Drug Resistants Impact on Tuberculosis Transmission

Transcription

1 Dug Resistants Impact on Tubeculosis Tansmission SILVIA MARTORANO RAIMUNDO Univesidade de S ao Paulo Faculdade de Medicina Rua Teodoo Sampaio 115, S ao Paulo, SP BRASIL silviam@dim.fm.usp.b EZIO VENTURINO Univesità di Toino Dipatimento di Matematica via Calo Albeto 1, 1123 Toino ITALY ezio.ventuino@unito.it Abstact: In epidemiology, measues fo pophilaxis of infectious diseases ae taken often using mathematical methods and statistical tools to evaluate possible futue scenaios of the evolution of tansmissible diseases. Hee we pesent two models fo TB tansitions among diffeent stages of the disease. We analyze them assuming to have a lage basin of susceptible individuals available. The models account fo immigation and demogaphic effects. A flow of infected membes into the population is assumed. Pat of it is made by a specified faction is dug-sensitive latent individuals, while the othe pat consists of dug-esistant latent individuals. Ou aim is the desciption and analysis of all the possible ways the infected individuals move. In paticula, fom the classes of latent to infectious, and possibly back upon successful teatment, o towad acute stages of the disease fo dug esistant cases due to impope, incomplete o ineffective healing measues. Some conditions fo the eadication of the disease ae extacted fom analytical consideations and simulation esults and might be useful fo epidemiological implementations. Key Wods: Dug esistant, tansmissible disease, tubeculosis. 1 Intoduction Tubeculosis (TB) is a bacteial disease that is caused by Mycobacteium tubeculosis (MTB). Although many people believe that TB is a scouge of the past, the disease is one of the most pevalent infections among humans and contibutes consideably to illness and death aound the wold [29]. It is estimated that appoximately one-thid of the global population is infected with MTB and that seven to eight million new cases of tubeculosis occu each yea [2, 3, 25]. Annual tubeculosis motality is between two and thee million people, making this disease the most common infectious cause of death in the wold. Accoding to the Wold Health Oganization, moe people died fom TB in 1995 than in any othe yea in histoy. It has been estimated that, at cuent ates, up to one-half of a billion people will suffe fom TB in the next 5 yeas. Humans ae the natual host fo this pathogen, which is tansmitted by espiatoy oute. Unlike most othe infectious diseases, the couse of MTB infection is unusual fo bacteial pathogens, because it involves a delay between infection and disease that is extemely vaiable, anging fom a few weeks to a lifetime. Thus, TB is typically descibed as a slow Wok pefomed with the suppot of the WWS pogam of the Univesity of Toino disease. Following pimay infection, only a small popotion of individuals develops the TB disease, most people emain in the latent stage of the disease. The infection can thus show no clinical symptoms fo the lifetime of the host. Infected individuals develop active TB as a consequence of endogenous eactivation of latent bacilli o exogenous einfection. Howeve, the factos elated to the disease pogession ae not still well descibed and a development of active TB aises the question whethe that epesents a eactivation of the initial infection (endogenous eactivation) o a new infection with MTB (exogenous einfection, sometimes called supe-infection o einfection) [21, 27]. The elative impotance of these two pathways to the development of active disease has significant implications fo teatment and contol stategies, most notably in deciding whethe latently infected and teated individuals ae at isk of einfection [12]. The Wold Health Oganization cuently ecommends, fo all new cases of TB, standadized shotcouse chemotheapy based on a egimen of fou fistline dugs taken fo 6-8 months [18]. It is the mainstay of tubeculosis contol woldwide. Howeve, despite apid pogess in dug development, micobial infections in geneal ae becoming inceasingly difficult to teat as a esult of the emegence of dugesistant stains. Resistance to TB dug has been epoted since the ealy days of the intoduction of ISSN:

2 chemotheapy. In spite of its global magnitude, the poblem has not yet been adequately addessed. In the past decade it was evealed that some MTB stains ae able to disseminate moe quickly than othes and, also, that multidug-esistant tubeculosis (MDR-TB) may be one of the most impotant theats to TB contol pogams. The Wold Health Oganization estimates that up to 5 million pesons woldwide may be infected with dug-esistant stains of TB. Dug-esistant tubeculosis is poduced by the selection of esistant stains in patients who fail to complete chemotheapy with the coect combination of dugs. MDR-TB is a fom of TB that is esistant to at least two dugs used fo the teatment of TB, isoniazid and ifampicin. It is oiginally developed when a case of dug-susceptible TB is impopely o incompletely teated. Dug-esistant tubeculosis occus when dug-esistant bacilli outgow dugsusceptible bacilli. The dug-esistant oganisms ae poduced by andom mutations in the bacteial chomosomes. These mutations occu spontaneously in wild-type pansensitive stains even befoe the stains come in contact with an antitubeculosis dug [8]. Studies have found that dug-esistance ates incease with a polongued duation of the pevious teatments [7]. Although pesons peviously teated fo dugsensitive tubeculosis can be einfected with dugesistant stains, as shown in [6, 11, 17, 24], in most instances dug esistance develops because of inadequate o eatic theapy. In the eal wold, the teatment of patients with active-tb equies a multiple dug egimen; teatment is highly effective (with a 95% cue ate) if the patient habos dug-sensitive oganisms and is compliant with the teatment egimen [4]. Ou goal is to look at some of these featues fo TB pathogenesis. We popose a mathematical model to assess the tansmission dynamics of both dugsensitive and dug-esistant TB consideing slow TB, i.e., cases that esult fom endogenous eactivation of dug-sensitive and dug-esistant latent infections. The pape is oganized as follows. In the next Section we pesent the model fomulation, in Section 3 we analyze the basic model, its equilibia and thei stability. Section 4 contains anothe vesion of the basic model, with newly infected ecuited via a diffeent mechanism. Conditions on the system paametes ensuing the endemicity of the disease ae identified. Also, a conjectue involving the deteminant of the Jacobian of the model matix ensuing disease eadication is highlighted. Some numeical simulations and a final discussion of the esults conclude the pape. 2 Model Fomulation In contast to othe models of simila natue, fo othe types of diseases [4, 5, 19, 2, 23, 26], ou mathematical model monitos the tempoal dynamics of latent individuals (infected but unable to infect othes) and infectious individuals (active-tb infectious, i.e., infectious subjects which ae also able to infect othes). The eason fo consideing these simplified models is to ignoe the infection mechanism and concentate only on the tansition mechanisms among classes of infected individuals. In ode to accomplish this task, we assume that new infected individuals ae intoduced by exogenous tems into ou system. Common to ou whole discussion is the implicit assumption of an infinite pool of susceptibles, which ae not explicitly modelled and thus do not appea in the ecuitment tems fo new latent individuals. This implicitly lineaizes the equations in the system, since the elevant mechanism we model consists then in the tansitions to and fom diffeent compatments of infection o disease. Unde ou assumptions, the infection cannot be eliminated fom the population because of the constant inflow of newly infected individuals; in ode to eadicate the disease it would be necessay to isolate the faction of incoming latent individuals. In paticula, we will thus make two distinct assumptions. Fistly, that the newly infected, named latent individuals, ae ecuited via constant immigation ates (1). Secondly, that they ae geneated via ates that ae popotional to the numbe of people that ae aleady actively infected by the disease and can theefoe spead it, i.e., the infectious individuals (1). Since the model assesses the dug-esistant and dug-sensitive tubeculosis cases, two subclasses of latent and infectious individuals ae equied to build it. Hence, the total diseased population ( ), is divided into fou classes, namely,, the dug-sensitive latent individuals;, the dug-esistant latent individuals;, the dug-sensitive infectious TB individuals, and, the dug-esistant infectious TB individuals. Hee, both active-tb and case-tb mean active TB infectious case, and the subscipts and stand fo dug-sensitive and dug-esistant types. The dynamic of dug-esistant TB is due to two independent but inteacting pocesses: (i) endogenous eactivation of latent dug-esistant and, (ii) convesion of sensitive cases (wild-type pansensitive) to dug-esistant cases duing the teatment (acquied esistance). The pogession ates fom latent TB to active- TB ae assumed to be popotional to the latent-tb cases, i.e., they ae given by and, such that TB cases ( o ) aising as a esult of en- ISSN:

3 > dogenous eactivation (slow pogession) of the pimay complex with the sensitive o esistant stain. Thus, we define,, as the popotion of dug-esistant latent TB individuals who develop dug-sensitive TB. Consequently! is the popotion of dug-esistant latent TB individuals who develop dug-esistant TB. Although we assume that all the dug-sensitive TB individuals ( " ) ae teated with a multiple dug egimen (isoniazid and ifampin), the teatment has opposing effects at the population-level: teatment cues dug-sensitive cases, but an acquied dugesistance quickly emeges among patients who eceive ineffective o inappopiate teatment egimes o which ae noncompliant with a multiple dug teatment egimen. In the model effective teatment of dug-sensitive TB individuals # occus at a ate $, whee $&%(' is the aveage peiod of teatment. Acquied dug esistance can aise diectly, fo whateve eason (with pobability )*, duing the teatment of a dug-sensitive case. Consequently, ) epesents the pobability that teatment failue occus due to the development of dug esistance, +,)-.. Thus, the model includes the possibility that the teatment of a dug-sensitive case can esult in one of thee outcomes; (a) teatment can cue the patients (cases ae emoved fom the classes at a ate equal to /)*1$, and ente the 2 classes), (b) teatment failue can occu, and the patient acquies dug-esistant TB (cases ae emoved fom the classes at a ate equal to )3$, and ente the classes), o can be emoved fom the " classes at a ate equal to >, and ente the classes, (b) teatment can cue the patient, and cases ae emoved fom the " classes at a ate equal to, and ente the classes, o (c) teatment failue occus (teated cases emain in the classes). Theefoe, dug-esistant TB cases ae unteatable (and/o unteated) if 4 4@ ; dug-esistant cases ae teated, with equal effectiveness if A4 A4B ; and dug-esistant cases ae only patially effectively teated if C8 D8E and C8 8E. The model incopoates ecuitment (F ) and natual death (G ), as well as disease-elated death (H ), so that the total population size may vay in time. We define I, as the popotion DJIK of latent dugesistant individuals who ente into the / class and LMIN as the popotion of latent dug-sensitive individuals who ente into class. Based on the above assumptions and definitions we pesent next the analysis of the models. 3 Constant Immigation Model Fist of all the tansfe diagam of the basic model with constant immigation is shown in Figue 1. This model is govened by the following non- (c) teatment failue can occu, yet the patient emains infected with dug-sensitive TB (teated cases emain in the classes). Theefoe, if )546 dug-sensitive cases ae teated with effectiveness; if )+47, the theapeutic methods of teatment ae inefficient; and fo 98:);8< dugsensitive cases ae only patially effectively teated. It is also assumed that dug-esistant TB cases # = can be teated at ate >, whee > %(' is the aveage peiod of teatment, but teatment efficacy is educed. The elative teatment efficacy of dugesistant cases (in compaasion with teatment of dug-sensitive cases) is specified by the paamete ( fo dug-sensitive, and fo dug-esistant cases). Thus, the model includes the possibility that teatment of a dug-esistant case can esult in one of thee outcomes: (a) teatment can cue the patient, and a few cases Figue 1: Model with constant immigation: tansfe diagam ISSN:

4 RPPPPPPPPPPPPPPP } RPPPPPPPPPPPPPPPPPPP 4 ' d Ž l l \ l homogeneous linea system of diffeential equations: PPPPPPPPPPPPPPPO Q SUTWV SYX 4 Z[INWF-\]Z)*1$3 \^>,_G`\abc SYdfe V SYX 4 g "\9! g h,ig`\ahd\k$jk SUTWl SYX 4 IfF-\a> N 9,iG`\,bc 2 SYdfebl SmX 4 ZM(Y 2 `\5)3$3 n,o G`\aH \^>C +\ 1pq s (1) whee 4t \u \. v \u is the total population and the dynamics of the whole population is expessed by Sxw SYX 4 FDyG Hz# v^ {\ f} (2) Thus the total population is constant only if F9[G ~HJ# "\K s4jc} When thee is no initial infection in the population, - v, equation (2) becomes SUw SYX 4 FDyG } (3) This equation has a single stable equilibium 4<, fo any initial value of. Thus, in absence of the disease the total population will stabilize at the value fom any initial conditions and all individuals will be latent. In fact, by setting H54:, then equation (3) implies iƒm, when ƒ. So, when Hz4 and F94JG, we have Sxw SYX 4 c and the total population is a constant. Howeve, if the disease has been peviously established in a population the total population is given by Sxw SYX 4 FDyG ~H ˆ \ 2,F;yG This equation has a single equilibium 4 5 # \J v, fo any initial value of. Thus, once iƒm is less than, it emains so fo all futue times, and when iƒy is geate than, iƒy declines until eventually iƒm fo all sufficiently lage ƒ. In the next section, the analysis of the model (1) is pefomed. We identify the steady state of the system (1) to analyse the dependence on the model paametes of the equilibium and its stability. 3.1 Endemic equilibium To find the equilibium point, let us set the deivatives of (1) to zeo. This esults in a unique non-tivial steady state Š 4 ˆ v m with components given by whee PPPPPPPPPPPPPPPPPPPO Q 4 3!Ž Œ \u!ž Ž dfe 4 3!Ž 'Y%j Œ 3!Ž š 1 œ dfe 4!Ž ŽŸ ž š 3 Œ W 3!Ž m 'Y%j 1 š ž š 4 9 ig+\ fš) $ \I (1-p)[$ (1-q)G \nig`\ f ig+\ah=kp_ * >sig`\ b \5> 2iG`\h!f \nig`\ah= ig+\ fx Ÿ 4 > G igm\ah+\5$* \" ªo G`\ H+\K$ZY ~)œšp_ \CiG`\aHD\K$j ig`\ b o GM\aH+\ &> kp«4 > NiG`\,b o G ig+\ HM\$jN\, ig`\ah=kp \^> G($o G`\ œ \ ZM!š) p \nig`\ f ig`\ah= ig+\ œ!u \nig`\ah= ig+\ f o ig+\ahm\k$j ig+\k$j \ )*1$ pª c} (4) Clealy,. Note that the vanishing of implies the infeasibility of Š, since 8< and 8. Fo feasibility we must impose also v ±, and. These give the lowe bound ²C³3 µ ' o {Y M!jI!Ffp«o IfF {ožig`\ahm\k$j /D! (\hb)3$3p p_ ' o IfF {ožig`\ahm\k$j /D! (\hb)3$3p p 3.2 Dynamical tajectoies We will now pefom the stability analysis fo the nonhomogeneous linea system (1). Letting ¹B4, in matix fom it can be ewit- Y I(šF=mcI!F=mj ten as º» SY¼ SmX 4 ½» \K¹= (5) ISSN:

5 º ÆÅ Æ Æ Ç Í ² whee» iƒy +¾], and ¹À¾]^ is a constant vecto, while ½ is a Á`ÂDÁ constant matix. In ode to make a eal phenomena compatible to its mathematical desciption we will study the behavio of tajectoies in the neighbohood of the tivial solution of the system (1). Geneally, the stability of some solutions of the system» 4» \a¹ is econducted to the stability of a tivial solution, i.e., investigating the natue of the oigin. This point is asymptotically stable when all oots à of the chaacteistic equation ae negative (if eal) o have negative eal pats (if complex). The matix ½ is given by ½Ä4 ½ 'Y' /)*1$ > ½~ÈYÈ ^ ½~ÉYÉ > )3$ ZM( & ½ Y with ½ 'Y' 4 ~ˆG`\aff ½ ÈYÈ 4 "ig`\ahd\k$j ½ ÉYÉ 4 ~ˆG`\aff ½ Y 4 ~o G`\aHD\5>C +\ ÊÌË Ë Ë kp /Î9} (6) We can compute the chaacteistic equation by expanding Ï&ÐqÑ _½ aò&óc "4 along the fist column and then explicitly evaluating the thid ode deteminants appeaing in this expansion. Upon suitable collection of tems as indicated below, we ae led to the following equation whee Š Š ' ˆÔÕ (\aökˆôõ (7)  o l 4 Š ' ' Š É*ˆÔWt4 o ig`\ f =Ô pjožig`\ HM\5$* (\aô p G`\aHD\5>C M\ kp&\5ôc L ÖKˆÔW Ž U 'Y% Y!Ž 'Y%j «%Øš ˆÔWt4 ÙÛÚs\aÙ Ô& with Ù Ú 4 o $œ>()& \ (\5> \v$z)*½ Y p,c Ù ' 4 > ZD!½~ÈYÈ gzd! (\K$j /)* c} (8) Notice that the cubic polynomial has always two negative oots, Ô ' 4ܽ ÈYÈ and Ô È 4Ý/½ Y and a positive one Ô ÉM4ÞG \ß. The value at the oigin is ŠªÉ*ˆjZ4AiGD\ f½ ÈYÈ ½ Y Ē. Futhemoe it tends to v as ƒm \. We must intesect it with the function given by the ight hand side of (7), which is the sum of the staight line Š with positive slope ' and the hypebola Ö5ˆÔÕ. The latte has the vetical asymptote located at Ô-4ÀÔ, it is positive fo ' Ô9ĒÔ ' and tends to zeo as Ô" t\. Thus as Ô{ t\, the function Ö \5Š is asymptotic to the staight line ' Š. ' Howeve, Ö5ˆj \]Š ' ˆj has no definite sign. Intesecting Ö \5Š with ' ŠªÉ, thee ae fou possible cases that can aise, depending on the signs of the two functions at Ô;4à. Given that one function has a zeo at Ôy4AÔ and the othe one in the same location has a ' vetical asymptote, with ážâ Ö5ˆÔÕL4: z ØmãØmäå we can conclude that the two functions intelace, giving an intesection fo a positive value of Ô if Ö5ˆjN\5Š ' ˆjæ ŠªÉ*ˆjx} (9) In such case then the inteio equilibium is unstable, since we would have a positive eigenvalue. To futhe assess the system s equilibium behavio in this 11-dimensional paamete space is athe difficult. Nevetheless, it would be inteesting to detemine whethe the equilibium can at all be destabilized. In Figue 2 we show the simulations fo a combination of ealistic paamete values, given by I;4Jc}Ìç, Fy4Ec} &éè*ê, )4ßc} *ë, $n4ì }, >[4ßc}Ìë, 4.c}Ìíœë, À4Ýc} *çœë, GA4Üc} &éè*ê, ì4tc} *çœë, 4tc} *ë, H~4ßc} œ&. The solutions dop vey fast, but a longe integation inteval coupled with a semilogaithmic scale shows that as claimed above the disease emains endemic, since the tajectoies appoach the inteio coexistence equilibium at the values Z 4î} ê ëœïœç,.4ð} &éïœñ, 2 4ð} &éñœíœí, 4 } *ç*ê3, see Figues 3 and 4. 4 Model with Linea Immigation We conside now a modification of the pevious system (1). The ationale fo this modification is again to bypass the infection pocess of the moe complicated models and concentate only on the tansmission mechanisms among infected individuals. The pevious model (1) has a constant immigation ate. Hee, again ignoing the infection mechanism by contact among susceptibles and infectious individuals, we state that the newly infected individuals ae geneated popotionally to the sizes of the infectious ones (TB disease individuals), of thei espective types. This assumption may be moe ealistic fo mimicking the speading of the disease via inteactions among susceptibles and infectious individuals. In fact in com- ISSN:

6 sigma=.5,q=.5,xi=1,mu=.167, lambda=.25,p=.5,alpha= sigma=.5,q=.5,xi=1,mu=.167, lambda=.25,p=.5,alpha= Figue 4: Model with constant immigation: blown up plot of inteio coexistence equilibium Figue 2: Model with constant immigation: solutions appoach vey fast the steady state solution sigma=.5,q=.5,xi=1,mu=.167, lambda=.25,p=.5,alpha=.1 mon mathematical epidemiology, the models ae assumed to ecuit new infected by eithe the mass action law, ie. via a bilinea tem involving the sizes of each subpopulation of infected and susceptibles, leading to the so-called homogeneous mixing models, o via the standad incidence, i.e. the above bilinea poduct divided by the total population size. The latte is moe apt to descibe the disease spead by a few infectives in a lage population, and leads to the popotionate mixing models. The bilinea poduct tems in both situations aise by counting the contacts that an infected can have with susceptibles, which lead to new cases of the disease, oiginated by the chosen infected individual. Multiplying the latte by the total numbe of contagious people, one obtains the ate at which the disease speads, i.e. the ate at which new infected aise by contact among the two subpopulations. Fo a faily ecent eview pape on these topics, and much moe infomation on othe moe advanced models, the eade is efeed to [14] Figue 3: Model with constant immigation: semilogaithmic plot of inteio coexistence equilibium On the basis of the above consideations, in Figue 5 we epesent the tansfe diagam of the model with linea immigation. It is simila to the one of Figue 1, but fo the immigation ates on the left. If we conside thus the model with new infectives given by tems popotional to the size of infected, espectively and, namely with immigation ates given by ò" Õ and ò!, we obtain the ISSN:

7 RPPPPPPPPPPPPPPP ó ½ ö ù à ½ l.9.8 sigma=.5,q=.5,xi=1,mu=.167,lambda=.25,p=.5,alpha= Figue 5: Model with linea immigation: tansfe diagam following linea system SmTWV SmX 4 ò Õ v^ {\]Z)*1$3 \^> W ; ig`\ b & g PPPPPPPPPPPPPPPO SdfefV Q SYX 4 g \h( 2 9,iG`\ HM\5$*k v^ SmTWl SYX 4 ò \a>,ˆg`\,bw Sdfefl SYX 4 Z+(Y 2 +\5)3$3 n,o G`\ H \^>C +\ 1pq v^ s} (1) In contast to (5), this system can thus be ewitten in matix notation as SY¼ SYX 4 ½ ó» (11) whee ½ó coincides with ½ except fo the following two enties ½ ' È 4 ½ ' È \ ò -4À/)*1$Z\,ò f ½É ó 4 ½É \ ò [4J> M\ ò s} Substituting al the possible combinations of vanishing and nonvanishing components, it is easily checked that all bounday solutions of (1), i.e. those equilibia with at least one zeo component, ae eithe impossible o infeasible, except fo the oigin. Since in seeking nontivial equilibia of (1) we ae led to the homogeneous system ó» 4:ô with nonsingula matix, then only the tivial solution is found. Thus (1) Figue 6: Model with linea immigation: tajectoies decaying towad the oigin when stable does not allow inteio coexistence solutions. Its only possible equilibium point is thus given by the oigin. Howeve, this means that all the solutions of the system modeled via (1) vanish if such equilibium is stable, and that depends on the eigenvalues of ½ ó. Thus the oigin õ being a stable equilibium means that the infection disappeas, an impotant esult. The conditions unde which this happens guaantee then the disease eadication. This is appaent also fom the simulation esults, compae Figue 6. The latte is obtained fo the following paamete values: G~4.c} &éè*ê, ;4uc} *çœë, $n4à, >4ßc}Ìë, H54.c} œ&, ID4]c}Ìç, )4Jc} *ë, +4Jc} *çœë, M4Jc} *ë, y4jc}ìíœë, 4]c} *ñ ö, -4Jc} Á. The chaacteistic equation of (1) is a quatic Þø Úfù whee ù Ú 4<Ï&ÐqÑ!ú «õ"4àï&ðqñ ó Ã Ò 4Jc. On the othe hand, by computing the deteminant of the Jacobian, we find whee with ÚÄ4 o G ig`\ahd\k$j (\ igm\ah+\a) $jkp Â/o ig`\ f ig+\ah=n\5> 2iG`\h!fY\ ig`\ bš> p  / / "ig`\ fš) $j>s D\ x 4 lû û(ü òý 4 Ž ' o igm\ b ig`\ Hª 'Y%j \^> gig+\!b(\eig`\ bš> ( ISSN:

8 V sigma=.5,q=.5,xi=1,mu=.167,lambda=.25,p=.5,alpha= Figue 7: Model with linea immigation: tajectoies gowing unbounded when oigin is unstable and with 4 Vû û(ü ò ý 4 Ž' o G ig+\ahm\k$jn\ ig`\ahd\5)3$jkp«thus a sufficient condition fo instability amounts to ù Úy8Þ. Howeve, notice that if ò",4þò :4ÿ then ù ÚK ±. We conjectue then that in such case since no new infected join the system, the disease dies away, since motality will exhaust the compatments g,, 2,. In such case then the oigin, coesponding to disease disappeaance, will become a stable equilibium. On the othe hand, if the oigin is unstable, since no othe equilibium exists, the numeical simulations show that in this situation the system tajectoies gow unbounded, see Figue 7, meaning that the disease speads. The paamete values fo the latte ae the same as fo Figue 6, but fo the highe ecuitment ates, namely ö 4Jc}Ìñ, ö -4Jc} Á. Note that in these simulations, the two cuves coesponding to the dug-esistant cases, solid lines, gow moe apidly than the othe ones. This indicates a tendence fo which in the futue moe and moe cases fo the disease that ae not teatable with cuent means ae to be expected. Although the latent esistant cases, asymptomatic as such, dominate the diseased ones, this scenaio does not diminish its unwished implications, since the pool of infected individuals constitutes always a esevoi fo the disease, and these people may always develop the full disease in thei lifetimes. 5 Discussion In classical deteministic epidemiological models fo the tansmission dynamics of an infectious disease thee is a theshold behavio, with eithe a disease-fee equilibium o an endemic equilibium appoached by the solutions. Howeve, as shown in the model (1), if thee is a constant flow of infected into the population, it is not possible to have a disease-fee equilibium. Thee is always an endemic equilibium appoached by all solutions and the disease emain in the population at an endemic level. We can educe the pevalence of the disease by consideing the immigation of infected individuals popotional to the numbe of people that ae aleady infectious (TB diseased individuals) as shown in model (1). The pevalence of the disease can be educed by deceasing the pobability that teatment failue can occus (inceasing and ). Numbe of TB esistant individuals k S = k R = 1 k S = k R = k S =.25, k R = yeas Figue 8: Model with linea immigation: evolution of the disease, dug-esistant population,. Top to bottom, the cuves ae fo the paamete values 4 ì4þ, ]4 ì4 ; J4±c} *çœë, ì4±c}ìíœë ; 4: }, 4: }. Figue 8 shows that even if all dug-esistant cases wee teated ( 4t and 4Ü ), the disease emains endemically in the population. To eadicate the disease fom the population is necessay to decease the pobability that the teatment failue occus duing the teatment of a dug-sensitive case, i.e. we need to decease ). Figue 9 descibes the pofile of the dugesistant population,, with deceasing values of )4: Lc}ÌëWL fom top to bottom. Finally, in Figue 1 we show a futhe possible scenaio wose that that of Figue 7 in which fo the ISSN:

9 ö.18 8 sigma=.5,q=.5,xi=1,mu=.167,lambda=.25,p=.5,alpha=.1 numbe of TB dug esistant individuals yeas Figue 9: Model with linea immigation: dugesistant population, v. The cuves coespond to )4: Lc}ÌëWL fom top to bottom following paamete values, indicating highe ecuitment ates, the disease cases would much outpass the numbe of latent caies. It is obtained via the following paamete values, G 4ßc} &éè*ê, D4ßc} *çœë, $ 4ì, > 4ßc}Ìë, H4]c} œ&, Ih4Ec}Ìç, ){4Ec} *ë, +4ßc} *çœë, M4Jc} *ë, [4Jc}Ìíœë, ö y4zc}ìçœñ -4Jc} Áéë. Common to both models pesented hee (1) and (1) is the assumption of ignoing the class of susceptible individuals. This is easonable if we assume it to be a lage basin, among which the infected can pick up thei new ecuits. We have concentated instead on the way the infected move fom the classes of latent to diseased, and possibly back upon successful teatment, o towad acute stages of the disease fo dug esistant cases due to impope, incomplete o ineffective healing measues. Ou conclusions give conditions fo which the disease emain endemic, and a conjectue fo disease eadication. 6 Conclusions In epidemiology, measues fo pophilaxis of infectious diseases ae taken often using mathematical methods and statistical tools to evaluate possible futue scenaios of the evolution of tansmissible diseases. Fo instance, design of systems to monito patients ae pesented and discussed in seveal situations, such as [1, 1, 15, 16, 22]. Also, evaluation of biological data fo clinical puposes ae investigated in [9]. The basic epoduction numbe discussed in [13] is an impotant quantity to assess the futue evolution Figue 1: Model with linea immigation: scenaio with disease cases much outnumbeing the latent caies of infections speading in a population, whee individuals inteact and by contact the disease is tasmitted fom a geneally asymptomatic caie to a susceptible individual. This basic epoduction numbe is elated to the incidence ate of the disease and allows to detemine unde which conditions the epidemics possibly vanishes. On this basis the Wold Health Oganization in 198 discontinued the vaccination against smallpox, since this centuy-long viulent disease could be assessed to be woldwide extinguished. In these consideations, the isk factos evaluation plays always an essential ole also fo othe diseases which cay elevant costs fo the community, [28]. The consideations pesented in this pape ae to be undestood along the same lines, as theoetical tools to help in the fight against this disease, which has caused much damage in the past, and with the employment of moden medical teatments seemed to be eadicated, fo which howeve in ecent yeas a ecudescence has unfotunately been obseved. An undestanding of its causes may theefoe help in futue epidemiological policies. Refeences [1] V. Alves, J. Machado, A. Abelha, J. Neves, Agent Based Decision Suppot Systems in Medicine Wseas Tansactions on Biology and Biomedicine, 2, , 25. [2] Bleed, D., Watt, C. and Dye, C. Wold Health Repot 21: Global Tubeculosis Contol, tech. ISSN:

10 ep., Wold Health Oganization, WHO /CDS /TB /21.287, 21. ( gtb/ publications/ globep1/ index.html). [3] Bloom, B. ed., Tubeculosis: Pathogenesis, Potection and Contol. Washington D.C.: ASM Pess, [4] Blowe, S.M., Gebeding, J.L., (1998), Undestanding, pedicting and contolling the emegence of dug-esistant tubeculosis: a theoetical famewok. J. Mol. Med, 76: and the amplification dynamics of dug esistance. Natue medicine 1 (1): [5] Castillo-Chavez, C., and Song, B. (24) Dynamical models of tubeculosis and thei applications. Math. Biosc. and Engineeing, 1(2), [6] CDC, (1985), Dug esistant tubeculosis among the homeless Boston, MMWR, 34: [7] Costello H.D., Caas G.J., Snide D.E. J., (198), Dug esistance among peviously teated tubeculosis patients, a bief epot. Am. Rev. Respi. Dis., 121: [8] David H.L., (197), Pobability distibution of dug-esistant mutants in unselected populations of Mycobacteia tubeculosis, Applied Micobiology 2: [9] M. Doepp, G. Edelmann, Fequency Distibution Analysis of Biological Data: Clinical Evaluation Wseas Tansactions on Biology and Biomedicine, 1, , 24. [1] M. Doepp, G. Edelmann, Chaos in Human Being: An Aid fo Medical Decision Making Wseas Tansactions on Biology and Biomedicine, 1, 43-49, 24. [11] Espinal, M.A., (23), The global situation of MDR-TB, Tubeculosis, 83: [12] Fine, P., and Small, P. (1999), Exogenous Reinfection in Tubeculosis, The New England Jounal of Medicine, 341, [13] J. Guadiola, A. Vecchio, The Basic Repoduction Numbe fo Infections Dynamics Models and the Global Stability of Stationay Points, Wseas Tansactions on Biology and Biomedicine, 2, 337, 25. [14] H.W. Hethcote, The mathematics of infectious diseases, SIAM Review 42 (2) [15] C.-L. Hsu, S.-Y. Yang, D.-L. Lee, L. Y. Deng, Pactical design of intelligent eminde system of having medicine fo chonic patients, Wseas Tansactions on Biology and Biomedicine, 4, , 27. [16] P.-H. Huang, Medical healthy infomation system-based on IPv6 platfom, Wseas Tansactions on Biology and Biomedicine, 3, , 26. [17] Nadell E, McInnes B., Thomas B., Weidhaas S., (1986), Exogenous einfection with tubeculosis in a shelte fo the homeless. N. Engl. J. Med., 315: [18] Pio, A., Chaulet, P., (1998), Tubeculosis handbook (Wold FHealth Oganization, Geneva) [19] Raimundo, S.M., Yang, H.M; Bassanezi, R.C. and Feeia, M.A.C., (22), The Attaction Basins and the Assessment of the Tansmission Coefficients fo HIV and M. tubeculosis Infections among Women Inmates. J. Biol. Syst. 1 (1), [2] Raimundo, S.M.; Yang, H.M.; Engel, A.B. and Bassanezi, R.C., (23), An appoach to estimating the Tansmission Coefficients fo AIDS and fo Tubeculosis. Systems Analysis Modelling Simulation 43 (4), [21] Romeyn, J., (197), Exogenous einfection in tubeculosis, Ameican Rewiew of Respiatoy Diseases, 11(6), [22] B.-Y. Shih, W.-I. Lee, Y.-S. Chung, A.-W. Chen, The development of a mobile emegency healthcae infomation system in Taiwan, Wseas Tansactions on Biology and Biomedicine, 4, 54-58, 27. [23] Singe, B. H., and Kischine, D. (24) Influence of backwad bifucation on intepetation of RÚ in a model of epidemic tubeculosis with einfection. Math. Biosc. and Engineeing, 1(1), [24] Small P.M., Shafe R.W., Hopewell P.C., et al., (1993),) Exogenous einfection with multidugesistant Mycobacteium tubeculosis in patients with advanced HIV infection, N. Engl. J. Med., 328 (16): [25] Snide, D. J., Raviglione, M., and Kochi, A., (1994), Global Buden of Tubeculosis, pp In Bloom [2]. ISSN:

11 [26] Song, B., Castillo-Chavez, C., and Apaicio, J. P. (22) Tubeculosis models with fast and slow dynamics: the ole of close and casual conatcts. Math. Biosc. 18, [27] Stead, W. (1967), Phatogenesis of a fist episode of chonic pulmonay tubeculosis in man: Recudescence of esiduals of the pimay infection o exogenous einfection, Ameican Review of Respiatoy Diseases, 95, [28] L. S. Waldbese, M. Abudiab, S. Ommani, Risk Factos Assessment and Mathematical Modeling of Neisseia Meningitidis Caiage in a Univesity Student Population, Wseas Tansactions on Biology and Biomedicine, 3, , 26. [29] WHO (26) Global tubeculosis contol: suveillance, planning, financing, p.77. Wold Health Oganization, Geneva, Switzeland. ISSN: