=0, weha ve/,c-oT.1(1_

:181/J()¢J(-T/I)2:0 .Therefore,(2. 18) iscoopera t.ivo.Thenwecons ide rt.hcnuxiliury system

By(2.1G),t.hc eige nvalue,\;, of system(2.19)is/,c-^{O T}'\(I-381/1\')-(II.",

+

(Jo.E:o)

>

O.

ByTheorem 1.1..1,weget'\0

>

0 ifandonlyif

A;, >

O.Therefore,'\0

>

O.

Wcca ll chooseI>0sma lleno ug hsuchthat.IcA,,1::::11'1(1),111E[11./1

+ T i d.

Clearl.\',teA"1sa t is fies(2.18 )for all/2:II'The il hy thecom parison pri nciple .we~cl

coutrn d ictio n.

Cla im 2.Tlw1'l~exists(J()2

>

0,such Ihfll!ol' fl ny4'>EXo,liIllSllP,_oo

II

(I)(/)4~-EI112:

(,'irstweconsider t.hcloll owinglinear«ooporativo syste m

27

(J",e-"".T"(

* -

^{£)i(t-T",)-(/1",}

⁺ £".)Ii!~(t ).

(J"e-/",T" (I-£)\ ir~ (t-T,,)-(/1/,

+ £/,+

r)i(l).

(2.20)

Forsullicicntlysmall

e >

O.lcl'\1(£ )betheprin cipl eeige nvalueofsvstcm(2.20).

It is easyto secthat

f

is cont inuouslydiilcromiuh lc cooperat ive in the s('nsethat.for

anyt/~EC([-T,O],IR~) ,thelinearoperat orL:=I({(¢)snt.islio»1,;(0)

2':

0whcunvor

o

^EC([-T,0],IR~ )wit h'1;;(0)=0forson icI:Si:S2.Theilweconsidert.hcauxilinry system

(2.21)

Let,\;(£)hethepri ncipal eigenva lue ofeige nvaluepro blemofsystciu(2.21).n.\,

calcu lat ion, it followsthat ,\;(s)

>

0if andonly if0(£):=

I(;:.:'\'~~.(;;;::'~:::':~; C~; -

£)(1-e)

>

1.Whene

=

0,0(£)

=

R~

>

I.Since0(£)iscou t.inuous withresped toe,wecancho oseesmallenoughsuch that0(£)

>

I,th us,\;(£)

>

O.B~'Theo rem 1.1.4.'\1(c)

>

0 ifandon lyif X_I(£)

>

O. Thus ,we('lU Irestrict.esnia lleno ughsuch

Forthissmalle,there exists(5~=c5~ (E:)

>

0, suchtha t

bl

+

b,)

~

^lin

⁺

^b7

>

I-e

>

0,atul b

l

~~~~~ 117 ^> *

^{- rE}

^>

^0,\;j

^{1 (I)l , II~,}

^..,b7)-B1

^{1 < c5~.}

28

Assum e,hy contra d ictio n,thatlimsllp,~oo

II

(1)(/)~b-E,

11<

^')2lorsome~)EXo.

Th(~11thereexists alargenum ber12,suchth a t.for alli

2:

12,

Theil we callfurtherch oose1;\

>

121a rge ellollgh,suchtha t for all12:I;"

Thatis,whe nt.

2:

l;"wehave

/3",,,-I''''T'''(* -E)/(I-T",)-(11",+E",)IVAl), (3"c-^{I I}"T/,(I-E)1V2(1.-T,,)-(II"

+

E"

+

,)/(1. ).

Letp=(VI,V2f'hethepositi verighteigenvecto rassociatedwith,\ \(E)forsyst em

(2.20),cho ose!

> o

sma llenoughsuchtha t

Clear ly.h) q(E)I( p" v2f'satisfies(2.20) for12:1;1'Theilhythe corup nriso u prin ci- p1(\\v(~get

Sillce,\ ,(E)>0,lett illgl->00,wooln.ain

i~~IIW )

=

00,i~~1(1)

=

(X) n con t.rudictio n.

Letw(</J )he(he omega limit set oft.hcorb itof <1>(1)through

4 )

E,1.;1.

2D

Cla im3.U¢E.lJ"w(rP)=EllUEI•

Forall~'rPEMil, i.o.'1>(I)rPEDXII•WC111\\'cIVAI ,rP)

==

0,orI(I.~?)

==

O. If Il'AI.rP)

==

O.t.hcnfromthoequatio nsofS.EundI.weha velilll /_ ooS(t.6)=N'. lilll,__ooE(I,4» =0and lilll/__oo1(1, 4»=

n.

l.ot.'()(I)hethesolut.ionscinillowofthe modelsystem,which isdefinedas

FollowingfromTht'or cIll2 .2.1,(1)(1)is com pact.forallYI>T.Letw

=

w((/J)hetho olllegalim itsorof<!>(I)rP.It.then follows fromTheorem 1.:1.1t.hat.wis all iutcrun llv chailltralls it.i\'esct for«)(I).lIcllce,wchavc

w=WIX{(O,N'.O,O)}xW2

forsomeWIE C([-T,nj,IRt)andW2EC([-T,O].IR+).Itis easyto soct.hat

where~'I,~'2'~)7EC([- T.0]' IR+) ,II>,(I)isthesoluti onscmillowassoci ated withthe following system

1111',(1)

-111-

alld^l\'2(I)isthc so lu t.ionseiu iflowassocintc d with

(2.2,1)

SinceWisanintcruullvchaint.runsitivc set for1(>(/).it. t.hcnfollowst.hntWI.W2 arealsoiut.crually chaintrunsit.ivo setsfor'h(l ).1\)2(1),respccti vely. For systeu:

30

(2.24).{O}istheuniqueequilibrium point.andgloballyasvmpt.oticnllvstahlc.Let

\\f=l\fl(l.)+lh (l.).syst.elll(2.22)(2 .2:3)isequivalentto dl\!(I.)

d I d\\',

(I) -dl -

n:-^OT

"[1- \\/( II~

^{T,d]+(\\,(I.-TA)_11..,\\/(1),}

-11..,\\1,(1.).

(2.25)

By[.I:!,Prop osit.iou-I.I],W'and0are globallya.~ymJlt.ot. icallyst.ah l«fo rsystem(2.25) and(2.23),rcspcct.ivcly.So(jV', Orris glohallynsviup to t.icall,'stahlcforsvstcu:

(2.22)nn d(2.23)illC([-T,O],IR~)\{(O,O)}.'I'hcro lorc,byTheoremI.:l.2,weget WI={(W',O)}or{(O,O)}, andW2={O}.Thus,wehave w={(W',O,O,N ',O,O,O)}

or{(O,O,O,N ' ,O,O,O)}.

t.luu\\'2(1.,(,6)

>

0for allI;:::1o,andJ(I,4))

==

O.Fro mtheeq uatio nsofW"W2 and

u.

weha ve

w=w:(X{(O,O)}X WIX{(O,O)}

forsomew:(EC([-T,0]' IR+)andWIEC([-T,OJ,IR~).It. is easyt.oseet.hut

where4)1,4'104'5EC([-T,0]' IR+),'1':«(1.)isthesoluti onscmiflowasso c-iutr-dwit h the Iollowing syst.cru

and '1>'1(1. )ist.hc solut.iouscurillowasso ciatcdwith

(2.27) (2.28) 31

By[4:1,Prop osition

'1.1],

W'is globallv asytn pto t.icallvstnhlc forsyste m(2.2G)in C([-T,OI,lR+)\{O}.Clea rly,N'am i 0 are gloha llyasym p to t ica lly st nblcforsyste m (2.27)am i(2.28),resp ecti vely.Theorem1.:1,2implies tha tW;\={W'}or {O},WI= {(N',O)}.Theref or e,wehavew={(W',O,O,N',O,O,O)} or {(O,O,O,N',O,O,O)} . Co nseq uently,wehaveU1>EM"W (</;)=Eo

u

E,.

Definea couti nuous Iunctionu :'Yo-;lR+hy

Clearly,1)-'(0,00)CXo. It followsfrom(2.17)tha tI!ha,~theprop ertythatif eithe rp(q))=0andqJEX o,orp(lp)

>

0,thenp«I>(I)<p)

>

0,^f(JI'all

t >

O.Thus I) is a generalized distancefuncti onfor thescmi flow <T>(I):

,y,\ -; ,y,\.

By clai m :3,we get tha tany forward orb itof<I>(t)iniliaconver gestoEoorE"hy cla im 1 andclnimZ,we concludetha tEoandE,are twoisola tedinvarian t.sets in

,y,\,

and (IV'( Eo)UIV'(E Jl)

n

Xo=0.Mo reover, it iseasytosec that nosubse tof{Eo,E,}

formsacycleinDXo.I3y Theo rem1.2.3,itthen followstha tthere existsII

>

(lsuch tha tIimi n f'~ool)(I>(t 14))::::I} 1'01'allq?E'\0,which impliestheuniformpcrsistcn co

stated in thetheorem.

o

Thesuhscq uc ntresultshowsthat thediseas ediesou t il'Ro

<

I,providedt.hcr«

isonlyasmall invasionintheIV~andIclassc». Foranygiven1\ 1

>

0,deno te

XiI)=

{< I)

EC([-T,0],[0,MI'):

t ^¢l~)

^{::::8,'<:/sE[-T,OJ,}

¢~(O)

⁼

. f" c" ""'f)"'~L~ ~'ii::~

^ds,

</;(0)

= } . o

,1'1" fJ</;.,(s)</;;\(s)I'}

" -T"(. "L::.\<Pi(S)l s.

Thcllwchavethel'ollowin gresnlt..

32

Theorem2.3.2.Let (1/1) hold.IfRo

<

1.thenfOI'eIlCI~/1M

>

max{~,'·'·-I;,~".'/\ }.

there crisisII(=((M)

>

0sttcltlluit.fOI"IIl!yII>E

<' (ill \

Eo with(~~As) .4)(i(s))E

[0. (f.

fOI"lillyS

E [-T,OI.

llu:SOllltioll ll(/., ¢)oftI/I:modelsyste miliro uqh¢sutisjics lim/_ DO

II

II(/..¢)-E,

11=

O.

Proo].Let.M

>

IIIax {~,....~I;,~:q\ }he givon.FWIIItheprov eofTheorem2.2.1.w«

sectha t

x,t

is positivelyinva r iant for thesolut ionscmillowof t.hcmodel system.We t.hcn havo

II(/.,¢J )E

[ O,M]',

VI;:::0, 11>

e s] ',

Cons ider thefoll owin glinear and monotonesystem

(i",c-f,,,,r,,(

:~: ~;

^{)1(/.-T",)-(/1",}

⁺ f",) \l!~(/.),

fJ"c-f",r/ll'At-T,,)- (/1"

+ f" +

,)1(/.).

(2.29)

Forsufficicut.lysmall

e >

0,letAAf)hetheprincip leeigenvalue of this eige nva lue probl em.Clear ly,syste m(2.2!J)is coo pc ru t ivc,Thenwe cons ide r thcnext aux ilia rv syste m

(2.30)

LetA~ (f)hetheprincipaleigcnvnlucof theCOIT('SpOlHling eige nvalue pro blem of syste m(2.30).Thenwe get

A~(f) <

0 ifandonlyif(i (f):=

I(;;':'~~~~,I;;;;:'~:::/:~; ~~:::: <

1.

Whene=0,(i(O)=R~

<

I.Since(i(e)is cont.in no uswithresp ect. toE,wecun('hoosc fsm allenoug h,suchtluu/i (f)

<

1,there fore,A~(f)

<

O.Following1'1'011IThe-orem 1.1.4.we get.A~ (f)

<

0 if andon lyifA~(f)

<

O.Thus,we can restrict.Esma llenollgh

33

Nowweconsid ert.hoIollowingcq un tio ns:

rc-"TA[I-

\\/(tl~

^{TA)]+I\/(I_T;\)-}tl",\H I.)-

E"'~I,

^(2.:11)

II-11"l 1(1)-E"~l' (2.:12)

Choose small~I

>

0and largeT=T(M)

>

0such t.hnt.for anv solu tio nsof (1\!(I,rjJ),N(I,cjJ)).WeI,hellha ve

1\/(1)

<

W'

+

E,iV(t)>N'-E,'It:;:0:T.

Denotethesolut.ionofsystem(2.29)byii(l ,r/J)=(11/AI),l(1)) withrespect.to initialdat ar~=(¢I,r~2)EC([-T,0], [0,M]2).Theilforsystem(2.2!J) ,for~I

>

0,

Foreverysolutionofthemodel systemthrough

4 J,

thereexistsa( =((JIl)

>

0 suc h that

provid edt.luu(cjJ:l~), cjJ(;(s) )

< ((,() .

WoIurthcr claimf,hat.(2.:H)hold sfor allt:;:o:O.Sup pose,hyco llt.ra d ict ioll,t ha t

principl e,for

t

E['1'), 7

2],

wehave

:34

a con tra d ictio n.So(2.:14) holdsforall/:::::O.From(2. 3:1) an d(2.:15),woSl'Cthat lilnt_oo(W~(I,¢),1(1,tl)))=(0 , 0) .Let<1>(1)hethe solutionscm illowof t.homodel system. andlet.W=w(</))he theonwgalimitset of<I>(I)¢,whichisaniutcrn ally chain transitive setfor<1>(1).Hence.wehave

w=w"x

{O}

x W(ix

{O}

XW7

for someW,,,WuEC([-T,0],IR:~)andW7EC([-T,0],IR:+).Itiseasyto secthat

wit hthofollowiug systo m

tlll~~(t)

/"c-"T\[I_llfl(l-

Tid ~

^li/,(I-

T ,dl+(lIW _ T,d +

11/,.(1-

T,d)

-/I",ll/j(l), tlll/,.(I.) -tl / -

<!)/;(I)ist.hc solut.iouscmiflow associa tedwith

and<1>7(1)isthe solu t.ionscmiflow asso cia tedwith

Itt.honfollowsthatw",W(i,W7arciutorn nlly chain transitiveset s for<1>,,(1),<I>(i(l).

alld(1)7(1),respecti vely.Bythean alysis ofsvstcrn(2.22),(2.23),(2.2 7)and(2.2,1), we getw"

=

{(W',O)} ,Wu

=

{(N ',O)}.andW7

=

{O}. Thus ,we'haveW= {(IV' ,O,O,N ',O,O,O)}.Itfollowsthatlilllt_oou (l,~&)=(W', O,O,N',O,O,O). 0

2.4 Glob al at t rac t iv ity

Inth is section,II'Cst udythe globa l attra ctivityin themodel system inthecase where thediscns c-iuducoddeathrutcsof infectedmosqu itosaud luu nan ind ividua lsarczero, andthe Iocuuditv ofiufcct cdmosqu itosistho samo asthe suscep ti blemosqu itos.In this case,themod elsyst.cm becomes

(1lI:;~(t)

^l'c-aTA[1_

IV(ll~

^{T,d]+IV(t-T,d-}jl",IVI(I)- /J",

j~T~/)

^IVI(/)

dl~;(I)

^13",

J~~))

^IlW)-jl",IV,.(I)-13",C-I,,,.T,,.

~((I-=- :::;)

IIW-T",).

dl~;~(/ )

/J",C-I,,,.T,,,

~\tl-=- :: :;)

IVI(I-T",)-

jlwlV~(t),

d~~~t)

^If-jl"S(t)-13"

IV~(I) J~~'/)'

^(2.:lG)

d/~;i) 13"IV~(t) ~~'/)

^-jl" E(I.)-IJ"C-I'hT/'IlW-T,,)

j~~II -=- ~;,))' d~/;l) IJ"C-I'I,T/'IV~(I -

^T,,)

~~II -=-~;,)) -

(jl"

+, )/(/),

dH(I) ,1(1)-jl/,l?(t).

--;Jf

It is clear tha twhen Ro

<

0,system(2.3G)has onlytwoeq uilibriaEo aud E1•However,system(2.3 G)ad m itsaequilib ri umE':=(IVt,IV,7,IV;,5',E',I',

tr ,

whcuRo

>

I,whcre

5' ll(jl"/J", +jl",(jl"+,)c',/,Th ) jl/,(jl/,(J",+jl",(jl"+,)CI",ThRfi)'

r =

jl"I)",

:~::,:'(;~~-,;;I"'TI'Rfi'

:lG

am i

IF;

11'(

IF,:

E'

(/I"+,)N'c',,,T,,f'

#,,5'

II",N ' d ,,,·T,,.!!,;

- -/,,,,-/,- (c',,,.T,,.-1)1F; , (d",T',-1)(/lh+ ,)f'

Thefollowingtwo resu ltssho wthe globalat.tr. uti vitv of system(2.:lG).

Thcorcm2.4.1. Lei (111) hold.

If

Ro

<

I.(J= I.andf",=flo=O.then1//(,

P1"IIoj.If(J=0ande; =c"=0,thewholemosquit osandhu m a n populationsnd m it thefoll owingtwo cquat.iou s:

dll'(l )

d t

dN(!}

J f

I'c-"T"[1-

1F(t1~

T,tll+lF(t-T,,)-11".IF(t ).

1I- 'lf,N (I).

Since\1"andN'is globa llyasym ptotica llystah lcforthe above two eq ual.io ns.rc- spoctivcly,the reexis tsT=T(f)

>

0such tha t

lV(t )::;IF'

+

c,N(t)2:N'-c.VI2:T.

ThllH,whenI2:T,weIllIV(~

dIFA!}

-dI- d/

Ji

f'",I·-/,,,,T,,,(

:~: ~ ;

^{)/(1.-T".)-(,I",}

+ c",)\ F~(t),

f'Io C-I,/, T/'II'~(t-TIo)-(/110

+

Clo

+

,)/(1.).

37

WhenRo

<

I.ssma lleno ugh.hytheana lysisofsystem(2.29)and thc com pnrisou principl e.wethenhave

It.the n followsfromthetheor y of asy m p totica llysemi flows(see[32])that

,I~~(IFIU).lF,.U ),SU ) .EU). HU))=(11",0,N',O.O).

This completestheproof. D

'Io oht.ain the gloh al at t ract ivityoftheend em ieeqnilihr illm.weneedthefollowing udditio ualnssurupt.ion:

Theor em2.4.2.Let(Ill)1I11d(1/2) hold.IIRo

>

Iawle.;=Eh=0,a=1.llicu [or1I11y¢ E,yountl:¢a(0)

f=

0,¢(;(O)

f=

O.weluuu:Iilll ,_ooIIU. ¢)=E'.

Proof.WhenE",=Eh=O.a=I.wehave

rllFU)

---;/l

rlN U )

----;i/

I'(;-" T.-I[I-

IV( /l~

^{T··!l]+WU-T}i,)-'1",11'(/).

II- I{{,N(/).

(2.:17)

When(III) hold s.(W' .N ')is gloha llyasy mp tot icallysta blcfill'syst em (2.:17).

Hence.weha v« t.hcfollowinglimit.ing syst cm:

dl~~(i)

^A-/1,,,11',(1)-/:I:,'/(I)W1(I),

d\~;//)

(3;"l (l)W1(I)-/1",lV,,(I )-(:I;"c-I,,,,r"'I(I-T",)W,(I-T",),

d\ \~~(I) /:I:,,f~-I,,,, r"'l (I

_T",)W₁(1-T",)-/1",W₂(1),

d~;t)

^II-/1,,8(1)-

^{/3 ;,}

W2(1)8 (1), (2.:18)

d~;')

(j;,W2(1)8 (1)-/I"E (I)-/j;,c-ll"T"11'2(1-T,,)8 (1-T,,),

d~/~/)

/3;,c-I",r"W2(1-T/,)8(1-T,,)-(/1),

+

,)1(1), dl?(I ) ,1(1)-/I" I?(I).

---;[t

whe re

A

=W'/I""/j;"=

/3",/N' ,

/:1;,=

/h/N' .

Itfollowsthat

.'1' (1) W{(I) + c', ,,r"'\V~(I+T,,, )

;\-/1",.'1(1).

Theilt.hc equilibri umA//I",=IV'is globa llyns yrupto ticu llvst.nhlo. For~y~tclll (2.:18),wet.hc n considerthefollowinglimiti ngsystem:

dlh(l) -d/ - dS(I)

---;u

dI (I)

dJ

(2.:HJ) (2.'lll) (2.'11)

Cla im. The setV:=C([-T,

O ],

[0,W' e-I,,,,r,,,] xlR~Jis]!ositiwlyin var ian tJar sys /i;/11(2.:19)-(2.41).

:m

Toprovethis claim,wcdclincF(I/)):=

(

/-J:,'C-I''''T'''I/):l(-TW)(W.-cl,,,,T"'I/)I(O))-1/",1/)1(0)]

11-11,,1/)2(0)-/-J;,I/)I(O)I/)2(O) ,l;fI/!ED, P;,c-I,/,T/,1/)1(-Th)I/)2(-T,,))-(II"

+

,)1/):1(0)

Note thatDisrelati velyclosedille([-T,0],JR:l),.uu]FCI/')is cont.iuuous and Lipschitz ill1/'illeachcompactscl.illJRxD,By [IG,'I'hcorcm2,:1],itfollowsthatfor all1/'ED, there isallunique solutio n ofsystem(2.:19)-(2..11)through (0,1/))Oilitsruaximnl intervalofexistence.SinceI~(I/');:::0whenever1/)EDwith1/),(0)=0,Theorem 1.1.1iJllpliestha t t he solut iouo f(2.a9)-(2.4 1)arcnonn egati veI'orullrin itsmnxiuurl interv al ofoxistcucc.Furt.hcnu orc,ifI/)I((J)=W'el,,,,T,,,,thenFI(I/'):::::O.It follows by[29, Remark5.2.11tha t l\!2(1,1/)):::::W·c-I,,,·T,,,forali i>O.Thus,'Dispositively invari ant.

Bytheargumentssimilar tothoseill Thcorcmz.d.l ,itcusilyfollowsthutsvstcm (2.a9) -(2 .41) isunifon ulvpersistentill the sense thatthereexistsa'II>0such

li:~!:1,d'(I1/AI, I/)),[(I, I/)) );:::(111,1/1).

Foranvgivon1/) EDwit h1/'1(0)

i'

0and1/):1(0)

i'

0,let(II/AI),3(1),[(I))= (11/₂(1,1/)),3(1, 1/)),[(1,1/'))' IIIorder tousethemeth od of Iluct.uat.ious (sec,c.g.,[18, :l:1,:19]) forsystem(2.:l9)-(2.'1l),wcdefille

11'2⁰⁰ liJf~~I]J11'2(1),11/₂₀₀=li:~!:1,lfII/Al);

3⁰⁰ li;~~IP 3(1 ),300=li:~!:1,Jf' 3(1) ;

[00 li;I~~P[(I),[oo =li:~!:1,Jf·[(I).

40

exis t.scqucnrcs

t:,

---t00ami

a ;,

---t00,i=1,2,3,sucht.luu

,,'~~

!lJ 2(t:.l 117°,

I1f~ (I :, )

=

0,Vn

2:

^I;

"l~~\\fAa:,) \\f2:>o ,\\f~(a},)=0, V/I

2:

I;

,!~~5(t~, )

5

⁰⁰

,5'(1;' )

=0,VII

2:

1;

"I~~5(a;,)

5

00 ,5'( a;,)=0, VII

2:

I;

,,'~~J(t;~) Joo.J' (t;~)=0,VII

2:

I;

"I~~l( a;~ )

1

00,

l'(a; ;)

=0,VII

2:

I.

thc abovcclaimt.lmt

nud hcn cc,

By(2AO),weha ve

wh ich im pliesthat

ill

(2"12)

(2.,1:3)

Invicw of (2.41),we obtai n

nnd hcnrc ,

There fore,combining(2.'13)and(2.'14) together,w« get.

(2.-1,1)

Comparing(2.'12)with(2.'1G),woobt.ain

Silllplifyingthe abovctwo inc<jllalit ics, wc get.

Sincecond itio n(1/2)holds,wehaveIVt'

=

1V200. By(2.,1:3) and(2..IG),we get SOO=Sooand[00=[00'Itfollowsthutliml~00(l1!2(t ),S(l ),r(l))=(lVi,S ' ,!')for any1/) ED wit ht!',(O)i'Oand1/):1(0)

i' 0.

Noll' wcdclin cIo rsys tcru (2.:38)that

13y sim ilarargumonts<t';in'I'hcorcm2.2.1, it.foll ow sthatco"isanon cmp tv and cOlllpactsllbsetofC ([- T,O], ~~ ) .Since for allY I" ,

Thus vwulurvc

Therefor e,weob tain thatw'CD,

Bytheabove claimand thocon t.inuo us-t.iruo versionof['12,Lcnnu a1.2.21,itfollows that w'is au iutcrnnllychnintransit.ivoscrfort.he solu t.ionsom illowolsvstci u(2.:19 )- (2.'11)Oilthe posi ti vel yinvarian tsetV.ThenbyTheore m1.:1.2,w'

=

{(lV;,S',I' )}

or{(O,N',O)}. 13ysim ilarargu ruc u tof Theorem2.:1.1,w'

# {to,

N',O)}.'l' hc rc forc.

w'={(Wi,S',I')} .Hence,weha ve,forsystem(2.:l8) ,

IV'-el,,,,T"'IV;=1\7,VI"- t00asII- t00.

Let<1>'(1)bet.ho solu tio nscu iiilowofsys tem(2.:38). Bysim ila rargu me ntas TheorcnI 2.2.1fort.humod elsyste m,wo obtuint.hat.'1>'(1)iscom pac t for anyI

>

T.

LetWi=w' ((jJ)bet.hc omegalimitset of(1)I(I)c/>.It.thoufoll owstroiuTheoreml.:l.l tha tWiis alliutcrua llycha int.ransit.ivcsct for'1>'(1).Hen ce,wehuvc

Wi

=

Wtx

w ;

x{(W;,S' )}xw~x{I'}XW~l' forSOIllC

w;

EC([-T,O],~+),i=1,2,:1.It is easytosec tha t

(1)1(1)

I",' (wt,

1'2,W;,S',1)f>,1', 4), )=(Wt,\!J'(4h ).

w; ,

S',\!J'(1,iJ,,),1',\!J'(1', )).

4:3

where4'"4'~,4)7EC([-T,Oj,IR+),<1>',(1)isthe solu tio nscm illowasso cia te dwit h

<l>~(I)ist.hosolu t io nscm illowassocia te dwit h

(2.47)

and <1>(1(1)is thesolutio nscmi llowassociate dwit h

(2.48)

Since

w'

is au in ternally cha intrans it iveset for (1)'(1), it thenfoll owsthat

w;,

w~,W~1 arcalsointerna llycha in tran si ti ve sets for <I)',(I),<I>~(I) ,(1):1(1),rospcctivcly.Clea rly,

11'/ ,

E',R"aretheuniqu e equi lib rium po in t an dglo ba llyasyin ptot.icallvsta b le fOJ (2.4G),(2.47)and(2.48),respectively. There fo re,byTheo rem1.:1.2,weget

w ;

= {W,:},w~

=

{E ' },andw(l

=

{H'}.Thus,wehavew'

=

E'.

2.5 Numeri cal sim ula t io ns

o

Inthissection ,wecarry out numcri cnl sim u lnt.ionstoillust rat e0111'ana lyticresults . Inviewof[3SJ, wefixTA

=

10,T",

=

10,T"

=

S,aw l thentak ethree sets ofvalues ofotlw r parallle te rs to per for lllt hen u nwr iealsiln u la t io lls.

First,wetake

Ii", =

(l.OG,{.;"

=

O.IS,r

=

I,a

=

0.2,'Y

=

O.IS,/1",

=

0.1,

/1"

=

0.()()01,II

=

0.001,f(

=

10,IT

=

0.8,lOw

=

(Ull,10"

=

0.0001.It is easyto

veri fythatcon d itio n(HI)holds,and Ro

=

0.17,1,11"

=

2.GI ,N'

=

10.Itfollows fromTheorem3.2 th atwhenW~(s )andl(s),sE[-T,O),arcsmal l,thedise asewill die out(secFiguro2. 1).