The Global Analysis of a Stochastic Two-Scale Network Epidemic Dynamic Model with Varying Immunity Period

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1 Joal of Apple Mahemac a Phyc hp:// ISSN Ole: ISSN P: The Global Aaly of a Sochac Two-Scale Newok Epemc Dyamc Moel wh Vayg Immy Peo Dve Wak G. S. Lae Depame of Mahemacal Scece Geog Sohe Uvey Saeboo USA Depame of Mahemac a Sac Uvey of Soh Floa Tampa USA How o ce h pape: Wak D. a Lae G.S. (7 The Global Aaly of a Sochac Two-Scale Newok Epemc Dyamc Moel wh Vayg Immy Peo. Joal of Apple Mahemac a Phyc hp://o.og/.436/amp.7.55 Receve: Apl 3 7 Accepe: May 3 7 Pblhe: May 7 7 Copygh 7 by aho a Scefc Reeach Pblhg Ic. Th wok lcee e he Ceave Commo Abo Ieaoal Lcee (CC BY 4.. hp://ceavecommo.og/lcee/by/4./ Ope Acce Abac A ochac SIR epemc yamc moel wh be-me-elay fo a wo-cale yamc poplao eve. The be me elay he vayg aally ace mmy peo of he emoval cla of val who have ecovee fom he feco a have ace aal mmy o he eae. We vegae he ochac aympoc ably of he eae fee elbm of he epemc yamc moel a vefy he mpac o he eacao of he eae. Keywo Deae-Fee Seay Sae Sochac Aympoc Sably Thehol Vale Povely Self Iva Se Lyapov Fcoal Teche. Ioco The ece ave of hgh echology he aea of commcao apoao a bac evce mllaeal eaco have affoe effce global ma flow of hma beg amal goo epme a ea o he eah ml-pache face. A a el of h he wol ha become lke a eghbohoo. Fhemoe he aoal a baoal poblem have become he mlaoal poblem. Th ha geeae a ee of coopeao a eag abo he bac ee of hma pece he global commy. I ho he ea of globalzao peag almo all apec of he hma pece o he face of eah. The wol oay face he challege of ceagly hgh ae of globalzao of ew hma feco eae a eae DOI:.436/amp.7.55 May 7 7

2 D. Wak G. S. Lae a [] aoce wh he hgh mbe of e-pach coeco moe effce global hma apoao. Fo ace he ece 9 HN fl paemc [] cloely eelae wh he may e-pach coeco faclae hma apoao of he eae. Aemp o y hma feco eae yamc flece by hma mobly poce complex hma mea-poplao ce ae mae [3]-[5]. The clo of he effec of eae laecy o mmy o he epemc yamc moelg poce lea o moe ealc epemc yamc moel. Fhemoe epemc yamc pocee poplao exhbg vayg me eae laecy o mmy elay peo ae epeee by ffeel eao moel wh be me elay. Seveal e [6] [7] [8] [9] [] copoag be elay ecbg he effec of eae laecy o mmy he yamc of hma feco eae have bee oe. A mahemacal SIR (cepble-fecve-emoval epemc yamc moel wh be me elay epeeg he vayg me empoal mmy peo he mme poplao cla e by Bly a Kyychko [9]. I he y he exece of pove olo exhbe. Fhemoe he global aympoc ably of he eae fee a eemc elb ae how by g Lyapov fcoal eche. Moeove hey peee mecal mlao el fo a pecl cae SIR epemc wh empoal mmy. The empoal mmy wa epeee he epemc yamc moel by leg he Dac ela-fco be he egal keel o he pobably ey fco of he be me elay. Sochac moel alo offe a bee epeeao of he ealy. Seveal ochac yamc moel ecbg gle a ml-gop eae yamc have bee vegae []-[9]. I [] a ochac ml-gop SIRS epemc yamc moel eve a e. The aom evomeal flcao mafe a vably he eae amo poce. I ao he global pove olo exece exhbe by he Lyapov eegy fco meho a a povely elf va e efe. Moeove he he ochac aympoc a mea ae ably of he eae fee elbm ae exhbe by applyg Lyapov eco meho. I [] D. Wak a G.S. Lae eve a e a ochac wo-cale ewok coa empoay elaye SIR epemc moel. The empoay mmy peo acco fo he me lag g whch ewly ecovee val fom he eae wh cofee feco ace o aal mmy loe he mmy a ega he cepble ae. They lze he Lyapov eegy fco meho o pove he global pove olo poce exece a efe a povely elf va e. Moeove he he ochac aympoc a mea ae ably of he eae fee elbm ae exhbe by applyg Lyapov fcoal eche. I [] a ochac SIR epemc yamc moel wh be me elay e. Moeove he ochac aympoc ably of he eae fee elbm alo exhbe by applyg he Lyapov fcoal ech- 5

3 D. Wak G. S. Lae e. I h pape we exe he wo-cale ewok SIR empoay elaye epe- mc yamc moel [] o a wo-cale ewok SIR elaye epemc ya- mc moel wh vayg aal mmy peo. The vayg mmy peo acco fo he vayg me legh of mmy wh he mme popla- o cla. Th mea ha val ecoveg fom he eae ace aal mmy aga he eae. Moeove he mmy peo vae fo val he mme poplao cla. Fhemoe he ace m- my wae wh me a he empoay mme val ae ecovee o he cepble ae. Th wok ogaze a follow. I Seco we eve he be me ace mmy elay epemc yamc moel. I Seco 3 we pee he moel valao el of he epemc moel. I Seco 4 we how he ochac aympoc ably of he eae fee elbm.. Devao of he SIR Dbe Delay Sochac Dyamc Moel I h eco we eve he vayg mmy elay effec he SIR eae yamc of ee of e ego C of he wo-cale poplao. We ecall he geeal lage cale wo level ochac SIR coa empoay elaye epemc yamc moel e gve ([] (.7-(.9. We exe he coa empoay mmy effec [] o he vayg me empoay mmy effec a follow: we ame ha fo each I( M I( I ee of e a feco ( ego C vg e ego C ecove fom he eae a ace aal mmy a aga he eae mmeely afe ecovey. The ecovee val fhe looe mmy a become cepble o he eae afe a peo of me whee he mmy peo a fe aom vable wh vale bewee a fo he ffee val he mme poplao cla. Ug ea fom [9] we eve a copoae he vayg me ace mmy elay effec o he epemc yamc moel ((.7-(.9 [] by ocg he em e δ a ( ( a ρ δ a I f e whee he pobably ha a val who ecovee fom eae a a eale me ll alve a me. Fhemoe f ( he egal keel [9] epeeg he pobably ey of he me o looe ace mmy by ee of e a he vg e ego a ego C who wee pevoly feco C a who have ecovee fom eae f acg aal mmy wh vayg me legh. Moeove ( a f. The wo level lage cale ochac SIR elaye epemc yamc moel wh vayg aal mmy peo a whch flece by he hma mobly poce [3] a follow: 5

4 D. Wak G. S. Lae M δ B ρ ( ( k Sk ρ S I f k a e ( γ σ δ S β a S Ia v ( a S Ia w a S δ σ S I ( ( f e ( ρ δ S β ( a S I a v a S Ia w a δ ( ( e ( γl S l I l fl ρl δl Sl M β a l l al ( a l l al l (. M ρ k Ik ρ I I k a ( γ σ δ I β a S Ia v ( a S Ia w a I σ I I ( ρ δ I β S I a a M v ( a S Ia w a M γl I l Il ( ρl δl l Il β a l Sl I al v ( a l Sl Ial wl a l (. R M δ ρ ( ( k Rk ρ l Rl I I f k a e ( γ σ δ R a δ ( ( σ R I I f e ( ρ δ R a δl γ l R ( ( l Il l I l fl e ( ρl δl R l a l (.3 whee all paamee ae pevoly efe. Fhemoe fo each I( M a I( we have he followg l coo ( S ( I ( R ( ( ϕ ( ϕ( ϕ3( [ ] ϕk ([ ] k 3 I( M a I( I( ϕ ( > k 3 k (.4 whee ([ ] he pace of coo fco wh he pemm om ϕ Sp ϕ(. (.5 a w a Wee poce. Fhemoe he aom coo fco ϕ k k 3 ae -meaable o epee of ( w fo all. 53

5 D. Wak G. S. Lae We expe he ae of yem (.-(.3 veco fom a e be- ely. We eoe T 3 ( T ( T ( ( x S I R x x x x T T T 3 x x x x T T T 3 T M T T MT 3 x x x x T M M 3( ( x x x x M ( whee I( M I( a I (. We e M (.6. Defo p-om : Le z be a abay veco efe (.6 whee ( T a I z z z z3 wheeve I( M I( 3 (. The p-om o efe a follow wheeve p < a wheeve p. Le z 3 M M p p z p a (.7 z z max z (.8 3 p M a 3 k k m k. (.9 m M a 3. Cloe Ball : Le z be fxe. The cloe ball wh cee a z a a B z he e 3 > eoe ( ; 3 3 ( { } z ; z : z z p 3 B (. I ao fom (.-(.3 efe he veco ( l I ( I( M a I ( M y S I R [ I T ( T ( ( y y y y T T T T T M T ( y y y y T M T T MT y y y y T y a follow: Fo M M ( ( y y y y (. a oba ( M B ρk yk ρ fo y γ σ δ y I k a l yl σ y ( ρ δ y I fo a a γl y ( ρl δl yl l I l fo yl ( (. 54

6 D. Wak G. S. Lae 3. Moel Valao Rel I he followg we ae a pove a pove olo poce exece heoem fo he elaye yem (.-(.3. We lze he Lyapov eegy fco meho o eale y [] o eablh he el of h heoem. We obeve fom (.-(.3 ha (.3 ecople fom he f wo eao he yem. Theefoe ffce o pove he exece of pove olo poce S I. We lze he oao (.6 a keep m ha fo ( X ( T S I.. Gve ay l coo (.4 a (.5 hee ex a e olo poce Theoem 3.. Le I( M I( a a I( ( T ( ( ( X w S w I w afyg (. a (. fo all. Moeove he olo poce pove fo all a.. Tha ( ( S w > I > a.. w Poof: I eay o ee ha he coeffce of (. a (. afy he local Lpchz coo fo he gve l aa (.4. Theefoe hee ex a e maxmal local olo X ( w o τ e ( w whee ( e w me o he exploo me [3]. We how beely ha ( ( fo all τ e ( w almo ely. We efe he followg oppg me τ p { ( τe ( w : S > a I [ ] [ ] } > τ ( m ( τ fo. τ he f hg a we how ha τ ( τe ( w a.. Sppoe o he coay ha P( τ ( < τe ( w >. Le w { τ ( < τe ( w } a τ ( We ewe (3. a follow M ( M ( a ( ( ( τ ( V X V X V X l S l I. V X V X V X V X l M M ( ( ( ( l A (3.3 fhe mple ha S w I w >. Defe (3. (3. (3.3 M M V ( X V ( X V ( X V ( Xl l (3.4 whee V he Io-Doob ffeel opeao wh epec o he yem (.-(.3. We expe he em o he gh-ha-e of (3.4 he followg: Se Level: Fom (3. he em o he gh-ha-e of (3.4 fo he cae of a 55

7 D. Wak G. S. Lae M B Sk S ρ δ ( ( ( V X k ρ ρ I f e S k S l S S I v I ( γ σ δ β a ( ( a a a M Ik I ρk ρ ( γ σ δ k S l S S I v I ( ( S ( ( I β a a a I a v Ia w ( v Ia w a a I S ( (3.5 Ia-egoal Level: Fom (3. he em o he gh-ha-e of (3.4 fo he cae of a S δ ( ( ( V X σ I f e S S I v I ( ρ δ β a ( ( a a a I σ ρ δ I S M β a I I ( a a v I ( ( S ( a ( I S v I w v I w M a a a a I ( ( (3.6 Regoal Level: Fom (3. he em o he gh-ha-e of (3.4 fo he cae of a l S l δl V ( Xl γ l I ( ( l fl e S S l I v I ( ρl δl βl al ( l ( al a a I γ ρ δ l l l l l Il S Ial l βl a Il ( a v I ( ( S l l ( al ( Il S v I w v I w l al l l al l a a Il l ( ( I follow fom (3.5-(3.7 (3.4 a (3. ha fo ( < τ (3.7 56

8 ( ( ( ( V X V X M S δ ( ( ( γ σ δ D. Wak G. S. Lae ( ( ( M β Ia v Ia γ σ δ a a a β S I I f e ( ( ( ( ( S M a a a a I a I v I v I w M M M S v Ia w I a I S ( I v I ( ρ δ β a ( ( a a a ( ( ( ( ( S ρ δ ( ( I M M M a a a a I f e M v I a a S v I w v I w ( ( M M l l S l δ ( ( ( l ρ δ I l fl e l l δ M M βl Ial vl Ial a l ρl δl l a l ( ( ( ( ( S l vl ( I al vl Ial wl ( ( I l M M M a l a M M Sl l al l l a Il ( v I w (3.8 Takg he lm o (3.8 a τ ( follow fom (3. a (3. ha he lef-ha-e V ( X ( V ( X ( (ce fom (3. a (3. V ( X ( ( l ( ( l τ S τ I ( τ (. Th coac he fee of he gh-ha-e of he ealy (3.8. Hece τ ( τe ( w a.. We how beely ha τ ( w. e Le k > be a pove ege ch ha ϕ k whee he veco of ϕ ϕ efe (.4. Fhemoe l vale ( M a. he p-m om (.7 fo he cae of p. We efe he oppg me whee fom (.7 I eay o ee ha a k p { [ τ ( [ ]} e : X k ( m ( τ. τk τk k ( ( ( ( M M a (3.9 X S I. (3. τ ceae. Se lm k τ k ( k τ. The 57

9 D. Wak G. S. Lae V ( X ( τ τ τ a.. We how he followg ha: ( e τe ( τ a.. P ( τ. Sppoe o he coay ha P ( τ τ e I he ame ce fom a (3. a (3.4 efe a.. ( τ < >. Le { } w τ τ e M M V ( X V ( X a ( δ ( a V X e S I τ k (. P τ e τ < a τ. (3. Fom (3. g he expeo (3.4 he Io-Doob ffeel V wh epec o he yem (.-(.3 gve a follow: Se Level: Fom (3. he em of he gh-ha-e of (3.4 fo he cae of a V ( X e B S S M δ ρk k ρ k l δ ( ( ( I f e γ σ S M δ e ρk Ik ρ I I ( γ σ I k l (3. Ia-egoal Level: Fom (3. he em of he gh-ha-e of (3.4 fo he cae of a δ ( σ ( ( e δ V X S I f e ρ S ( ρ δ e σ I I I (3.3 Regoal Level: Fom (3. he em of he gh-ha-e of (3.4 fo he cae of a l δl ( γ ( ( δl V Xl e l S l I l fl e ρl S l ( ρ δl e γl I l Il l l I l Fom (3.-(3.4 (3.4 egag (3.4 ove [ ] (3.4 τ lea o he followg M τ M V ( X ( e δ B k Sk S I ( ( f e δ ρ ρ k l M M τ ( γ σ S e δ ρ k Ik ρ I I ( γ σ I k l M τ δ δ σ ( ( ρ e S I f e S M τ ( ρ δ e σ I I I M τ δl δl γl l ( ( l l ρ l l l M τ l e S I f e S ( ρ δl e γl I l Il l l I l (3.5 58

10 D. Wak G. S. Lae τ k efe (3.9. I eay o ee fom (3.5 (3.9 (3. a (3. ha Fom (3.5 we le τ τ k ( whee ( ( ( τ ( ( τ ( k X V X (3.6 k k Takg he lm o (3.6 a k lea o a coaco becae he lef-ha-e of he ealy (3.6 fe a he gh-ha-e fe. Hece τe τ a.. I he followg we how ha τe τ a.. We le w { τ e < }. Applyg ome algebac maplao a mplfcao o (3.5 we have he followg ( { } ( τ τ < I V X e M B δτ ( { } ( τ e I V X I e δ { τ } ( e< < M M l τ l { } ( δ δ τ ( ( e< l l l l τ l l I f I e I e I e e S M τ δ δ { e } σ σ τ < ( I M τ M δ { } τ δ τe< { τ } e< { τe<} l M τ M δl δ I{ } γ ( e e γl e S I τ < l I I e I I e M M τ δ l l l I I e whee I A he cao fco of he e A. We ecall [3] M σ σ a γ γ l l ffh em o he gh-ha-e of (3.7 ae ch ha a γ e δ σ δ e We ow le τ τ k ( T (3.7. Hece he foh a σ e δ δ δ δ δl l I( M l γ l l e δ (3.7 T (3.9. The expece vale of (3.7 emae a follow ( { } ( τ τ k ( < EI V X T e B V X e δ δτ ( ( ( k M M l > whee ( T l fl ( δ l ϕl ( e τ efe k (3.8 Fhemoe fom (3. (3. a he efo of he cao fco I A follow ha I { ( } X ( τ ( k I { } V ( X T ( τ ( T τ < τ τe< k (3.9 e k 59

11 D. Wak G. S. Lae I follow fom (3.8 (3.9 a (3.9 ha ({ τe < τk ( } E I { ( } X ( τk ( T E I{ } V X ( τ ( e k T τ < τ τ < P T k ( e k M M B δ T δl V ( X ( e fl ( l ϕl ( e δ l (3. I follow mmeely fom (3. ha P( { τe < τ T} a k. Fhemoe ce T < abay we cocle ha P ({ τe τ } Fally by he oal pobably pcple < <. ({ τe < } ({ τe < τ } ({ τe < τ <} P( { τe τ} P( { τe < τ < }. P P P (3. Th fom (3. τe τ a.. a wa ee o how. Remak 3.. Fo ay I( M a I( Theoem 3. gfe ha he mbe of ee of e of all caegoe pee a home e o vg a a e-egoal e a l epecvely ae oe gave. Th mple ha he oal mbe of ee of e pee a home e a alo vg e ego he a a e-egoal acceble oma [] gve by he m N M ( y oega- a ve. Moeove he oal effecve poplao [] efe by M ( ( a a ay e eff N y a ego C alo oegave a all me. The followg el efe a ppe bo fo he olo poce of he yem (.-(.3. We lze Theoem 3. o eablh h el. Theoem 3.. Sppoe he hypohee of Theoem 3. afe. Le ( m M a a δ. If he Poof: See ([] Lemma 3. M M M y ( B (3. a M M M y ( B fo a.. (3.3 a Remak 3.. Fom Theoem 3. a Theoem 3. we cocle ha a cloe 3 B 3 ; R e he m om ceee a he og R 3 M R wh a B elf-va wh ega o a wo- ball ( cale ewok yamc of hma epemc poce (.-(.3 ha e he flece of hma mobly poce [3]. Tha M M M B ( ; ( : ( a ( 3 S I R y x y B (3.4 R a a pove elf-va e fo yem (.-(.3. We hall eoe M B B (3.5 6

12 D. Wak G. S. Lae 4. Exece a Aympoc Behavo of Deae Fee Elbm I h eco we y he exece a he aympoc behavo of he eae fee elbm ae of he yem (.-(.3. The eae fee elbm obae by olvg he yem of algebac eao obae by eg he f a he ffo pa of he yem of ochac ffeel eao o zeo. I ao we lze he coo ha I R he eve whe hee o eae he poplao. We mmaze he el he followg. Fo ay I( M I( a a I( le D ρ σ ρ γ γ σ δ δ. ρ δ ρ δ > (4. Fhemoe le ( S I R a a a a be he elbm ae of he elaye yem (.-(.3. Oe ca ee ha he eae fee elbm ae gve by E ( S whee B fo a D B σ S fo a (4. D ρ δ B γ fo. D ρ δa The aympoc ably popey of E wll be eablhe by vefyg he coo of he ochac veo of he Lyapov eco meho gve ([3] Theoem.4 [3] a ([3] Theoem 4.4 [3] epecvely. I oe o y he alave popee of (.-(.3 wh epec o he elbm ae ( S f we e he chage of vable ha hf he elbm o he og. Fo h ppoe we e he followg afomao: U S S V I (4.3 W R By employg h afomao yem (.-(.3 afome o he followg fom M δ ρ ( ( U V f e a ( γ σ δ U β ( S U V a a M v ( ( fo a S U Va w l δ σ U V ( ( f e ( ρ δ U Ul (4.4 β ( S U Va a v ( ( fo a S U Va w l δ ( ( ( l M γl U l V l fl e ρl δl Ul β a l Sl I al v ( ( fo a l Sl Ul Val wl 6

13 D. Wak G. S. Lae ρ V ( γ σ δ W β ( S U V a a v ( S U Va w ( fo a ( β l ( ( ( a a ( a ( vl ( Sl Ul Val wl ( fo M a M M Vl σ V ρ δ V a S U V a M v S U V w fo l γl V l ρl δl l Vl βl Sl Ul V al M a (4.5 a M ρ fo l Wl V V f e γ σ δ W a l δ ( ( Wl σ W V V ( fo f e ρ δ W l (4.6 δl γ l W ( ( l Vl l V ( fo l fl e ρl δl W l δ ( ( ( We ae a pove he followg lemma ha wol be efl he poof of he ably el. Lemma 4.. Le : 3 V be a fco efe by M M ( ( a ( ( ( ( T ( a. V x V x V x S S I c I R x U V W c 3 The V ( whee a afe ( ( ( ( (4.7 b x V x a x (4.8 M M c ( m ( ( ( b x U V W M a c a M M ( { } ( ( ( a x max c U V W. M a a (4.9 Poof: See ([] Lemma 4.. Remak 4.. Lemma 4. how ha he Lyapov fco V efe (4.7 pove efe ecece a ally boe (4.8 fco [3] [3]. We ow ae he followg lemma. Lemma 4.. Ame ha he hypohe of Lemma 4. afe. Defe a Lyapov fcoal whee V efe by (4.7 a V V V (4. 6

14 D. Wak G. S. Lae U V ( a V 3 f e V a M M δ ( ( a ( ( θ θ Fhemoe le M ( ( σ M γ a a a a fo a ( γ σ δ ( ρ 3 fo a ( ρ δa ( ρ 3 fo ( ρ δa ( S M v v β b a v b fo γ σ δ ( S M v v β ab v b a fo a ρ δa a a M v v v β b ab ( S a fo. a ρ δa a a (4. (4. (4.3 W a M ( ( M γ σ a a a fo a ( γ σ δ ( ρ fo a ( ρ δa ( ρ fo ( ρ δa (4.4 fo ome ably efe pove mbe a a whee epe o δ a fo all I ( M I( a a I (. Ame ha U V < a W. Thee ex pove mbe φ ψ a ϕ ch ha he ffeel opeao LV aoce wh Io-Doob ype ochac yem (.-(.3 afe he followg ealy 63

15 D. Wak G. S. Lae M ( φ ( ψ ( ϕ ( LV x U V W a ( U ( V ( W φ ψ ϕ φ ( U ψ ( V ϕ ( W. a (4.5 Moeove whee a pove coa c efe by Poof: ( ( LV x cv x (4.6 { φ ψ ϕ } { C } m c max M a M a (4.7 The compao of ffeel opeao [3] [3] apple o he Lyapov fco V (4.7 wh epec o he lage-cale yem of Io-Doob ype ochac ffeel Eao (.-(.3 a follow: whee M M LV ( x LV ( x LV ( x LV ( x (4.8 a a ( ( LV x C ρ V V ρ U U ρ V U ρ U V ρ W W a δ ( ( ( ( δ U V f e V V f e W V ( ( δ f e V W ( ( γ σ δ V U ( γ σ δ ( U ( c ( c ( γ σ δ ( V ( γ σ α δ ( β ( W c S U V V a a ( ( ( a a c v S U V fo a ( { ( LV x c σ V V σ U U σ V U σ U V σ W W a δa ( ( ρ ( ( a a δa U V f e V V f e δ aw V a f e c a c ρ δ a V ( ( ( ( ( ( ( ρ δ ( ( ρ δ ( a U a W av W M v ( ( ρ δ } β ( V U c S U V V a v v a a a ab ba a v b ( ( ( M v v v ab ba v b c v S U V fo a (4.9 (4. 64

16 D. Wak G. S. Lae ( ( { M LV x c γ V V γ U U γ V U γ U V a a δa γ W W au V ( ( f e δa av V ( f ( e aw ( ( ( c a ( c ( ρ δa a ( V ( ρ δa ( U ( ρ αa δa ( W av W ( a a ( ρ δa V U } M v v v c βab S U Vba V a v b M M v v v ( ab ( ( ba a v b δa V f e ( c v S U V fo By g (3.5 a he algebac ealy whee ab a g c ( ( (4. b g c (4. abc a he fco g ch ha g( c > em (4.9-(4. emae a follow: a M v v b M v v b ( c β S U V V v v b b ( ( ( ( c β S g δ g δ V v b S M v v B v c β b ( Vb v b g M v a v b M v a v b ( δ g ( δ ( c β S U V V v v ab ba ( ( ( ( c β S g δ g δ V v ab a a S M v v B v c β ab ( Vb a v b g a a M v a v b M v a v b ( δ g ( δ ( c β S U V V v v ab ba ( ( ( ( c β S g δ g δ V v ab a a S M v v B v c β ab ( Vba a v b g a a ( δ g ( δ. The foeeh (4.3 Fhemoe by g Cachy-Swaz a Höle eale a (4. he xh eveh a eghh em (4.9-(4. ae emae a follow: ( a ( ( a δa A V f e δa ( V ( f ( ( A e ( ( ( { } I M I a I A U V W. (4.4 65

17 D. Wak G. S. Lae Fom (4.9-(4.3 (4.8 epeae age of (3.5 a ealy (4. cople wh ome algebac maplao a mplfcao we have he followg ealy ( σ ( γ LV ( x U ( ( ( ( ( M σ γ c c c a a a M M 4 ( γ σ δ ( a a a ( ( γ σ δ ( M v 4 c β b v b ( c ( γ σ δ ( V M ( σ M ( γ ( S ( γ σ δ ( W a a a M ( ρ 3 ( ρ δa ( U a ( ( ρ ( a a ( ρ δa c ( c 4 M v c βab ( S ( c ( a ρ δa a ( V v b ( ρ ( ρ δa ( W M M ( ρ 3 ( ρ δa ( U a ( c ( ρ ( a a ( ρ δa ( c 4 M v c β ab ( S ( c ( a ρ δa a ( V v b ( ρ ( ρ δa ( W M ( a δ 3 ( ( ( a V f e a M M v S B v v c βb ( vb ( S B ( Vb v b M M v S B v v c βab ( vab ( S B ( Vba a v b M M M v S B v v c βab ( vab ( S B ( Vba a v b (4.5 66

18 whee ( g δa D. Wak G. S. Lae g appopely efe by (4.. The ffeel opeao LV [3] [3] apple o he Lyapov fcoal (4. a (4. lea o he followg ( ( ( a ( a M δ 3 ( ( ( a a LV x LV x V f e We oe ha ( M δ 3 ( ( ( a V. f e a (4.6 δa f e. Fhemoe follow fom (4.6 (4.5 a ome fhe algebac maplao a mplfcao ha { M ( φ ( ψ ( ϕ ( a a LV x U V W ( U ( V ( W φ ψ ϕ ( U ( V ( W } φ ψ ϕ whee fo each I( M I( a a I( (4.3 a (4.4 we efe he coa a φ. (4.7 g (4. ψ a ϕ a follow: c S B c v S B M v v M v v v ba v v v a ba βab v ba ( ab ( ba v b ba v b fo ome pove mbe a I. ( (4.8 c fo all I ( M I( ( γ σ δ ( U ( ( U ( ρ δa ( U fo a φ ρ δa fo a fo a (4.9 ( γ σ δ c ( V E fo a ψ ( a ρ δa a c ( V E fo a ( a ρ δa a c ( V E fo (4.3 a ( γ σ δ ( W ( ( W ( ρ δa ( W fo a ϕ ρ δa fo a fo (4.3 moeove U V W ae gve (4. (4.3 (4.4 a 67

19 D. Wak G. S. Lae E M a a a ( γ σ δ ( ( γ σ δ ( ( 4 3 fo a ( γ σ δ ( ( ( ( ( ρ ρa a ρ δa a c 4 3 fo a ( a ρ δa a ( ( ( ( ( ρ ρ a a ρ δa a c 4 3 fo ( a ρ δa a ( ( ( ( σ γ c c Ue he ampo o U V a ϕ ae pove fo able choce of he coa W clea ha φ ψ a c >. Th h pove he ealy (4.5. Now he valy of (4.6 follow fom (4.5 a (4.8 ha whee LV ( x cv ( x { φ ψ ϕ } { C } m c max M a M a. Th complee he poof. We ow fomally ae he ochac ably heoem fo he eae fee elb.. Le ame ha he hypohee of Lemma 4. ae afe. The he eae fee Theoem 4.. Gve I( M I( a a I( olo E ae aympocally able he lage. Moeove he olo E ae expoelly mea ae able. Poof: Fom he applcao of compao el [3] [3] he poof of ochac aympoc ably follow mmeely. Moeove he eae fee elbm ae expoelly mea ae able. We ow coe he followg coollay o Theoem 4.. Coollay 4.. Le I( M a I( fo all I( M a I(.. Ame ha σ γ δ fo a a ( ρ 3 U (4.3 fo a ( ρ δa ( ρ 3 fo ( ρ δ a 68

20 D. Wak G. S. Lae V ( S M v v β b a v b fo δ ( S M v v β ab a v b fo a ρ δa a M v v β v b ab ( S a a ρ δa a a fo. a (4.33 a δ fo a a ( ρ W (4.34 fo a ( ρ δa ( ρ fo ( ρ δa The elbm ae E ochacally aympocally able pove ha U W a V < fo all I ( M a a I (. Poof: Follow mmeely fom he hypohee of Lemma 4. (leg σ γ he coclo of Theoem 4. a ome algebac maplao. Remak 4... The peee el abo he wo-level lage cale elaye SIR eae yamc moel epe o he elyg yem paamee. I pacla he ffce coo ae algebacally mple compaoally aacve a explc em of he ae paamee. A a el of h eveal ceao ca be ce a exhb paccal coe of aco o cool he eae. Fo mplcy we pee a llao a follow: he coo of σ γ Coollay 4. gfy ha he abay e a k he coex of compameal yem [33] [34] fo all ohe e he e a aegoal acceble oma. Th ceao playe Fge. The co o U a W exhb ha he aveage lfe pa malle ha he o aveage lfe pa of val he a a e-egoal acceble oma of e. Fhemoe he coo V < I( M a I( a U W a a a I( gfy ha he mage of eae hboy pocee fo example he mage of he ecovey poce geae ha he eae amo poce. A fe eale y of he eae yamc he wo cale ewok yamc 69

21 D. Wak G. S. Lae Fge. Show ha ee of e ae pee oly a he home e hey olae evey e fom he e a a egoal acceble oma C(. Hece. Se a k he coex of he compameal yem [33] [34]. The aow epee a apo ewok bewee ay wo e a ego. Fhemoe he oe le a aow cae coeco wh ohe e a ego. ce fo may eal lfe ceao g he peee wo level lage-cale elay SIR eae yamc moel wll appea elewhee.. The ochac elaye epemc moel (.-(.3 a geeal epeeao of feco ace mmy elay a wo-cale ewok poplao eae yamc. The ochac elaye epemc moel wh empoay mmy peo ((.7-(.9 [] a he mecal mlao el (Sec- o 5 [] ae pecl cae of (.-(.3 whe we le he pobably ey fco of he mmy peo f ( δ ( T I( I( 3 whee δ he Dac δ-fco [9]. 5. Coclo 3 The peee wo-cale ewok elaye epemc yamc moel wh vayg mmy peo chaaceze he yamc of a SIR epemc a poplao wh vao cale level ceae by he heeogeee he poplao. Moeove he eae yamc bec o aom evomeal pebao a he eae amo age of he eae. Fhemoe he SIR epemc cofe vayg me empoay ace mmy o ecovee val mmeely afe ecovey. Th wok pove a mahemacal a pobablc algohmc ool o evelop ffee level ee ype eae amo ae he vably he amo poce a well a he be me elay he famewok of he ewok-cec Io-Doob ype yamc eao. I ao he cocep of be elay cae by he ace mmy peo he yamc of hma epemc exploe fo he f me he coex of complex cale-ce ype hma meapoplao. The moel valao el ae evelope a a povely elf va e fo he yamc moel efe. Moeove he globalzao of he pove olo exece obae by applyg a eegy fco meho. I ao g he Lyapov fcoal eche he eale ochac aym- 7

22 D. Wak G. S. Lae poc ably el of he eae fee elb ae alo exhbe h pape. Moeove he yem paamee vale epee hehol vale coollg he ochac aympoc ably of he eae fee elbm ae alo efe. Fhemoe a eco o he ochac aympoc ably el fo a mple eal lfe ceao llae. We oe fhe eal y of he ochac SIR hma epemc yamc moel wh vayg mmy peo fo wo cale ewok moble poplao exhbg eveal eal lfe hma mobly pae wll appea elewhee. We oe ha he eae yamc bec o aom evomeal pebao fom ohe elae b-pocee ch a he mobly ecovey bh a eah pocee. The vably e o he eae amo copoae he epemc yamc moel wll be exee o he vably he mobly ecovey a bh a eah pocee. A fhe eale y of he ocllao of he epemc poce abo he eal eemc elbm of he yamc epemc moel wll alo appea ele whee. I ao a eale y of he heeay feae of he feco age ch a he me-lag o fecoe of expoe val he poplao cely eway a wll alo appea elewhee. Ackowlegeme Th eeach wa ppoe by he Mahemacal Scece Dvo US Amy Reeach Offce Ga No. W9NF---9. Refeece [] Koble S. Mahmo A. Lemo S. a Pay L. (6 The Impac of Globalzao o Ifeco Deae Emegece a Cool: Explog he Coeece a Oppoe Wokhop Smmay Fom o Mcobl Thea. The Naoal Acaeme Pe Wahgo DC. [] 9 HN Fl: Ieaoal Sao Upae. hp:// [3] Rvachev L. a Log I. (985 A Mahemacal Moel fo he Global Spea of Ifleza. Mahemacal Bocece [4] Log I. (988 A Mahemacal Moel fo Pecg he Geogaphc Spea of New Ifeco Age. Mahemacal Bocece [5] Wag W. a Zhao X.-Q. (4 A Epemc Moel a Pachy Evome. Mahemacal Bocece [6] Wag W. a Zhao X.-Q. (5 A Age-Sce Epemc Moel a Pachy Evome. SIAM Joal o Apple Mahemac hp://o.og/.37/s [7] Coe C. Bee J.C. Caell R.S. Impovl D. Kapak L. Po M.D. Toyo A. a Ra S. (9 The Effec of Hma Moveme o he Peece of Veco Boe Deae. Joal of Theoecal Bology [8] Ao J. a Va e Deche P. (3 A Ml-Cy Epemc Moel. Mahemacal Poplao Se hp://o.og/.8/ [9] Ao J. Dav J.R. Haley D. Joa R. Mlle J.M. a va e Deche P. (5 A Ml-Spece Epemc Moel wh Spal Dyamc. Mahemacal 7

23 D. Wak G. S. Lae Mece a Bology 9-4. hp://o.og/.93/mammb/3 [] Saepel L. a Dez K. (995 A Sce Epemc Moel Icopoag Geogaphc Mobly amog Rego. Mahemacal Bocece [] Saepel L. a Heg D.A. (998 Sce Epemc Moel a he Spea of Ifleza he Ceal Caaa Sbacc. Hma Bology [] Saepel L. a Heg D.A. (3 Smlag he Effec of Qaae o he Spea of he Fl Ceal Caaa. Blle of Mahemacal Bology hp://o.og/.6/blm..37 [3] J Y. Wag W. a Xo S. (7 A SIRS Moel wh a Nolea Icece Rae. Chao Solo Facal [4] Ra S. Wag W. a Lev S.A. e al. (6 The Effec of Global Tavel o he Spea of SARS. Mahemacal Bocece a Egeeg hp://o.og/.3934/mbe [5] Rogez D.J. a Toe-Soao L. ( Moel fo Ifeco Deae Spally Heeogeeo Evome. Blle of Mahemacal Bology hp://o.og/.6/blm..3 [6] Beea E. Haa T. Ma W. a Takech Y. ( Global Aympoc Sably of a SIR Epemc Moel wh Dbe Tme Delay. Nolea Aaly [7] Gao S. Teg Z. Neo J.J. a Toe A. (7 Aaly of a SIR Epemc Moel wh Ple Vaccao a Dbe Tme Delay. Joal of Bomece a Boechology 7 Acle ID: hp://o.og/.55/7/6487 [8] Ao J. Cooke K.L. va e Deche P. a Velaco-Heez J. (4 A Epemology Moel Tha Icle a Leaky Vacce wh a Geeal Wag Fco. Dcee a Coo Dyamcal Syem [9] Bly K.B. a Kyychko Y.N. ( Sably a Bfcao a Epemc Moel wh Vayg Immy Peo. Blle of Mahemacal Bology hp://o.og/.7/ y [] Toaoe E. Bccellao S.M. a Veo P. (5 Sably of a Sochac SIR Syem. Joal of Phyc A [] Wak D. a Lae G.S. ( Fameal Popee of a Two-Scale Newok Sochac Hma Epemc Dyamc Moel. Neal Paallel a Scefc Compao [] Wak D. a Lae G.S. ( Global Popee of a Two-Scale Newok Sochac Delaye Hma Epemc Dyamc Moel. Nolea Aaly: Real Wol Applcao [3] Che G. a L T. (9 Sably of Sochac Delaye SIR Moel. Sochac a Dyamc hp://o.og/.4/s [4] Gay A. Geehalgh D. H L. Mao X. a Pa J. ( A Sochac Dffeel Eao SIS Epemc Moel. SIAM Joal o Apple Mahemac hp://o.og/.37/8856x [5] L Q. (9 Sably of SIRS Syem wh Raom Pebao. Phyca A [6] Lahoz A. Oma L. a Koach D. ( Global Aaly of a Deemc a Sochac Nolea SIRS Epemc Moel. Nolea Aaly: Moellg a Cool [7] Dalala N. Geehalgh D. a Mao X. (7 A Sochac Moel of AIDS a Coom Ue. Joal of Mahemacal Aaly a Applcao

24 D. Wak G. S. Lae [8] Dalala N. Geehalgh D. a Mao X. (8 A Sochac Moel fo Ieal HIV Dyamc. Joal of Mahemacal Aaly a Applcao [9] Y J. Jg D. a Sh N. (9 Global Sably of Two-Gop SIR Moel wh Raom Pebao. Joal of Mahemacal Aaly a Applcao [3] Wak D. a Lae G.S. ( A Two-Scale Newok Dyamc Moel fo Hma Mobly Poce. Mahemacal Bocece 9-5. [3] Mao X. (8 Sochac Dffeel Eao a Applcao. Eo Howoo Pblhg L. Sawo. hp://o.og/.533/ [3] Lae G.S. a Lakhmkaham V. (98 Raom Dffeel Ieale. Acaemc Pe New Yok. [33] Lae G.S. (976 Cellla Syem-II. Sably of Campameal Syem. Mahemacal Bocece 3 -. [34] Lae G.S. (976 Cellla Syem-I. Sably of Chemcal Syem. Mahemacal Bocece Sbm o ecomme ex macp o SCIRP a we wll pove be evce fo yo: Accepg pe-bmo e hogh Emal Facebook LkeI Twe ec. A we eleco of oal (clve of 9 bec moe ha oal Povg 4-ho hgh-aly evce Ue-fely ole bmo yem Fa a wf pee-evew yem Effce ypeeg a poofeag pocee Dplay of he el of owloa a v a well a he mbe of ce acle Maxmm emao of yo eeach wok Sbm yo macp a: hp://papebmo.cp.og/ O coac amp@cp.og 73