Mathematical Analysis and Simulation of an Age-Structured Model of Two-Patch for Tuberculosis (TB)

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1 Apple Mthemtcs, 6, 7, ISSN Onlne: ISSN Prnt: Mthemtcl Anlyss n Smulton of n Age-Structure Moel of Two-Ptch for Tuberculoss (TB Bjo Kmb Aboul Wh, Sley Bsso Deprtment of Mthemtcs n Computer Scence, Abou Moumoun Unversty, Nmey, Nger How to cte ths pper: Wh, B.K.A. n Bsso, S. (6 Mthemtcl Anlyss n Smulton of n Age-Structure Moel of Two-Ptch for Tuberculoss (TB. Apple Mthemtcs, 7, Receve: August 8, 6 Accepte: September 7, 6 Publshe: September 3, 6 Copyrght 6 by uthors n Scentfc Reserch Publshng Inc. Ths work s lcense uner the Cretve Commons Attrbuton Interntonl Lcense (CC BY Open Access Abstrct Ths pper stue structure moel by ge of tuberculoss. A populton ve nto two prts ws consere for the stuy. Ech subpopulton s submtte to progrm of vccnton. It ws llowe the mgrton of vccnte people only between the two ptches. After the etermnton of R ( ψ n R, the locl n globl stblty of the sese-free equlbrum ws stue. It showe the exstence of three enemc equlbrum ponts. The theoretcl results were llustrte by numerc smulton. Keywors Age-Structure, Reprouctve Number, Two-Ptch, TB, Stblty, Smulton. Introucton Tuberculoss (TB (short for tubercle bcllus s wespre, nfectous sese cuse by vrous strns of mycobcter, usully Mycobcterum tuberculoss (MTB. Tuberculoss typclly ttcks the lungs, but cn lso ffect other prts of the boy []. To be nfecte bcll must penetrte eep nto the lveol, but the contgousness of the sese s reltvely low n epens on the mmune system of subjects. Invuls t hghest rsk re young chlren, ults, efcent elerly, n people lvng n precrous soco-economc contons, n nursng or whose mmunty s efcent (AIDS, mmunosuppressve therpy... []. Ths s one of the most common ol nfectous seses [3] [4], wth bout two bllon people beng currently nfecte. There re DOI:.436/m September 3, 6

2 B. K. A. Wh, S. Bsso bout nne mllon new cses of nfecton ech yer n two mllon eths per yer ccorng to WHO estmtons [3] [5]. For more nformton, mny uthors hve worke on the epemology of tuberculoss []-[3] [5]-[3]. In mny evelopng countres n generl n sub-shrn Afrc prtculrly, TB s the leng cuse of eth, ccountng for bout two mllon eths n qurter of voble ult eths []. It s well known tht fctors such s the emergence of rug resstnce gnst tuberculoss, the growth of the ncence of HIV n recent yers, s well s other seses fvor the evelopment of Koch bcllus n the boy cll for mprove strteges to control ths ely sese [] [] [4]. Lst My, the Worl Helth Assembly pprove n mbtous strtegy for yers (6-35 to put n en to Worl TB epemc (Worl Dy of fght gnst tuberculoss Mrch 4, 5. In lterture, severl rtcles scusse bout confecton: TB-HIV/AIDS n the most recent s []. Nowys, t s not secret for everyone tht fghtng gnst nfectous seses s lso fght gnst poverty. Humns re trtonlly orgnze nto well-efne socl unts, such s fmles, trbes, vllges, ctes, countres or regons re goo exmples of ptches [] []. For ths stuy, two subpopultons were consere n ech ws subjecte to vccnton progrm. However, only the vccnte nvuls cn mgrte from one ptch to nother. Despte tht we hve neglecte the relpse rte, to vo ny rsk of trete nvuls rectvton, ny mgrton between ptches ws llowe. After provng tht the problem s well efne n t hs unque soluton f the ntl conton s gven, we re ble to clculte the reproucton of numbers R ( ψ n R. We hve estblshe the exstence contons for three enemc equlbrum ponts, n the contons of locl n globl stblty of the equlbrum pont wthout sese. Fnlly, numercl smultons llustrte clncl outcomes. Ths pper s orgnze s follows: Secton ntrouces the two-ptch moel structure n ge to stuy the ynmcs of TB trnsmsson. The exstence of postve n unque solutons s emonstrte n Secton 3. The pont of equlbrum wthout sese, reprouctve numbers R ( ψ n R re efne n the secton 4 wth the locl n globl stblty of the sese-free equlbrum pont. The exstence of three enemc equlbrum ponts s proven n Secton 5. Some numercl smulton results re gven n Secton 6. In Secton 7, we hve scusson, concluson n further work.. Prmeters n Mthemtcl Moel Formulton Two-ptch ge structure moel of tuberculoss ws consere. The moel s to splt the populton nto two subpopultons. The recrutment s only possble n the clss of susceptble n the vccnte nvuls were ble to mgrte between the two subpopultons. Ech subpopulton s ve nto fve clsses bse on ther epemologcl sttus: susceptble, vccnte, ltent, nfectous or trete. We enote these subgroups S ( t,, v ( t,, L ( t,, I ( t, n J ( t, respectvely. The brth µ enote the mortlty rte relte to the rte of the ptch s b ; µ n 883

3 B. K. A. Wh, S. Bsso sese reltve to the ptch n the rte of nturl mortlty. The tme n ge epene of the force of nfecton of the subpopulton s λ ( t, n vccnton rte s ψ ; p (, s the probblty tht n nfectve nvul of ge wll hve c s the contct wth n successfully nfect susceptble nvul of ge, ge-specc per-cpt contct/ctvty rte (ll of these functons re ssume to be contnuous n to be zero beyon some mxmum ge. A frcton φ of newly nfecte nvuls of the sub-populton s ssume to unergo fst progresson rectly to the nfectous clss I. Rtes of mgrton, of susceptble pssge to ltent nfectous stte n tretment re respectvely ρ ; k n r. Rsk reucton rtes of tretment n vccnton re σ n δ respectvely, σ ( φ, δ ( φ, n ths pper =,. The ge-structure moel for the trnsmsson of TB (see Fgure s escrbe by the followng system of prtl fferentl equtons: + S( t, = b N( t, ( t, S( t, t λ + ψ + µ + L( t, = λ( t, ( φ S( t, + σj( t, + δ V( t, ( k+ µ L( t, t + I( t, = kl ( t, ( r+ µ + µ I( t, + φλ ( t, S( t, t + J( t, = ri ( t, ( σλ ( t, + µ J ( t, t + V( t, = ψ S( t, + ρv( t, ( ρ + µ + δλ ( t, V( t, t + S( t, = b N( t, λ( t, + ψ + µ S( t, t + L( t, = λ( t, ( φ S( t, + σj( t, + δ V( t, k + µ L t, t + I( t, = kl ( t, ( r + µ + µ I( t, + φλ ( t, S( t, t + J( t, = ri (, t ( σλ ( t, + µ J( t, t + V( t, = ψ S( t, + ρv( t, ( ρ + µ + δλ ( t, V( t, t wth ntl n bounry contons: n λ = β S t = b N t L t = V t = I t = J t = S = S L = L V = V I I J J (, (, (, (, (, (, (, ; (, ; (, (, = ; (, = ( t, (, I + t, c p,, ssume tht ssume tht N t (, ˆ β ( ( p = g ( 884

4 B. K. A. Wh, S. Bsso Fgure. Flow chrt of the two-ptch moel for tuberculoss sese trnsmsson. (see Greenhlgh, 988 [5] n Detz Schenzle, 985 [6], n (, = (, + (, + (, + (, + (, + S ( t, + L( t, + I ( t, + J ( t, + V( t, N t S t L t I t J t V t By summng equtons of system ( n (, we obtn the followng equtons for the totl populton N( t, : + N t = b N t I t I t t N( t, = b N( t, where b b b (, ( µ (, µ (, µ (, = + ; n re respectvely the mnmum n mxmum ge of procreton n + s the mxmum ge of n nvul, wth + < +. Let ( t, (, S t, L t, s( t, = ; l( t, = N t, N t, I ( t, = N t J t, V( t, j( t, = ; v( t, =. N t, N( t, The system ( cn be normlze s the followng system:. (3 (4 885

5 B. K. A. Wh, S. Bsso + s( t, = b ( t, b ( ( t, ( ( t, s( t, t λ + ψ + µ µ + l( t, = λ( t, ( s( t, j( t, v( t, ( k b ( ( t, ( ( t, l( t, t φ + σ + δ + µ µ + ( t, = ( r+ µ + b µ ( ( t, µ ( ( t, ( t, + φλ ( t, s( t, + kl ( t, t + j( t, = r ( t, ( σλ ( t, + b µ ( ( t, µ ( ( t, j( t, t + v( t, = ( b + ρ+ δλ ( t, µ ( ( t, µ ( ( t, v( t, + ψ s( t, + ρv( t, t + s( t, = b λ( t, + ψ + b µ ( ( t, µ ( ( t, s ( t, t + l( t, = λ( t, ( s( t, j( t, v( t, k b ( ( t, ( ( t, lt, t φ + σ + δ + µ µ + ( t, = ( r+ µ + b µ ( ( t, µ ( ( t, ( t, + φλ ( t, s( t, + kl ( t, t + j( t, = r ( t, ( σλ ( t, + b µ ( ( t, µ ( ( t, j( t, t + v( t, = ( b + ρ+ δλ ( t, µ ( ( t, µ ( ( t, v( t, + ψ s( t, + ρv( t, t wth bounry contons s t, = Λ ; v t, = l t, = t, = j t, = wth Λ +Λ =. The problem s well-poseness, the methoe of proof s the sme use n [8]. 3. Exstence of Postve Solutons In ths secton we wll prove tht the system (5 hs unque postve soluton, n to cheve ths we wll wrte the system (5 n compct form (bstrct Cuchy problem. Conser the Bnch spce X efne by norm 5 = j= (5 X = L, +, enowe wth the ϕ = ϕ (6 where ϕ ( ϕ, ϕ, ϕ, ϕ, ϕ, ϕ, ϕ, ϕ, ϕ, ϕ = X L +. Let n. s the norm of (, {( s, l,, j, v, s, l,, j, v X+ \ s l j, v s l j v } Ω= (7 The stte spce of system (5, where X+ = L+ (, +, n L (, postve cone of L (, +. Let A be lner opertor efne by j + + enotes the ( Aϕ = ( A, A, A, A, A, A, A, A, A, A T. (8 To etermne the components A, we neglect terms of orer two n those whch j 886

6 B. K. A. Wh, S. Bsso re not multple by s, l,, j or v n system (5 (see [7], we obtn: ( t, = ( t, ( r + µ + b ( t, + kl ( t, t j( t, = j( t, + r ( t, b j( t, t v( t, = v( t, ( b + ρ v( t, + ψ s( t, + ρv( t, t s( t, = s( t, ψ + b s( t, t l( t, = l( t, ( k + b l( t, t ( t, = ( t, ( r+ µ + b ( t, + kl ( t, t j( t, = j( t, + r ( t, b j( t, t v( t, = v( t, ( b + ρ v( t, + ψ s( t, + ρv( t,. t After replcng s, l,, j, v, s, l,, j n ϕ, ϕ 3, ϕ 4, ϕ 5, ϕ (, ϕ, ϕ 3, ϕ 4, ϕ 5 n the system ( respectvely, the coorntes of A j re obtne from strght expressons (note tht ech Aj = ( f ( ϕ, f ( ϕ, f ( ϕ3, f ( ϕ4, f ( ϕ5, f ( ϕ, f ( ϕ, f ( ϕ3, f ( ϕ4, f ( ϕ5 wth respect to ϕ re gven by: Wth j v by ϕ, A =, ϕ ( b + k ϕ,,,,,,,, A3 =, kϕ, ϕ 3 ( r + µ + b ϕ3,,,,,,, A4 =,, r ϕ3, ϕ4 b ϕ4,,,,,, A5 = ψ ϕ,,,, ϕ5 ( ρ + b ϕ5,,,,, ρϕ 5 A =,,,,, ϕ ( ψ + b ϕ,,,,. (9 A =,,,,,, ϕ ( b + k ϕ,,, A3 =,,,,,, kϕ, ϕ 3 ( r + µ + b ϕ3,, A4 =,,,,,,, rϕ3, ϕ4 b ϕ4, A5 =,,,, ρϕ 5, ψ ϕ,,,, ϕ5 ( ρ + b ϕ5 T = (,, 3, 4, 5,,, 3, 4, 5 D( A ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ 887

7 B. K. A. Wh, S. Bsso where D( A s the omn gven by: { T j + } = ϕ ϕ [ ϕ = ( Λ Λ D A X \ AC,,,,,,,,,,,,. An AC [, + enotes the set of bsolutely contnuous functons on [ lso efne nonlner opertor F : X X by: ( Fϕ where = b (( Qϕ3 ϕ + ( µ ϕ 3 + µ ϕ3 ϕ (( Q (( ( + ϕ (( Qϕ3 ϕ + ( µ ϕ 3 + µ ϕ3 ϕ3 ( µ ϕ 3 + µ ϕ3 δ (( Qϕ3 ϕ4 ( µ ϕ 3 + µ ϕ3 σ (( Qϕ3 ϕ5 b (( Qϕ3 ϕ + ( µ ϕ 3 + µ ϕ3 ϕ (( Qϕ3 (( ϕ ϕ ( + ϕ (( Qϕ3 ϕ + ( µ ϕ 3 + µ ϕ3 ϕ3 ( µ ϕ 3 + µ ϕ3 δ (( Qϕ3 ϕ4 ( µ ϕ 3 + µ ϕ3 σ (( Qϕ3 ϕ5 ϕ ϕ ϕ σϕ δϕ µ ϕ µ ϕ ϕ Q s boune lner opertor on (, σϕ δϕ µ ϕ µ ϕ ϕ L + gven by β ˆ β, +. We ( Qf = c g + f. ( Let u( t = ( s(., t, l(., t, (., t, j(., t, v(., t, s(., t, l(., t, (., t, j(., t, v(., t thus, we cn rewrte the system (5 s n bstrct Cuchy problem: where u t Au t F u t t u = u = + ( = ( T u s, l,, j, v, s, l,, j, v. Accorng to these results we hve the followng results (see [7]-[9]: Lemm. The opertor F s contnuously Fréchet fferentble on X. Lemm. The opertor A genertes C -semgroup of the boune lner opertors e ta n the spce Ω s postvely nvrnt by e ta. Theorem. For ech u X + there re mxml ntervl of exstence [,t mx n unque contnuous ml soluton u( tu, X +, t [, tmx for ( such tht ta t At ( ξ u( t ue e F( u( ξ ξ = + Proof. The proof of ths theorem cn be foun n [8]-[]. 4. The Dsese-Free Stey Stte 4.. Determnton of the Dsese-Free Equlbrum A stey stte ( s, l,, j, v, s, l,, j, v( of sys- ( 888

8 B. K. A. Wh, S. Bsso tem (5 must stsfy the followng tme-nepenent system of ornry fferentl equtons: s = b c g b s β + ψ + µ µ l = β c g ( φ s + δv + σj ( b + k µ µ l = kl + φβ c g s r + b + µ µ µ j = r ( σβ c g + b µ µ j v = ψ s + ρv ( δβ c g + b µ µ + ρ v s = b c g b s β + ψ + µ µ l = β c g ( φ s δv σj + + ( b + k µ µ l = kl + φβ c g s r+ b + µ µ µ ( j = r ( σβ c g + b µ µ j v = ψ s + ρv δβ c g + b µ µ + ρ v + ˆ = β wth ntl vlue contons s = Λ ; l = = j = v =. Therefore, we obtn the sese-free stey stte s b v = Λ s l = = j = (3 ( b( τ + ψ ( τ τ b τ + ψ τ τ = Λ e + e. (4 ; 4.. Clculton of the Reproucton Numbers R( ψ - R To stuy the stblty of the sese-free stey stte, we enote the perturbtons of system by s( t, s( t, s l( t, = l( t, + l ( t, = ( t, + j( t, = j( t, + j v( t, = v( t, + v = +. (5 889

9 B. K. A. Wh, S. Bsso The perturbtons stsfy the followng equtons: + s( t, = ( t ( c g ( t, ( t, s t γ β µ µ ( b + ψ s( t, + l( t, = ( b + k l( t, t + γ( t β c g ( φ s + δv + ( t, = kl ( t, + φβ ( c g γ( t s t ( r+ µ + b ( t, + j( t, = r ( t, b j( t, t + v( t, = ψ s( t, + ρv( t, ( ρ+ b v( t, t + s t = t c g t t s t ( b + ψ s( t, + l ( t, = ( b + k l( t, t + γ( t β c g ( φ s + δv + ( t, = kl ( t, + φβ ( c g γ( t s t ( δβ ( c g γ ( t µ ( t, µ ( t, v (, γ β µ (, µ (, ( r + µ + b ( t, + j( t, = r ( t, b j( t, t + v t = s t + v t + b v t t γ ( ˆ t = + β (, t (, ψ (, ρ (, ( ρ (, ( δβ c g γ ( t µ ( t, µ ( t, v (6 wth bounry contons: s t, = l t, = t, = j t, = v t, =, we conser the exponentl solutons of system (6 of the form: The system (6 becomes: λt λt (, = e λt = = s t, = s e ; l t, = l e v t v t, e ; j t, j e λt λt. (7 89

10 B. K. A. Wh, S. Bsso s = ( b + ψ + λ s c g s β µ µ l = ( b + k+ λ l +β c g ( φ s + δv = φβ c g + k l ( r + µ + b + λ j = r ( b + λ j v = ( ρ+ b + λ v ( δβ c g µ µ v + ψ s + ρv s = ( b + ψ + λ s c g s β µ µ l = ( b + k+ λ l +β c g ( s v φ + δ = φβ c g + kl ( r+ µ + b + λ j = r ( b + λ j v = ( ρ + b + λ v δβ c g µ µ v + ψ s + ρv + ˆ = β (8 wth bounry contons: s = l = = j = v = Let = ( +. (9 Nψ φ s δ v From Equton (8, we obtn: ( e b τ + k + λ τ l = β c g Nψ ( ( b τ + r + λ + µ τ = kl +φβ c g. ( e Hence, by Equtons (( n ( fter chngng orer of ntegrton, we obtn: ( b r ( r k τ + µ τ + + λ τ µ τ τ α c g s kn ψ = + + e β φ e α. ( Injectng ( n the expresson of, n vng both ses the expresson by (snce, we get the chrcterstc equton: ( = e + e + b τ µ ˆ τ r λ τ r k c g s kn µ τ τ α β β φ ψ + α. (3 Denote the rght-hn se of Equton (3 by G ( λ.e.: ( G c g s kn b τ µ ˆ τ r λ τ r k e e µ τ τ α λ + β β φ ψ = + + α. (4 We efne the net reprouctve number s ( ψ R =,.e. G 89

11 B. K. A. Wh, S. Bsso ( R = + b τ µ ˆ τ r τ r k e c g s kn e µ τ τ α ψ + β + + β φ ψ + α. (5 We cn obtn n expresson for R n smlr wy s the ervton of R ( ψ by conserng Equton ( wthout vccnton;.e., by ssumng tht neglectng the equton of vccnte. It cn be shown tht ψ n R =R whch s clle the bsc reprouctve number (when purely susceptble populton s consere (see [8]. ( + b τ µ ˆ τ r τ r k e c g k ( e µ τ τ α β + + β φ φ + α (6 R = Λ + Let ( ψ ( ψ R = mx R n R = mx R Locl Stblty of the Dsese-Free Equlbrum Theorem. The nfecton-free stey-stte (5 s loclly symptotclly stble (l..s. f R ( ψ < n unstble f ( ψ Proof. Notcng tht R >. ( λ ( λ ( λ G < ; lm G = ; lm G = +. λ + λ We know tht Equton (3 hs unque negtve rel soluton λ f, n only f, G <, hence, R ( ψ < (Also, Equton (3 hs unque postve (zero rel soluton f R ( ψ > ( R ( ψ =. To show tht λ s the omnnt rel prt of roots of G ( λ, we let λ = x + y be n rbtrry complex soluton to Equton (3. Note tht nctng tht Re ( λ λ R ( ψ <, n unstble f ( ψ ( λ = G = G x+ y G x,. It follows tht the nfecton-free stey stte s l..s. f R >. In ths corollry, we hve the three cses of the unstblty of the sese free equlbrum. Corollry. whenever R ( ψ < n R symptotclly stble n the frst ptch n unstble n the secon. whenever R ( ψ > n R ptch n loclly symptotclly stble n the secon. 3 whenever R ( ψ > n R ptches. ψ >, the sese free s loclly ψ <, the sese free s unstble n the frst ψ >, the sese free s unstble n the two 4.4. Globl Stblty of the Dsese-Free Equlbrum Snce µ n (, t re boune, there exsts postve constnt c = Corollry. Assume tht r µ ( τ R tht stsfes µ τ t + ττ, τ R (. c + k, then we hve 89

12 B. K. A. Wh, S. Bsso ( r + µ ( τ + b( τ τ e φs( t +, + k( σj( t +, + δv( t +, ( k+ b( τ τ ( r k µ τ τ ξ + ( φ s( t +, e e ξ ( b( τ + µ ( τ + r τ ( µ ( τ + r k τ ξ Λ e φ + k( φ e ξ. Theorem 3. The sese-free equlbrum of system (5 s globlly symptotclly stble f R < n R c < ln. R Proof. The proof consst to show tht ( t, ; j ( t, ; l ( t, ; s( t, s n v( t, Λ s, when t +. Integrtng system (5 long chrcterstc lnes we get (,, e b τ = µ τ t ττ τ k τ = + + β λ + (7 l t c g t ( t, = e σ j t +, + δv t +, + φ s t +,, < t ( b( τ µ ( τ ( t τ, τ r µ ( τ τ ξ = (8 φβ ξ c ξ g ξ λ t + ξ + kl t + ξ, ξ ξ, < t Injectng (7 n (8, n chngng orer of ntegrton, we obtn: t, = µ τ + ττ τ, e t = β c g λ t + ( r + µ ( τ + b( τ τ e φs( t +, + kσj( t +, + δv( t +, ( k+ b( τ τ ( r µ ( τ k τ + ξ + ( φ s( t +, e e ξ Injectng (9 n λ ( t, n chngng orer of ntegrton, we obtn: +, ˆ t = µ τ + ττ τ e λ t = β β c g λ t + ( r + µ ( τ + b( τ τ e φs( t +, + kσj( t +, + δv( t +, ( k+ b( τ τ ( r µ ( τ k τ + ξ + ( φ s( t +, e e ξ By usng corollry, nequlty ( n Ftou s lemm, we hve Snce e R c R c t R λ ( t lm λ e lm sup. t + t + R <, lm supt + λ( t = ( t j( t l( t lm, = lm t +, = lm, = t + t + lm s( t, = s, lm v( t, = Λ s. t + t +. (9. (3 Corollry 3. The sese-free equlbrum s globlly symptotclly n: 893

13 B. K. A. Wh, S. Bsso the frst sub-populton f the secon sub-populton f µ τ ττ, τ + < ln. R R < n ( t µ τ ττ, τ + < ln. R R < n ( t For ths sese cn spper wthout ny form of nterventon, ccorng to these results we must ensure tht there s no new nfecte n the nfectous rte oes not rech certn spre. 5. Exstence of n Enemc Stte There exsts three enemc stey stte of system (5 whenever R ( ψ >. 5.. The Frst Bounry Enemc Equlbrum Theorem 4. A bounry enemc equlbrum of the form ( E = s, l,, j, v, s,,,, v whenever R ( ψ > n R ( ψ <. Ths mens tht the sese s enemc n the frst sub-populton n es out n the secon sub-populton. Proof. The metho commonly use to fn n enemc stey stte for ge-structure moels conssts of obtnng explct expressons for tme nepenent soluton of system (5 E ( = s, l,, j, v, s,,,, v stsfes the followng equtons: s = b c g b s β + ψ + µ l = β c g ( φ s + δv + σ j ( b + k µ l ( = k l + φβ c g s r + b + µ µ j = r ( σβ c g + b µ j (3 v = ψ s + ρv ( δβ c g + b µ v s = b ψ + b µ s( v = ψ s + ρ v ( b µ v = + ˆ β wth the ntl contons: Let Integrtng system (3, we obtn: s = Λ ; = l = v = j =. (, ( φ δ σ h = s + v + j. (3 894

14 B. K. A. Wh, S. Bsso β( τ c( τ g( τ µ ( τ + b + ψ s = Λe ( φ β( τ c( τ g( τ µ ( τ + b + ψ + b e (33 ( r µ ( τ + b( τ + µ ( τ τ = kl + φβ c g s e (34 ( c g b j e + τ = µ τ τ τ (35 ( c g b v s v e τ τ β τ + ψ ρ τ = + µ τ τ τ (36 ( b k l τ µ τ τ τ = + c g s h(, β (37 ( ψ τ + b τ µ τ τ ( ψ τ µ τ τ τ s = Λ e + b e (38 ( b( τ µ ( τ ( τ τ v = ψ s + ρv e. (39 By njectng (37 n (34, we obtn: ( β( τ µ τ τ + r + µ τ τ = β c g e. (4 ( r+ µ ( τ k τ ξ φs + kh (, e ξ Injectng (4 n the expresson of, n vng by (snce we obtn: + ( ˆ β τ µ τ τ + r + µ τ τ = β β c g e. (4 ( r+ µ ( τ k τ ξ φs + kh (, e ξ Let H, the functon efne by: + ( ˆ β τ µ τ τ + r + µ τ τ H = β β c g e. (4 ( r+ µ ( τ k τ ξ φs + kh (, e ξ, N ψ net reprouctve number s gven by.e. Snce h ( =.e. when s s v v H =, = n =, so the = R ( ψ ( + b τ r + µ τ + τ ( r k µ τ τ α + ψ R ψ = ˆ β e β c g φ s + kn e α. We now see tht n enemc stey stte exsts f Equton (4 hs postve soluton. Snce H = R ( ψ, hence H ( >. We know tht s + l + + v + j = Λ <. Hence <. (43 895

15 B. K. A. Wh, S. Bsso Snce >, from (4 n (43 we obtn: + ( ˆ β τ µ τ τ + r + µ τ τ H = β β c g e ( r+ µ ( τ k τ ξ φs + kh, e ξ + < ˆ β = β. + H s con- ˆ on In prtculr, for H β + <, but H >. Snce tnous functon of, we conclue tht =, hs postve soluton ; β +. Ths soluton my not be unque snce H my not be monotone ( ( pens on h(, whch s efne mplctly. It follows tht when = β +, we hve ( H H e- R ψ >, there exsts n enemc stey stte strbuton whch s gven by the unque soluton of Equton (4 corresponng to ˆ. 5.. The Secon Bounry Enemc Equlbrum Theorem 5. A bounry enemc equlbrum of the form E ( = s,,, v, s, l,, j, v whenever R ( ψ < n R ( ψ >. Ths mens tht the sese s es out n the frst subpopulton n s enemc n the secon sub-populton. Proof. (Ies of proof E ( = s,,, v, s, l,, j, v stsfes the followng equtons: s = b b s ψ + µ v = ψ s + ρ v ( b µ v s = b c g b s β + ψ + µ l = β c g ( φ s + δv + σ j ( b + k µ l (44 = kl + φβ c g s r+ b + µ µ ( j = r ( σβ c g + b µ j v = ψ s + ρv ( δβ c g + b µ v + ˆ = β wth the ntl contons: s = Λ ; = l = v = j =. (45 Let (, ( φ δ σ h = s + v + j. (46 Integrtng system (5, we obtn: ( ψτ + b τ µ τ τ ( ψ τ µ τ τ τ s = Λ e + b e (47 896

16 B. K. A. Wh, S. Bsso s ( b( τ µ ( τ ( τ τ = ψ + ρ e (48 v s v = Λ e + β( τ c( τ g( τ µ ( τ + b + ψ ( φ β( τ c( τ g( τ µ ( τ ( τ + b( τ + ψ( τ τ b e ( r µ ( τ + b( τ + µ ( τ τ = kl + φβ c g s e (5 (49 ( c g b e σ β τ τ τ + τ = µ τ τ τ (5 j ( c g b e δ τ τ β τ + ψ ρ τ = + µ τ τ τ (5 v s v ( b τ µ τ τ k τ = + β ( ( ( ( (. (53 l e c g s h, Hence, by the smlr metho usng n theorem 4, we obtn the result The Interor Enemc Equlbrum Theorem 6. An nteror enemc equlbrum of the form E = s, l,, j, v, s, l,, j, v ( whenever R ( ψ > n R ψ >, whch correspons to cse when the sese perssts n the two sub-popultons. Proof. E = ( s, l,, j, v, s, l,, j, v stsfes the followng equtons: s = b c g b s β + ψ + µ µ l = β c g ( φ s + δv + σ j ( b + k µ µ l( = kl + φβ c g s r + b + µ µ µ j = r ( σβ c g + b µ µ j v = ψ s + ρv ( δβ c g + b µ µ + ρ v s = b β c g ψ b µ µ s + + l = β c g ( φ s + δv + σj b + k µ µ l kl φβ c g s r b µ µ µ = j = r ( σβ c g + b µ µ j v = ψ s + ρv ( δβ c g + b µ µ + ρ v + ˆ = β (54 897

17 B. K. A. Wh, S. Bsso wth the ntl contons: s = Λ ; l = = j = v = (55 s = Λ e β ( τ c( τ g( τ + b( τ µ ( τ ( τ µ ( τ ( τ τ + β( τ c( τ g( τ + b( τ µ ( τ ( τ µ ( τ ( τ τ b e. (56 Let (, ( φ δ σ h = s + v + j (57 ( b( τ µ ( τ τ µ τ τ τ = β, e (58 l c g h ( b( τ + r+ µ ( τ µ ( τ ( τ µ ( τ ( τ τ = e + φβ (59 kl c g s ( c g b e σ β τ τ τ + τ µ τ τ = µ τ τ (6 j r ( c g ( b ψ ρ e = + δ β τ τ τ + τ µ τ τ µ τ τ τ (6 v s v ( ( c g b = δ β τ τ τ + τ µ τ τ µ τ τ τ + v ψ s ρ v e. (6 By njectng (58 n (59, we obtn: ( b( τ µ ( τ ( τ µ ( τ ( τ r+ µ ( τ τ = β e c g ( r+ µ ( τ k τ ξ φs +, e ξ. kh By njectng (63 n the expresson of obtn: Let, n vng by (snce + ( ˆ b τ µ τ τ µ τ τ r+ µ τ τ β β c g = e ( r+ µ ( τ k τ ξ φs + kh, e ξ. H, the functon efne by: + ( ˆ b τ µ τ τ µ τ τ r+ µ τ τ = β e β H c g ( r+ µ ( τ k τ ξ φs + kh, e ξ. Snce h (, =.e. when, s s n v v N ψ net reprouctve number s gven by H ( ψ = R,.e. (63 we (64 (65 = = =, so the + ( b( τ + µ ( τ + r τ ( µ ( τ + r k τ ˆ α R ψ = β e e. β c g φs kn ψ α + We now see tht n enemc stey stte exsts f Equton (64 hs postve soluton. Snce H = R ( ψ, hence H >. We know tht 898

18 B. K. A. Wh, S. Bsso s + l + + v + j = Λ <. Hence Snce < (66 >, from (65 n (66 we obtn: + ( ˆ β τ µ τ τ µ τ τ r+ µ τ τ H β e β c g ( r+ µ ( τ k τ ξ φs + kh (, e ξ = ˆ + < β = β. + In prtculr, for = β +, we hve ( H >. Snce H s contnous functon of, we conclue tht H ( =, hs postve soluton ˆ on ; β +. Ths soluton my not be unque snce H my not be monotone ( H ( epens on h(, whch s efne mplctly. It follows tht when R ( ψ >, there exsts n enemc stey stte strbuton whch s gven by the unque soluton of Equton (64 corresponng to Smulton In ths secton, when R ( ψ > n R ˆ H β + <, but ψ > we wll evlute the mpct of BCG vccne n the brth rte of the populton n the ynmcs of spre of TB. Assumng tht ll prmeters re the sme n both ptches except the vccne rte, we observe n ncrese n the number of nfecte f the vccnton rte ecreses (Fgure. Also tkng the sme prmeters except brth rtes, we see n ncrese number of nfecte f the rte ncreses (Fgure 3. Fgure. Evoluton of the number of ltents nvuls wth ψ = n.46 ψ =

19 B. K. A. Wh, S. Bsso Fgure 3. Evoluton of the number of ltents nvuls wth b ( =.46 n b =.38. Fgure 4. Evoluton of the number of nfectous nvuls when: R ( ψ =.435 ( ψ =.85 n ψ =.785. When R ( ψ =.57 n R ψ =.57 n R ψ =.435 ( ψ =.85 n ψ =.785, we hve the evoluton of the number of nfectous nvuls (Fgure Dscusson, Concluson n Future Work In ths pper, n ge structure moel of two-ptch for tuberculoss ws nlyze n 9

20 B. K. A. Wh, S. Bsso scusse. Ech sub-populton s subjecte to vccnton progrm. Aprt from ge; the vccnte comprtment, we ntrouce s clss of trete n the moel propose by Tew J. Jules n [] n llowe the mgrton of vccnte populton. The sme result ws foun f the most susceptble mgrte too. Although some stues hve shown n neffectveness of BCG n the preventon of tuberculoss [], our work emonstrte the contrbuton of BCG n the process of erctng TB. The negtve mpct of the ncrese n the brth rte ws shown. If we neglect the mortlty eth rte lnke to the sese, we obtn the only usul conton of globl stblty to the sese free equlbrum.e. R <. It remns for us mny chllenges such s the enemc equlbrum ponts of ths moel n the one of [8] to el wth. For future work, n orer to stuy the rel mpct of the tuberculoss mgrton n the ynmc of the expnson of the sese, we wll use ths moel n uthorze the mgrton of ll nvuls (.e. susceptble, nfecte, nfectous, vccnte n trete. Acknowlegements We thnk the Etor n the referee for ther comments. We woul lke to thnk Numercl Anlyss stuent group for ther vluble comments n the uthors whose works hve been use n ths rtcle. We lso thnk the mnstry of Hgher Eucton of Reserch n Innovton who knly supporte the costs of the publcton. References [] Rohet, E., Wrtun, S. n Anryt, A. (5 Stblty Anlyss Moel of Spreng n Controllng of Tuberculoss. Apple Mthemtcl Scences, 9, [] Slv, C.J. n Torres, D.F.M. (5 A TB-HIV/AIDS Confecton Moel n Optml Control Tretment. Dscrete n Contnuous Dynmcl Systems, 35, [3] Echeng, B.B. n Lebeev, K.A. (5 On Mthemtcl Moelng of the Effect of B- Therpeutc Tretment of Tuberculoss Epemc. Journl of Moern Mthemtcs n Sttstcs, 9, -7. [4] Roult, D. (6 Epéme et mles nfecteuse ns l hstore. [5] Worl Helth Orgnzton (5 Globl Tuberculoss Control: Survellnce, Plnnng, Fnncng. WHO/HTM/TB/5, Geneve, 349. [6] Ptel, A., Sskn, F.V., Abrhms, E. n Prker, J. (99 Cse-Control Evluton of School-Age BCG Vccnton Progrmme n Subtropcl Austrl. Bulletn of the Worl Helth Orgnzton, 69, [7] Zwerlng, A., Shresth, S. n Dv, W.D. (5 Mthemtcl Moelng n Tuberculoss Avnces n Dgnostcs n Novel Therpes. Avnces n Mecne, 5, Artcle ID: [8] Chvez, C.C. n Feng, Z. (998 Globl Stblty of n Age-Structure Moel for TB n Its Applctons to Optml Vccnton Strteges. Mthemtcl Boscences, 5, [9] Chvez, C.C., Hethcotte, H.W., Anresen, V., Levn, S.A. n Lu, W.M. (989 Epem- 9

21 B. K. A. Wh, S. Bsso ologcl Moels wth Age Structure, Proportonte Mxng, n Cross-Immunty. Journl of Mthemtcl Bology, 7, [] Agusto, F.B., Cook, J., Shelton, P.D. n Wckers, M.G. (5 Mthemtcl Moel of MDR-TB n XDR-TB wth Isolton n Lost to Follow-Up. Abstrct n Apple Anlyss, 5, Artcle ID: [] Tew, J.J., Bowong, S. n Mewol, B. ( Mthemtcl Anlyss of Two-Ptch Moel for the Dynmcl Trnsmsson of Tuberculoss. Apple Mthemtcl Moellng, 36, [] Tew, J.J., Bowong, S., Mewol, B. n Kurths, J. ( Two-Ptch Trnsmsson of Tuberculoss. Mthemtcl Populton Stues, 8, [3] Zhng, J. n Feng, G. (5 Globl Stblty for Tuberculoss Moel wth Isolton n Incomplete Tretment. Computton n Apple Mthemtcs, 34, [4] Mouleu, D.P., Bowong, S., Tew, J.J. n Emvuu, Y. ( Anlyss of the Impct of Dbetes on the Dynmcl Trnsmsson of Tuberculoss. Mthemtcl Moellng of Nturl Phenomen, 7, [5] Greenhlgh, D. (988 Threshol n Stblty Results for n Epemc Moel wth n Age- Structure Meetng Rte. IMA Journl of Mthemtcs Apple n Mecne n Bology, 5, [6] Detz, K. n Schenzle, D. (985 Proportonte Mxng Moels for Age-Depenent Infecton Trnsmsson. Journl of Mthemtcl Bology,, [7] Zou, L., Run, S. n Zhng, W. ( An Age-Struvture Moel for the Trnsmsson Dynmcs of Heptts B. SIAM Journl on Apple Mthemtcs, 7, [8] Webb, G.F. (8 Populton Moels Structure by Age, Sze, n Sptl Poston. In: Mgl, P. n Run, S., Es., Structure Populton Moels n Bology n Epemology, Sprnger-Verlg, Berln, [9] Inb, H. (6 Mthemtcl Anlyss of n Age-Structure SIR Epemc Moel wth Vertcl Trnsmsson. Dscrete n Contnuous Dynmcl Systems Seres B, 6, [] Djjou, R.D., Tew, J.J. n Bowong, S. (4 Anlyss of n Age-Structure SIL Moel. Wth Demogrphcs Process n Vertcl Trnsmsson. ARIMA Journl, 7, 3-5. [] Sylers, A.A. n Whtt, D.D. (994 Bcterl Pthogeness: A Moleculr Approch. ASM, Wshngton DC. 9

22 Submt or recommen next mnuscrpt to SCIRP n we wll prove best servce for you: Acceptng pre-submsson nqures through Eml, Fcebook, LnkeIn, Twtter, etc. A we selecton of journls (nclusve of 9 subjects, more thn journls Provng 4-hour hgh-qulty servce User-frenly onlne submsson system Fr n swft peer-revew system Effcent typesettng n proofreng proceure Dsply of the result of ownlos n vsts, s well s the number of cte rtcles Mxmum ssemnton of your reserch work Submt your mnuscrpt t: Or contct m@scrp.org