Modeling the Dynamics of Tuberculosis Transmission in Children and Adults

Transcription

1 Journl of Mthemtis nd Sttistis 8 (): 9-0, 0 ISSN Siene Publitions Modeling the Dynmis of Tuberulosis Trnsmission in Children nd Adults F. Nybdz nd M. Kgosimore Deprtment of Mthemtil Sienes, Fulty of Siene, University of Stellenbosh, 9 Jonkershoek Rod, Stellenbosh, 7600, South Afri Deprtment of Bsi Sienes, Botswn College of Agriulture, P. Bg 007, Gborone, Botswn Abstrt: Problem sttement: The fight Aginst Tuberulosis (TB) hs minly foused on the dult popultion beuse hildren were pereived to pose very low risk in TB trnsmission. This ssumption ignores the potentil risk hildren hd s reservoirs of ltent infetions from whih future ses evolve when they beome dults. It ws therefore importnt to investigte the dynmis of TB tking into onsidertion, hildren. Approh: We formulted omprtmentl model for TB with two ge lsses, hildren nd dults. Qulittive nlysis of the model ws done to investigte the stbility of the model equilibri in terms of the model reprodution number R 0 Numeril simultions were lso done to investigte the role plyed by some key epidemiologil prmeters in the dynmis of the disese. Results: The model hd two equilibri: The disese free equilibrium whih ws globlly stble for R 0 < nd the endemi equilibrium whih ws lolly symptotilly stble for R 0 >, for R 0 ner. The study showed inresed ltent infetions in the dult popultion s result of inresed ltently infeted hildren who mture to dulthood with ltent infetions. Conlusion/Reommenions: Progression to tive TB mong dults is epidemiologilly signifint nd interventions should fous on the dult popultion. Anti-tuberulosis, tretment of dults is ruil in ontrolling the epidemi nd should interventions be proposed, they should trget progression to tive TB for those ltently infeted. The fight ginst TB should lso tke into onsidertion tuberulosis mong hildren. Key words: Tuberulosis (TB), peditri tuberulosis, stbility nlysis, reprodution number, equilibri, ltently infeted INTRODUCTION Primry tuberulosis is the fountinhed of tuberulosis disese nd when quired during hildhood, it my develop into serious tuberulosis disese within short period of time or remin ltent during hildhood only to be retivted in dulthood (Wood et l., 00). Children usully progress to tive disese within months of primry infetion beuse hildren hve high risk of progression to disese following infetion (Brent et l., 008; Shromi et l., 008). There hs been limited interest in hildhood tuberulosis beuse more thn 95% of hildren with tive disese re sputum smer negtive nd therefore not infetious. Infeted hildren represent pool from whih lrge proportion of future ses of dult TB will rise nd thus perpetuting the TB epidemi (Bloh 00). The burden of hildhood tuberulosis is ler indition of the TB severity in the dult popultion (Miris et l., 006; Wrren et l., 00). The ontribution of hildren to TB selod is not well doumented in poor-resoure settings with high disese burden, but reserh hs shown tht in high disese burdened res, hildren less thn 3 yers of ge, ontribute bout 3.7% of the totl TB selod in prtiulr ommunity in South Afri (Murry nd Slomon, 998). The result ompred well with tht determined for low inome ountries for hildren less thn 5 yers of ge, whih ws 5% (Shf et l., 00). Children usully develop TB s diret omplition of the initil infetion. Children with household or ommunity exposure to TB re highly likely to be infeted with the disese. Children quire TB infetion from n dult who is in their immedite nd Snider, 985; Brent et l., 008; Vynnyky et l., environment (Shf et l., 006; Wood et l., 00). Corresponding Author: Nybdz, F., Deprtment of Mthemtil Sienes, Fulty of Siene, University of Stellenbosh, 9 Jonkershoek Rod, Stellenbosh, 7600, South Afri 9

2 Lk of ontgiousness is one of the resons tht publi helth progrms on TB prevention hve exluded hildren. They re regrded s the end of the trnsmission hin nd therefore pose no thret to the evolution of the TB epidemi in the popultion. This view ould hve ontributed signifintly to erly hildhood TB. Mthemtil models for the dynmis of TB hve been extensively developed nd hve been used in designing ontrol progrms nd s preditive tools (Bhunu et l., 008; Blower et l., 995; Blower nd Dley, 00; Cstillo-Chvez nd Feng, 997; Cstillo-Chvez nd Song, 00; Gomes et l., 00; Strke, 003). Models tht onsider the potentil impt of TB vines were studied in (Miris et l., 005). Reently, Bhunu et l. (008) onsidered model tht studied the impt of tretment nd hemoprophylxis in ombting TB. Most of the studies reviewed onentrted on dult TB nd ignored TB in hildren. It is upon this bkground tht we propose simple deterministi model for TB tht inorportes hildren nd dults, with the gol of investigting the dynmis of intertion of peditri nd dult TB epidemis. We develop mthemtil model with the objet of quntifying the underlying ftors tht drive the epidemi in hildren. This study ims to hllenge publi helth poliies regrding hildhood TB. We endevor to nswer the following questions: Wht is the role of inresed dult TB ontrol on the growth of hildhood TB? Wht is the proportion of dults tht need to be trgeted in order to ontrol TB in hildren? To the best of our knowledge, there is limited number of theoretil models of hildhood TB. MATERIALS AND METHODS A two lss ge-strutured model: The model onsiders onstnt popultion N, subdivided into three different subgroups: suseptible (S) ltently infeted (L) nd infeted with tive TB (I). Individuls move from one subgroup to the other s their sttus with respet to the disese hnges. We gin divide the popultion into two ge lsses, hildren nd dults. Individuls below 5 yers belong to the hildren s ge lss while those bove 5 yers belong to the dult s ge-lss. These ge groups re seprted primrily to distinguish between peditri TB nd dult TB. While the ext ge demrtion is somewht rbitrry, the min gol here is to divide the popultion into two resonble ge lsses. Dt from medil institutions lso shows different phenomen in TB development for the two lsses. J. Mth. & Stt., 8 (): 9-0, 0 We use the following nottion for individuls in the two ge lsses: we use the subsript nd to represent hildren nd dults respetively, so tht S, LC, I nd SA, L, I respetively represent suseptible, ltently infeted nd infets with tive TB in hildren nd dults. The model is built on the following ssumptions. All subgroups re subjeted the nturl mortlity rteµ. The effetive ontt rte of infeted dults is ssumed to be higher with suseptible dults thn hildren ( > ). An effetive ontt is one tht is suffiient to result in n infetion if the ontted individul hs never been infeted (Wlls nd Shingdi, 00). Suseptible hildren quire TB I infetion from dults with tive TB t rte B = β ; N while suseptible dults quire TB infetion from I other dults with tive TB t rte B = β. β is the N effetive ontt rte. The prmeter ounts for the inresed infetiousness mong the dults due to severl ftors relting to mixing, bteri propgtion nd environmentl settings. A proportion p (q) of hildren (dults) develops tive TB in the first yer fter primry infetion nd the reminder develops ltent TB. Progression to tive TB ours t rte r (r ) for hildren (dults). Reovery, nturlly or with hemotherpy, results in n individul reverting bk to the ltent lss t rte σ (σ ) for hildren (dults). The rte t whih hildren join the dult lsses is given by f. However, infeted hildren re ssumed not to grdute into dulthood being infetive. We knowledge here tht the onsidertion of being onstnt is merely for mthemtil onveniene. We shll ssume tht f is less thn or equl to the birth rte. Figure depits the intertion of the two gelsses. The model is thus desribed by the following set of ordinry differentil Eq. : ds I = µ N βs ( µ + f ) S N dl I = ( p)β S + σi ( µ + f ) L, N di I = pβ S L ( µ + σ ) I, N ds I = fs βs µ S N dl I = fl + ( q)β S + σi ( µ ) L, N di I = qβ S L ( µ + σ ) I N By setting: 30 ()

3 J. Mth. & Stt., 8 (): 9-0, 0 Fig. : Model digrm showing the trnsmission of TB between disese stges nd ge lsses x = S,x = L,x I S L I 3 =,x =,x5 = nd x6 = N N N N N N We n rewrite the system () s Eq. : dx =µ β xx6 ( µ+ f ) x, dx = ( p) β xx6 +σx3 ( µ + f ) x, dx3 dx = p β xx6 x ( µ+ σ ) x 3, = fx βx x6 µ x, dx5 = fx + ( q)β xx6 +σx6 ( µ ) x 5, dx6 = q β xx6 x5 ( µ+ σ ) x 6. () The model hs initil onditions given by x i (0) 0, =,,, 6. Biologil onsidertions entil tht we study systemtilly () in the following region: Suppose there exists first time t suh tht x (t ) = dx 0 nd the derivtive < 0 t t = t nd x i (t) >0, (i =,,6) for 0<t <t. The first eqution of the system () dx gives (t ) = µ > 0, whih is ontrdition? Thus x (t) will remin positive for ll t. Similr steps n be followed for the remining vribles. We onlude tht ll solutions of the system () re positive for ll time. Our popultion is lso bounded in G thus ll solutions strting with G will remin in the G. Thus G is positively invrint nd ttrting nd our model is thus epidemiologilly well posed. Determintion of equilibri: We nlyze the system () by finding the model equilibri nd rrying out their stbility nlysis. At the stedy stte we set the right hnd side of equtions of the system () to zero nd determine the stte vribles. From the first five equtions of system (), we hve: G = {(x,x,x,x,x,x ) R x + x + x + x + x + x } Proposition: Solutions of system () re positive for ll t 0 nd re bounded. The region G is thus positively invrint nd ll solutions in G remin in G for ll time. Proof: For the given initil onditions, we prove by ontrdition tht if x (t), x (t), x 3 (t), x (t), x 5 (t) nd x 6 (t) re solutions of system (.), then they re positive (see lso Bhunu et l., 008; Ziv et l., 00). µ x =,x = ( p) β x x + σ x, 6 3 ( µ + f ) + β x 6 ( µ + f ) fx x p x x r x,x, 3 = β 6 + = ( µ + σ ) µ + βx6 x = fx + {( q) β x + σ }x. 5 6 ( µ ) Solving for x nd x 3 simultneously, we hve Eq. : x = ω xx6 (3) 3

4 J. Mth. & Stt., 8 (): 9-0, 0 x = ω x x () 3 6 Where: And: ω = [( p) µ + σ ] β. ( µ + σ )( µ + f ) + µ r ω = p( µ + f ) β. ( µ + σ )( µ + f ) + µ r Substituting for x in x, we hve Eq. 5: 6 6 µ f x =. (5) ( µ + β x )( µ + f + β x ) Substituting Eq. 3 into the expression for x 5, we hve: x = f x {( q) x }x ( r ) ω + β + σ. (6) 6 µ + Substituting Eq. 6 into the lst eqution of system () we hve the solutions, x6 = 0 nd the solution of the qudrti polynomil: P(x ) = x + x +, (7) Where: And: =µ ( µ+ f)( µ+σ )[ R ], 0 0 = ( µ + ( µ+ f) )( µ+σ ) r f ω ], = ( µ+σ ) β R 0 = R + R (8) µ f E 0 =,0,0,,0,0 µ + f µ + f The endemi equilibrium is thus determined from the solutions of (7). As regrds the sign of in (7), we hve the following proposition. Corollry: If R 0 < then >0. Proof: Consider: Where: K = ( µ + σ )( µ + ( µ + f ) ) r fω, [ ] = ( µ + σ )( µ + ( µ + f ) ) K r f ( p) µ+σ = β µ+σ µ + ( µ+ f) ( µ+σ )( µ+ f) +µ r We note tht if R 0 < then K sine: f f <. µ + ( µ + f ) µ + f This implies tht >0. We thus hve the following theorem on the existene of the endemi equilibrium. Theorem : If R 0 < then P (x 6 ) hs no positive solution. However, if R 0 >, then P (x 6 ) hs one positive solution. We therefore onlude tht system () hs unique endemi equilibrium point E whenever R 0 >. Proof: The solution of (7) is given by: With: β f ( µ q ) R =, µ ( µ + f )( µ + σ ) R. [( p) µ + σ ] β r f = ( µ + σ )( µ + f ) + µ r ( µ + f )( µ + σ ) 0 ± x6 =. Here >0 nd 0 >0 if R 0 < nd 0 < 0 if R 0 >nd from (8), if R 0 <0, then we hve two negtive solutions nd for R 0 >, we hve one positive solution for ll. Here, R represents the ontribution of dults to seondry infetions while R is the ontribution of RESULTS AND DISCUSSION hildren to seondry infetion. The threshold prmeter R 0 defines the men number of seondry Qulittive results: Anlysis of the Model ses generted by introduing single infeted Reprodution Number: We onsider first, the effets of individul into wholly suseptible popultion in whih inresing the reovery rtes σ nd σ. We note tht s hildren re involved in the dynmis of the disese. σ, R 0 0 the implitions of this result is tht The se x6 = 0 gives the disese free equilibrium emphsis should be pled on preventing dult point: individuls from progressing to tive TB. A high rte 3

5 J. Mth. & Stt., 8 (): 9-0, 0 of reovery of individuls with tive TB plys ruil role in deresing the pool of infeted individuls. However, s σ, R R σ, where: 0 0 β f ( µ q ) βf r R σ 0 = +. µ ( µ + f )( µ + σ ) µ + f ( µ + f )( µ + σ ) This result shows tht the reovery rte of hildren does not ply ruil role in the dynmi of TB s ompred to tht of dults. The ontour plot in Fig. shows the reltionship between R 0, σ nd σ. Figure represents set of R 0 ontours for whih, for hosen set of prmeter vlues, hnges in the vlues of σ nd σ would give. We note tht s the vlue of σ inreses, the vlue of R 0 dereses. In ft, this result is similr to the one proven nlytilly, tht s σ, R 0 0. Reovery in this se my be due to tretment or nturl reovery. Sine the fight ginst TB hs been restrited to, vintion, identifying infetious ses nd treting them, one n onsider this result to imply tht the fight ginst TB must be direted mostly to the dult popultion sine they re the soure of infetion. It is ler from Fig. tht inresing the reovery rte σ for hildren does not ply ny signifint role in the trnsmission dynmis of the disese. We now investigte the role of the progression rtes r nd r. As r 0, R R σ, where: 0 0 Fig. : Shows the reltionship between the reovery rtes nd the reprodution number R 0 for the following set of prmeter vlues; β = 0.0, µ = 0.07, p = q = 0., σ = 0.8,σ = 0.6, = 6, = 0, f = 0.0 R βqf =. ( µ + f )( µ + σ ) r 0 r Note tht R 0 < R 0. Thus redution in the progression rtes leds to redution in the number of seondry infetions generted by n infeted individul. We lso note tht s r 0, we hve where: R R, r 0 0 r β f( µ q ) β f[( p) µ+σ] r R0 = +. µ ( µ+ f)( µ+σ ) ( µ+ f)( µ+σ ) ( µ+ f)( µ+σ ) Tht R > R. Redution in the progression of r 0 0 ltently infeted hildren to tive peditri TB leds to inresed seondry infetions in the dult popultion. The implitions of this senrio re tht there re inresed ltent infetions in the dult popultion s result of inresed ltently infeted hildren who mture to dulthood with ltent infetions. Fig. 3: Shows the reltionship between the progression rtes nd the reprodution number R0 for the following prmeter vlues; β = 0.005,p = 0.,q0.,r = 0.08,r = 0.5, = 6, = 0, f = 0.0. Children beome reservoir of ltent infetions from whih retivtion is highly likely to our when they beome dults. Figure 3 illustrtes the theoretil results relting vritions of progression r nd r to seondry infetions. Globl stbility of E 0 : Theorem : The disese free equilibrium point E 0 is globlly symptotilly stble in G whenever R 0 <. Proof: We onsider the following Lypunov funtion: Where: 33 V = α x + α x3 + α 3x5 + α x 6

6 J. Mth. & Stt., 8 (): 9-0, 0 α = fr ( µ + σ ), α = fr σ, α = r [( µ + f )( µ + σ ) + µ r ] nd α = ( µ )[( µ + f )( µ + σ ) + µ r ] The derivtive of V with respet to time is given by: dv dx dx3 dx5 dx 6 = α + α + α 3 + α, [( µ+ f)( µ +σ ) +µ r ][ µ ( µ ) +µσ ](R )x, 0 for R < Suppose β is the bifurtion prmeter. When R 0 = we hve: ( µ + f )[ µ ( µ + σ )][( µ + σ )( µ + f ) + µ r ] β = f ( µ q )[( µ + σ )( µ + f ) + µ r ] + µ r f [( p) µ + σ ] The Jobin mtrix of the system (7) t E 0 for β = β is given by: It follows tht dv 0 for R 0 <, sine the model prmeters re ssumed to be positive. The derivtive dv = 0, if nd only if R 0 =. Hene V is Lypunov funtion on G. The lrgest ompt set tht is dv invrint in {(x, x, x 3, x, x 5, x 6) G = 0} is the singleton {E 0 }. It follows from the lslle s invrine Priniple (Lietmn nd Blower, 000), tht every solution of () with initil onditions in G pprohes E 0 s t whenever R 0 <. Lol stbility of the endemi equilibrium: The size nd nture of the mtrix resulting from stndrd lineriztion of the system () mkes the determintion of eigenvlues nd their nture diffiult nd tedious exerise. We resort to the enter mnifold theory desribed in (Feng et l., 00) nd used in (Song et l., 00) to determine the stbility of the endemi equilibrium point. We note tht our system is of the form: Where: X = (x, x, x 3, x, x 5, x 6 ) T dx = F(X) nd F (f, f, f 3, f, f 5, f 6 ) T where (.) T denotes mtrix trnspose. System () beomes Eq. 7: dx = f = µ βx x6 ( µ + f ) x dx = f = ( p)β xx6 + σx3 ( µ + f ) x, dx3 = f3 = pβ xx6 x ( µ + σ ) x 3, dx = f = fx βxx6 µ x, dx5 = f5 = fx + ( q)β x x6 + σx6 ( µ ) x 5, dx6 = f6 = qβ x x6 x5 ( µ + σ ) x6 (9) 3 J β A B = C D where, A,B,C nd D re 3 3 mtries given by: ( µ + f ) 0 0 A = 0 ( µ + f ) σ 0 r ( µ + σ ) 0 0 β x f 0 0 B = 0 0 ( p)β x,c = 0 f pβ x µ 0 β x D = 0 ( µ ) ( q)β x + σ 0 r qβ x ( µ + σ ) J β hs zero simple eigenvlue nd the orresponding right eigenvetor ssoited with this simple eigenvlues is given by y = (y, y, y 3, y, y 5, y 6 ) T where: µ ( µ + σ )[( µ + σ )( µ + f ) + µ r ] y = f ( µ + f ) ( µ q )[( µ + σ )( µ + f ) + µ r ] + µ r [( p) µ + σ ] { } µ ( µ + σ )[( p) µ + σ] y = f ( q r )[( )( f ) r ] r [( p) ] { µ + µ + σ µ + + µ + µ µ + σ } µ [p( µ + f ) ][( µ + σ )( µ + f ) + µ r ] y3 = f ( q r )[( )( f ) r ] r [( p) ] y { µ + µ + σ µ + + µ + µ µ + σ } ( µ + ( µ + f ))( µ + σ )[( µ + σ )( µ + f ) + µ r ] ( f ) ( q r )[( )( f ) r ] r [( p) ] = µ + µ + µ + σ µ + + µ + µ µ + σ { } µ ( σ )[( p) µ + σ ] + [( µ + σ )( µ + f ) + µ r ][ µ ( q) + σ ] y =, y = 5 6 { ( µ q )[( µ + σ )( µ + f ) + µ r ] + µ r [( p) µ + σ] } The left eigenvetor of the trnspose of J β tht is, the left eigenvetor of J β is given by: Z = (z,z,z,z,z,z ) T 3 5 6

7 Where: z = 0,z = η fr ( µ + σ ),z = ηfr σ,z = 0, 3 z = η r [( µ + σ )( µ + f) +µ r ],z = η ( µ )[( µ + σ )( µ + f) +µ r ] 5 6 And η>0 is hosen so tht the ondition y z= is stisfied. Following Theorem. in (Feng et l., 00), we thus ompute nd b to determine the stbility of E round R 0 = 0, where: f = z y y b = z y f k k kij= k i j ki = k i x i x j x i β J. Mth. & Stt., 8 (): 9-0, 0 (0) And β the bifurtion prmeter. The non-zero prtil derivtives of F t the disese free equilibrium point re given by: f f f f x x x x x x x x = ( p) β, = p β, = ( q) β, = qβ To determine we substitute the bove expressions of the prtil derivtives into Eq. 0. We thus hve: = y y β [( p)z + pz ] + y y β [( q)z + qz ] < Sine y <0 nd y <0, the right hnd side of is lwys negtive. For the sign of b the following re the non-zero prtil derivtives of F: f f f f x β x β x β x β = ( p)x, = p x, = ( q) x, = q x An evlution of b shows tht b is lwys greter thn 0. We thus hve the following results bsed on Theorem. in (Feng et l., 00). Theorem 3: The endemi equilibrium point E is lolly symptotilly stble for R 0 > but with R 0 ner. Numeril results: We now give disussion of the numeril solutions of systems (), the prmeter vlues nd interprettions of the ses tht rise thereof. The dynmis of the system () re studied numerilly using the fourth order Runge-Kutt numeril sheme in MATLAB. Prmeter vlues: Some of the model prmeters were deided upon bsed on the following ssumptions: humn beings re between 0 nd 80 yers. We use the nturl mortlity rte in the rnge 0.05 µ The rte t whih hildren join the dult lss f is ssumed to be less thn or equl to the birth rte. We try to ommoe, the ft tht not every hild survives to dulthood. So we set f µ Suseptible hildren nd dults re infeted with probbility β upon ontt. Contt ptterns tht n result in infetion n be rndom, ssoitive, ge-speifi nd sex-speifi nd so on, but in this study we ssume rndom mixing. This mens tht every suseptible is eqully likely to be infeted by n infetive should ontt our While the probbility of infetion is tken to be the sme for hildren nd dults, we ssume tht dults hve inresed hnes of mixing. We thus onsider different ontt rtes Estimtes hve shown tht 0% (proportions p, q) of TB infetion progression fst to tive TB (Jung et l., 00). The risk for young hildren to develop tive TB if the infetion is not treted is up to 3% in hildren less thn yer old, pproximtely % for hildren between nd 5 yers of ge nd muh lower for those between 6 nd 3 yers (Shf et l., 003). The proportion of hildren tht develop tive TB fst n thus be ssumed resonbly to be higher thn 0% Ative TB hs lwys been tken s ontinued development of the primry infetion first quired or due to endogenous retivtion or exogenous reinfetion with seond or the sme strin. Seprtion nd quntifition of these mehnisms espeilly (endogenous retivtion nd exogenous re-infetion) requires tehnology tht n differentite between strins (Zhng et l., 007). We rudely lump the two proesses (Jung et l., 00) in this study to represent progression to tive TB, denoted by r nd r Reovery, my be due to tretment or nturl reovery In whih se individuls revert bk to the exposed lss t rtes σ nd σ. The rte t whih individuls reover usully depends on the immune sttus of the individul, types of drugs, geneti nd soioeonomi ftors. We knowledge tht quntifition of these rtes of reovery is diffiult s evidened by vritions in the prmeter vlues in reserh publitions The estimted prmeter vlues used re given in Tble. Numeril plots: Figure shows the hnges in the Sine the birth nd deth rtes re tken to be the proportion of the suseptible popultions for hildren sme, we ssume tht the verge lifespns of nd dults, with inresing time. 35

8 J. Mth. & Stt., 8 (): 9-0, 0 () (b) () Fig. : Shows the hnges in the proportion for eh popultion with time. The vlue of the model reprodution number is in this se 36

9 J. Mth. & Stt., 8 (): 9-0, 0 () (b) () Fig. 5: Shows the hnges in the proportion for eh popultion with time. The vlue of the model reprodution number is.6 37

10 Tble : Tble of prmeters nd numeril vlues Prmeter Vlue(s) Referene µ (Song et l., 00) β (0, ) Vrible p, q (0, ) Vrible σ, σ (0.5, ) (Bhunu et l., 008; LSlle, 976; Nelson nd Wells, 00) r 0.03 (Nelson nd Wells, 00), (, 00) Estimted F f µ Estimted r (0.5,) (LSlle, 976) The suseptible popultion of dults initilly flls s the infetive mong dults inrese, but lter inreses to stedy stte vlue over time. The suseptible hildren inrese to stedy stte. This is resonble sine in most popultion strutures, the proportion of hildren is higher thn tht of dults. In Fig. b the ltently infeted popultion for the two subgroups dereses over time to zero. In Fig. the grphs for the two sub-popultions show similr trend in whih their proportions fll over time to zero We observe tht infetion in the dult popultion ler lter thn tht of hildren minly beuse of the ltent reservoir of infetion from hildren. Overll, the Fig. -, represent the disese free equilibrium point E 0 over longer period of time thn the one presented in the simultions. In Fig. 5 shows the hnges in the proportion of the suseptible popultions for hildren nd dults, with inresing time. Both sub-popultions derese to stedy stte over time nd gin we hve the proportion of hildren being higher thn tht of dults s the epidemi evolves. In Fig. 5b, the ltently infeted popultion for the two subgroups initilly dereses nd then rises to stedy stte over time. Figure 5 shows the grphs for the proportions of infetives. The dult infets inresed rpidly nd settled in n endemi stedy stte over time. This senrio represents the endemi stedy stte E, whih we dedued nlytilly to be symptotilly stble for R 0 >. CONCLUSION J. Mth. & Stt., 8 (): 9-0, 0 the geing funtion f, ws hosen so tht it does not exeed the birth rte of the popultion bsed on the ssumption tht the individuls tht grdute into dulthood survive with some probbility less thn or equl to one. We knowledge tht it is diffiult to estimte nd n ge-strutured model would hve eliminted suh diffiulty. The lumping up of progression to tive TB to inlude reinfetion nd retivtion nd reovery to inlude both nturl reovery nd reovery due to the tretment represent signifint short omings to the model. The greement between the model output nd epidemiologil hs been restrited to very brod TB epidemiologil trends. Despite these indequies, this study provides some unique insights in the wy progression nd reovery prmeters influene the dynmis of the disese. Any form of intervention implemented should influene these prmeters nd quntify their hnges nd influene on the reprodution number. Our nlysis shows tht progression to tive TB mong dults is epidemiologilly signifint nd interventions should fous on the dult popultion. This perhps explins why TB mong hildren hs not been mjor subjet for reserh. Similr trends n be observed in the reovery rtes r nd r. The reltionship between the progression rtes to tive TB for hildren nd dults nd the model reprodution number drws out importnt informtion tht n help in prophylti tretment suh s the use of Isonizid in preventing progression to tive TB. Figure 3, lso shows tht hnge in the progression rte for dults, results in signifint hnges in the model reprodution number. Attempts to use hemoprophylxis for tive TB prevention should be trgeted to dults. The globl stbility of the model shows tht if R 0 < then every solution tends to the disese free equilibrium nd the disese lers from the popultion. We showed tht there exists unique endemi equilibrium point whenever R 0 >. If the TB is present in popultion then, s long s R 0 >, it will lwys persist. In onlusion, despite ll our results pointing out the need to fous interventions on dults, peditri TB forms n importnt inditor of how the TB epidemi is evolving in dults. Infeted hildren t s reservoir of ltent infetions tht n retivte into tive TB when they beome dults. One of the most importnt ftors to onsider with regrds to peditri TB is tht, it is possible to lerly distinguish between disese progression stges, exposure, infetion nd disese. This is prtiulrly importnt to publi helth poliy formultion, The study presented in this study is very simple model showing the dynmis of TB in hildren nd dults in the bsene of ny intervention. Mny deterministi models of TB hve been developed but few, to the best of our knowledge, hve been developed this wy without using the ge strutures. We presented model in whih there re no interventions. A more relisti pproh would be to inlude the urrent intervention strtegies. This forms our reserh tht is in progress. Although some of the model ssumptions re relisti, mny of them over simplifies the nturl evolution of TB. In prtiulr, designing interventions nd modeling in generl. 38

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