MATHEMATICALBIOSCIENCES
http://math.asu.edu/mbe/ANDENGINEERINGVolume1,Number2,September2004
pp.
361404DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONSCarlosCastillo-ChavezDepartmentofMathematicsandStatisticsArizonaStateUniversity,Tempe,AZ85287-1804BaojunSongDepartmentofMathematicalSciencesMontclairStateUniversity,UpperMontclair,NJ07043(CommunicatedbyYangKuang)Abstract.
Thereemergenceoftuberculosis(TB)fromthe1980stotheearly1990sinstigatedextensiveresearchesonthemechanismsbehindthetransmis-siondynamicsofTBepidemics.
Thisarticleprovidesadetailedreviewofthework on the dynamics and
control of TB. The earliest mathematical models de-scribing the TB
dynamics appeared in the 1960s and focused on the
predictionandcontrol strategiesusingsimulationapproaches.
Mostrecentlydevelopedmodelsnotonlypayattentiontosimulationsbutalsotakecareofdynamicalanalysisusingmodernknowledgeofdynamicalsystems.
QuestionsaddressedbythesemodelsmainlyconcentrateonTBcontrol
strategies, optimal vacci-nationpolicies,
approachestowardtheeliminationof TBintheU.S.A.,
TBco-infectionwithHIV/AIDS,drug-resistantTB,responsesoftheimmunesys-tem,
impacts of demography, the role of public transportation systems,
and theimpact of contact patterns. Model formulations involve a
variety of
mathemat-icalareas,suchasODEs(OrdinaryDierentialEquations)(bothautonomousandnon-autonomoussystems),PDEs(PartialDierentialEquations),systemofdierenceequations,systemofintegro-dierentialequations,Markovchainmodel,andsimulationmodels.1.
Introduction.
Tuberculosis(TB)isadiseasethataectshumanandanimalpopulations.
AncientEgyptianmummiesshowdeformitiesconsistentwithtuber-cular
decay [20, 23]. TB was probably transmitted from animals to humans
in areaswhereagriculturebecamedominantandanimalsweredomesticated.
ThegrowthofhumancommunitiesprobablyincreasedtherecurrenceofTBepidemicsleadingtoitscurrentlyoverwhelminglyhighlevelsof
endemicityinsomedevelopingna-tions. McGrath estimates that a social
network of 180 to 440 persons is required toachieve the stable host
pathogen relationship necessary for TB infection to
becomeendemicinacommunity[58]. Historicallytheterms phthisis,
consumptionandwhiteplaguewereusedassynonymforTB.TBwasafataldisease.
Inearliertimes,
somephysiciansrefusedtovisitthelate-stageTBtokeeptheirreputation.
TBwasresponsibleforatleastonebilliondeathsduringthenineteenthandearlytwentiethcenturyandtheleadingcauseof2000MathematicsSubjectClassication.
92D30.Keywordsandphrases. Tuberculosis, dynamical models,
preventionandcontrol, global
dy-namics,bifurcation,HIV,diseaseemergenceandreemergence.361362 C.
CASTILLO-CHAVEZANDB. SONGhuman death for centuries. Today, only 3
million deaths worldwide are attributedto TB every year. World
Health Organizations (WHO) data shows that most
casesofTBareindevelopingcountries.
TwentythreecountiesinEastAsiaandAfricaaccountforover80%ofallcasesaroundtheworld[84].It
was not clear how TB was transmitted until Robert Kochs brilliant
discoveryof thetuberclebacillus in1882(Kochalsoidentiedthecauseof
anthrax). HeidentiedMycobacteriumtuberculosisasthecausativeagentof
TB. Thetuberclebacilli liveinthelungsof infectedhosts.
Theyspreadintheairwheninfectiousindividuals sneeze, cough, speakor
sing. Asusceptible individual
maybecomeinfectedwithTBifheorsheinhalesbacillifromtheair.
TheparticlescontainingMycobacterium tuberculosis are so small that
normal air currents keep them
airborneandtransportthemthroughoutroomsorbuildings[83]. Hence,
individualswhoregularly share space with those with active TB (the
infectious stage of the disease)haveamuchhigherriskof
becominginfected. Thesebacilli becomeestablishedinthe alveoli of
the lungs fromwhere theyspreadthroughout the bodyif
notsuppressedbytheimmunesystem.The hosts immune responses
usuallylimit bacilli multiplicationand, conse-quently, the spread
that follows initial infections. About 10% of infected
individualseventually develop active TB. Most infected individuals
remain as latently infectedcarriers for their entire lives. The
average length of the latent period (noninfectiousstage) ranges
frommonths todecades. However, theriskof
progressiontowardactiveTBincreasesmarkedlyinthepresenceof
co-infectionsthatdebilitatetheimmunesystem.
PersonswithHIVco-infectionsprogressfastertowardstheactiveTBstatethanthosewithoutthem[68].Mostformsof
TBcanbetreated. Eectiveandwidespreadtreatmentforac-tive
andlatentlyinfectedindividuals has beenavailable for about ve
decades.Streptomycin is still used today to treat TB but in
combination with pyrazinamide.1950 1960 1970 1980 1990
2000123456789x 104 year annual casesFigure1. Annual new cases of TB
in the United States from 1953to2000.
Datatakenfrom[22].DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS
363Isoniazidandrifampinarethought tobethemost eectiveintheght
againstM. tuberculosis.
Thewidespreadintroductionofantibioticsreducedmortalityby70% from
1945 to 1955 in the U.S.A. albeit most major reductions in TB
mortalityrateshadalreadybeenachievedbeforetheirintroduction[4, 29,
54].
LatentTBcanbehandledwithisoniazidbuttreatmentiseectiveonlyifappliedforatleastsix
months. Active cases must be treated for nine months with multiple
drugs (iso-niazid,rifampin, pyrazinamide) and complex regimens.
Treatment covers over
95%ofthecasesintheU.S.A.despiteitshighcost[85].
Antibiotic-resistantstrainsareeasily generated when treatment is
not completed. The consequences of
incompletetreatmentmaybeserious[15]. Lackof
treatmentcompliancehasseriousconse-quencesduetoitsdramaticimpactontheevolutionofantibioticresistantstrains[49].
The expenses associated with treatment programs for those with
active TB
aresohighthattheireectiveimplementationisoutofthereachformostdevelopingnations.As
shown in Fig. 1,the mortality associated with TB in the U.S.A.
continues toexhibit a downwardtrend. The annual case rate of TB had
been declining
steadilybutraisedslightlyinthe1980sandearly1990sintheU.S.A..
ThechangeinthistrendhadbeenlabeledasaperiodofTBreemergence.
TBreemergenceoverthepast decade and a half has challenged existing
prevention and control TB programsindevelopingnations.Inthispaper,
wereviewsomeoftheliteratureassociatedwithTBmodelsandtheirtheoreticalimpactparticularlythoseaspectswheretheauthorsortheircol-laborators
havecontributed. Someresults appearinginthis paper
bySongandCastillo-Chavezhavenotbeenpublished.The paper is organized
as follows. Section 2 introduces the notation that we
trytousethroughoutthemanuscript. Section3reviewssomeof
theearliestknownTBmodels.
Section4dealswiththeexplorationoftheimpactofvariousepidemi-1860
1880 1900 1920 1940 1960 1980 200000.511.522.533.54x 103 year TB
mortalityFigure2.
TBmortalityoftheUnitedStatesfrom1860to2000.Datatakenfrom[78,79].364
C. CASTILLO-CHAVEZANDB. SONGological factorsaswell astheroleof
closeandcasual contactsonTBdynamics.section5looksattheimpactof
demographyonTBdynamics. Insection6,
wereviewsomecell-basedmodelsforTBtransmissionattheimmunesystemlevel.
AMarkovchainmodel onTBprojectionsisdescribedinsection7.
Modelsdealingwith TB control strategies are discussed in section 8.
A model dealing with the roleof public mass transportation on TB
evolution and control is reviewed in section
9.Finally,alistofchallengesassociatedwithmodelingTBdynamicsisoutlined.2.
Notation. The populationof interest is dividedinto several
compartments(classes, categories, or subpopulations) dictated by
the epidemiological stages (hoststatues). For the most part, inthe
context of TB, four or ve
epidemiologicalstagesareidentied(seeTable1).
Weshalldoourbesttodenotethesesubclassesusinguniformsymbolsaswediscussamultitudeof
models.
Tables1and2listthedenitionsandsymbols(subpopulationsandparameters)thatwetrytouse.Ifitisnecessarytosubdivideapopulationintosubpopulations,subscriptswillbeTable1.
Symbolsanddenitionsofsubpopulationssymbol name denitionS
Susceptible notinfectedbutsusceptibletoinfectionE Exposed
infectedbutunabletoinfectothers(latentorcarrier)I Infectious
active-TBinfections,i.e.,he/shecaninfectothersT Treated
treated(fromlatent-TBoractive-TBinfection)V Vaccinated
possiblyreducedsusceptibilitytoTBusedtodistinguishthem.
Forinstance,IsandIrrepresentthedrug-sensitiveanddrug-resistantinfectiousTBclasses,
respectively. ActiveTB, caseTB,
indexTB,matureTB,opencase,andlesioncaseallmeanactive-TBinfectiouscasehere.We
use to measure of the likelihood of transmission or the force of
infection.However, the meaning of or its interpretation often
changes from model to model.3. Earlydynamical models. Therstmodel
forthetransmissiondynamicsofTB was built in 1962 by Waaler [81],
the chief statistician of the Norwegian
Tuber-culosisControlServices.
Waalerdividedthepopulationintothreeepidemiologicalclasses:
noninfected (susceptible), infected non-cases (latent TB), and
infected cases(infectious).
Heformulatedtheinfectionrateasanunknownfunctionofthenum-ber of
infectious individuals. He used a particular linear function to
model infectionrates in the implementation of his model. The
incidence (new cases of infections
perunittime)wasassumedtodependonlyonthenumberofinfectious.
Furthermore,theequationsforthelatentandinfectiousclasseswereassumedtobeuncoupledfromtheequationforthesusceptibleclass.
Thecentralpartofthismodelisgivenbythefollowinglinearsystemofdierenceequations:Et+1=
Et +aIt +eEtd2EtgEt, (1)It+1= It +gEtd3IteEt,
(2)DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS 365Table2.
Symbolsanddenitionsofparameters.symbol explanation recruitmentrate
transmissionrate(meaningvaries)c
averagenumberofcontactsperpersonperunittimek
per-capitaregularprogressionrate per-capitanaturalmortalityrated
per-capitaexcessdeathrateduetoTBr0per-capitatreatmentrateforrecentlylatently-infectedr1per-capitatreatmentrateforlatently-infectedr2per-capitatreatmentrateforactively-infected
per-capitaprogressionrateforearlylatent-TBprogressionwheretheincidencerateaItisproportional
tothenumberof infectious;
eistheper-capitaprogressionratefromlatent-TBtoinfectious-TBcases; g
is
theper-capitatreatmentrate(treatedindividualswillbecomemembersoflatent-TBclassagain.);
d2istheper-capitadeathrateof thelatent-TBclass;
andd3istheper-capitadeathrateoftheinfectious-TBclass.
UsingdatafromaruralareainsouthIndiafortheperiodof1950to1955,Waaler[35]estimatedtheparametersofthislinearmodel
tobea=1, e=0.1, d2=0.014, g=0.10085, d3=0.07.
Becausetheeigenvaluesallhavenormscloseto1(1.04),Waalerpredictedthatthetimetrendof
TBisunlikelytoincrease(itmaydecrease, albeitslowly).
Thislinearmodeldidnotmodelthemechanicsoftransmission.
However,theparameters,estimatedfromaspecicareainIndia,setusefulrangesfortheestimationofparametersindevelopingnations.Broggerdevelopedamodel
[10] thatimprovedonWaalers.
Broggernotonlyintroducedheterogeneity(age)butalsochangedthemethodusedforcalculatinginfectionrates.
TheinfectionrateinBroggersmodelwasacombinationoflinearandnonlinearinfectionterms.
Infact,itwasgivenbythetermS(1 Z +ZIN),where Zwas an adjusting
parameter used to dierentiate between normal
infection,superinfection, and direct leaps (within a very short
period, an uninfected
individualbecomesalesioncaseoranactive-TBcase).
Twoextremecaseswerecoveredinthemodel: Z=1makingtheincidencebeSIN,
thefamiliar versionof today,and,
Z=0givinganinfectionrateproportional tothenumber of
susceptibles.TheprevalenceINwasusedtoadjustall
owratesincludingthosefrominfectedtoopencases.
ThiswasnotsurprisingasBroggerwantedtouseprevalenceasanindicatoroftheeectivenessofcontrolpolicies.
Hisaimwastocomparedierentcontrol
strategiesthatincludedndingandtreatingmorecases,
theutilizationofvaccination, and mass roentgenograph. The data of
two WHO/UNICEF projects inThailand from 1960 to 1963 were used to
estimate the parameters incorporated intohismodel.
Broggerchosethoseparametersthatbesttsavailabledata.
Controlstrategies (additional new parameters) were squeezed into
the model. Simulations366 C. CASTILLO-CHAVEZANDB.
SONGwererunandcomparisonsmadetoevaluatethevalueofdierentstrategiesofTBcontrol
programs. Thismodel
didnotformulateclearlytherelationshipbetweeninfection rate and
prevalence. ReVelle classied this important relationship in
1967.UsingBroggerandWaalersmodel asatemplate,
ReVelleintroducedtherstnonlinearsystemofordinarydierential
equationsthatmodelsTBdynamics[63,64]. Inmodelingtheinfectionrate,
hedidnotfollowthetypicalmassactionlaw,givenbythebilinearfunctionSIofKermackandMcKendrick[47].
ItwasReV-elle who rst, at least in the context of TB dynamics,
rigorously explained why
theinfectionratedependslinearlyontheprevalenceusingtheprobabilisticapproachthat
is common today (homogeneous mixing). The form SINfor the infection
rateis found in most epidemic models used today. Mathematically it
is well known thatif thetotal
populationsizeNremainsconstantovertimeorif
itasymptoticallyapproaches a constant then the use of an infection
rate proportional to SIdoes notchange the qualitative properties of
the model. However, when modeling epidemicsfor developing
countries, as Revell did with his model, SINseems a more
appropri-ate form of modeling the infection rate. ReVelle modeled
TB dynamics via a systemof non-linear dierential equations but he
ignored population structure. Nine
com-partmentswereintroducedinRevellsnonlinearmodel.
ThetotalpopulationwasgovernedbytheMalthusmodel
becausehewantedtoapplyittodevelopingna-tions.
Makingprojects(inWaalerswordstimetrendof
tuberculosis)wasnotRevellsmaintheme.
Infact,hismainobjectiveseemedtobeassociatedwiththeevaluationandimplementationofcontrolpolicesandtheircost.
Hedevelopedanoptimizationmodel andusedittoselectcontrol
strategiesthatcouldbecarriedoutataminimalcost.
ItisworthtomentionthatWaaleralsodevelopedamodelin1970thatwouldminimizethecostofalternativetuberculosiscontrol
measures[82].All dynamical
modelspriortoFerebeesworkweremotivatedbythestudyofTBindevelopingnations.
ThetwodatasetsusedwerefromThailandandIndia.NospecicmodelseemstohavebeendevelopedfortheU.S.A..
Ferebee,associatechief of theresearchsectionof
thePublicHealthServiceTuberculosis Programof U.S.A.,
changedthistrend. Shesetupadiscretemodel,
baseduponasetofsimpleassumptions, tomodel thedynamicsof
TBintheU.S.A. [34]. Sheusedthe same compartments as Waaler did,
that is, susceptible, infected and infectious.The basic time unit
was ayear. Hence, withinone year,
newinfectedpeoplewouldbecomeinfectiousandcontributetothepoolofnewinfected,namely,someindividuals
wouldmovefromthesusceptibletotheinfectedclass
andfromtheinfectedtotheinfectious class withinayear.
Ferebeedescribedher algorithm,methods of estimation of relevant
parameters, and the number of infected people intheU.S.A..
Theresultsshowedthatthenumberofnewcaseswoulddecrease,butslowly, if
vaccination were not applied to the US population. It is worth
mentioningthattheestimationof
demographicparameterswassolelybasedonthe1963USdata.
Despiteitsshortcomings,thisworkindeedgavetherstroughestimateandforecast
of TB cases in the U.S.A.. She stated that her assumptions were
checked
forconsistencywithbitsandpiecesofinformationobtainedfromavarietyofsources.These
assumptions have since played an important role theoretically and
practicallyinthecontextofTB.Inotherwords,therstUSstudydenedanappropriateparameterrange.
Weshalloutlineunderlyingassumptions:i.
Thereare25millioninfectedindividualsand125millionsusceptible;DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS
367ii.
Theper-capitaprogressionratefrominfectedtoinfectiousisk1=1/625peryear;iii.
Primary infected people exhibit a higher per-capita progression
rate in the rstyear(k2=1/12); thatis, oneoutof
every12newinfectionswill
progresstotheinfectiousstageduringtherstyear;iv.
Eachnewinfectiousindividualswillinfect3people;v.
Nosignicantadditionaldeathisascribedtotuberculosis.Herassumptionk2>>
k1hasdirectedmodelsimulationsandconstructions. Thisassumption has
been recently incorporated via the inclusion of additional
compart-mentsforfastTBandslowTB(seesection4.1).Earlier mathematical
models for TBtransmissionwere developedmostlybystatisticians.
Theirapproachfollowedapattern: buildamathematical model
forTBtransmission; withhelpfromadataset, estimateparameters;
ndnumericalsolutions; and predict or make inferences about the
relative value of alternative con-trolstrategies.
Therewasnoqualitativeanalysisofthemodels,andthelong-timebehavior(asymptoticproperties)ofthemodelswasalsonotstudied.The
continuous decline of TB incidence in developed nations and the
introductionof eective antibiotics suggested that elimination of
active TB in developed nationswaspossible.
Thisviewmayhavebeenthemainreasonwhytherewasalmostnotheoretical
workonTBdynamicsfromthe1970stotheearly1990s. Thestoryhas changed
over the last decade because of the reemergence of TB (new
outbreaksintheU.S.A. andinmanydevelopednations).
Inthefollowingsectionsweshallreviewsomeofthemostrecentmodelsandthetheoreticalresults.4.
Intrinsicmechanicsoftransmission.4.1. Slowandfastroutes.
Theinitiallyexposedindividuals(infectedindividu-als) have a higher
risk of developing active TB. With time passing, those
individualsstillfacethepossibilityofprogressingtoinfectiousTB,buttherateofprogressionslowsdown.
Inotherwords, thelikelihoodof
becominganactiveinfectiouscasedecreaseswiththeageof theinfection.
Bearingthisinmind, several researchersconstructedaseriesof
dynamical
modelsforTBprogressionandtransmissioninscenariosthattookthesefactorsintoconsideration[6,
7, 8, 19, 32, 60]. Weshallreviewsomeof thiswork. Inthesimplestmodel
(thatweknow),
thepopulationofinterestispartitionedintothreeepidemiologicalclasses:
susceptible,latent,andinfectious.
TheinfectionrategivenbySI(usingthemassactionlaw)isdivided.AportionpSI
givesrisetoimmediateactivecases(fastprogression), whiletherest(1
p)SIgivesrisetolatent-TBcaseswithalowriskofprogressingtoac-tiveTB(slowprogression).
TheprogressionratefromlatentTBtoactiveTBisassumed to be
proportional to the number of latent-TB cases, that is, it is given
bykE, where k ranges from 0.00256 to 0.00527 (slow progression).
The total incidencerateispSI +kE.
Theversionin[6]isgivenbyfollowingsystem:dSdt= SI S, (3)dEdt= (1
p)SI kE E, (4)dIdt= pSI +kE dI I, (5)368 C. CASTILLO-CHAVEZANDB.
SONGwheretheparametersaredenedinTable2. Thequalitativedynamicsof
model(35)aregovernedbythebasicreproductivenumberR0= p +d+ (1 p)k
+k.
(6)Thisdimensionlessquantitymeasurestheaveragenumberofsecondaryinfectiouscases
produced by a typical infectious individual in a population of
susceptibles ata demographic steady state. The rst term in (6)
gives the new cases resulting fromfast progression while the second
those resulting from slow progression.
Sensitivityanduncertaintyanalysis were carriedout.
Simulationresults showedthat
TBdynamicswerequiteslowforacceptableparameterranges. Waalersmodel
alsosupportedslowTBdynamics[81]. Model
(35)requiresthatpbeknownapriori(itisnotallowedtochange)and,the
R0derivedfromthemodeldependslinearlyonpopulationsize.
Amodelthatremovestheserestrictionsisreviewednext.4.2.
Variablelatentperiod.
Insteadofassumingexponentialdistributionofla-tencyperiod,
Fengstudiedamodel
withanarbitrarydistributionforthelatencyperiod[31].
Todescribethismodel,weletp(s)beafunctionrepresentingthepro-portionofthoseindividualsexposedsunitsoftimeagoandwho,ifalive,arestillinfected(butnotinfectious)attimes.
Theremovalrateofindividualsfromthe EclassintotheIclass
unitsoftimeafterexposureisgivenby p().
Hence,thetotalnumberofexposedindividualsfromtheinitialtimet =
0tothecurrenttimet,whoarestillintheEclass,isgivenbytheintegral_t0cS(s)I(s)N(s)p(t
s)e(+r1)(ts)dswhilethenumberofindividualswhodevelopinfectiousTBcasesfrom0tot,whoarestillaliveandintheIclass,isgivenbythedoubleintegral_t0_0cS(s)I(s)N(s)e(+r1)(s)_
p( s)e(+r2+d)(t)_dsd.The following system of integro-dierential
equations is used to model TB
dynamicswithavariablelatentperiod:dSdt= SIN S +r1E +r2I, (7)E(t) =
E0(t) +_t0S(s)I(s)N(s)p(t s)e(+r1)(ts)ds, (8)I(t)
=_t0_0S(s)I(s)N(s)e(+r1)(s)_ p(
s)e(+r2+d)(t)_dsd+I0e(+r2+d)t+I0(t), (9)N= S +E
+I,where=cistheforceof infectionperinfective;
r1andr2aretheper-capitatreatmentratesfortheEclassandIclass,respectively;isthenaturalmortalityrate;
d is the per-capita death rate due to TB; E0(t) denotes those
individuals in theEclassattimet =
0whoarestillinthelatentclassattimet;I0(t)denotesthoseindividualswhoareinitiallyintheEclasswhohavemovedintoclassI
andstillaliveattimet;andI0e(+r2+d)twithI0=
I(0)representsthoseindividualswhoareinfectiousattime0whoarestillaliveintheIclassattimet.
Mathematically,itisassumedthatE0(t)andI0(t)havecompactsupport.DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS
3690 0.5 1 1.5 200.511.522.5 R1 R2IIIIVIIIFigure 3.
Bifurcationdiagramfor model (1015) whenq
=0.InregionI,thedisease-freeequilibriumisgloballyasymptoticallystable;
inregions II andIV, one of strains disappears;
andIIIrepresentsthecoexistenceregion.0 0.5 1 1.5 200.511.522.5 R1
R2IIIIIIFigure 4. Bifurcationdiagramfor model (1015) whenq
>0.Thecoexistenceofthetwostrainsisimpossible.370 C.
CASTILLO-CHAVEZANDB.
SONGTheintroductionofanarbitrarydistributionoflatencyperioddidnotchangethequalitativedynamicsof
TB; thatis,
aforwardbifurcationdiagramcharacterizesthedynamicsofthelastmodel(79).4.3.
Multiple strains. Incomplete treatment, wrong therapy,
andco-infectionwithother diseases, for instance, HIV,
maygiverisetonewresistant strains ofTB (multipledrug resistant,or
MDR strains). Form First Lady
EleanorRooseveltwasoneofthevictimsofMDRTB[24, 62].
ModelsthatincludemultiplestrainsofTBhavebeendeveloped[7,15,17].
Arecentlypublishedtwo-strainTBmodelincludedrug-sensitiveanddrug-resistantstrains[15,17].
Hence,twosubclassesoflatent andinfectious individuals arerequired.
Thesubscriptssandrstandfordrug-sensitiveanddrug-resistanttypes.
Themodelisgivenbythefollowingsetofequations:dSdt= scSIsN rcSIrN S,
(10)dEsdt= scSIsN ( +ks)Esr1sEs +pr2sIsscEsIrN , (11)dIsdt= ksEs(
+ds)Isr2sIs, (12)dTdt= r1sEs + (1 p q)r2sscT IsN rcT IrN T,
(13)dErdt= qr2sIs( +kr)Er +rc(S +Es +T)IrN , (14)dIrdt= krEr(
+dr)Ir, (15)N= S +Es +Is +T+Er +Ir.It can be seen from Equation
(15) that the treatment rate for the Irclass is
equaltozero,meaningthatTBduetothisstrainisnottreatablebycurrentantibiotics.The
proportion of treated infectious individuals who did not complete
treatment isp+q. The proportion p modies the rate at which they
depart from the latent
class;qr2sIsgivestherateatwhichindividualsdevelopresistant-TBduetotheirlackofcompliancewithTBtreatment.
Theproportionofsuccessfullytreatedindividualsis1 p q.
ThedimensionlessquantitiesR1=_sc +pr2s +ds +r2s__ks +ks
+ds_andR2=_rc +dr__kr +kr_givethebasicreproductivenumberof
strainsj, wherej =1, 2. Theasymptoticbehavior of model (1015) is
determined by Rj. Fig.s 3 and 4 show the
bifurcationdiagramforthistwo-strainmodel.
Thesediagramsshowsthatnaturallyresistantandnaturaltypescanco-exist,albeittheregionofcoexistenceissmall(regionIVinFig.
3). Furthermore, in[15] it was shownthat
antibiotic-inducedresistanceresultsinthesubstantial expansionof
theregionof coexistence. Infact, regionsIVand
IIIbecomeasinglelargeregionofcoexistence.
Inotherwords,antibiotic-inducedresistanceguaranteesthesurvivalofresistantstrains.
Atwo-strainmodelthatincorporatestheeectsofmultipledrugresistancecanalsobefoundin[7].DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS
3714.4. Multiple strains and variable latent period. A model that
considers bothmultiple strains and variable latent period is
proposed by Feng [33]. Drug-resistantand drug-sensitive strains are
modeled, but only the age of the infection with
drug-sensitivestrainisconsidered.
Theyintroduceafunctionp()astheproportionofthesensitive-strainthatareactiveatinfection-agetodistinguishactiveTBandinactive
TB. The model does not make a dierence between active and inactive
TBfor the drug-resistant strain because after acquiring
drug-resistent TB, an individualdies quickly.
Consequentlyforthedrug-resistant straintheyonly countactive
TB.Thetotal populationis dividedintothreeclasses:
susceptible(S(t)),
infectionswithdrug-sensitivestrain(Is(t)),andinfectionswithdrug-resistantstrain(Ir(t)).Lettingis(,
t)betheinfection-agedensityof
infectedindividualswiththedrug-sensitivestrainattimet,themodelframeworktakesthefollowingsystem:dSdt=
b(N)N _ +1cIasN+2cIrN_S(t) + (1 r)r2Ias, (16)_ t+_is(, t) + ((1 r
+qr)r2p() + +d1) is(, t) = 0, (17)dIrdt= 2cSIrN ( +d2)Ir +qrr2Ias,
(18)is(0, t) = 1cSIasN , (19)whereIas(t) =_0p()is(,
t)disthetotalnumberofactiveTBofdrug-sensitivestrain; Is(t) =_0is(,
t)dthetotal number of infectedindividuals
withdrug-sensitiveTB(bothlatentTBandactiveTBareincluded);b(N)theper-capita00.20.40.60.81R0I*Rp1Figure
5. Backwardbifurcationdiagramwhenexogenous
rein-fectionisincludedinmodel(2023). When R0<
Rp,thedisease-freeequilibriumisgloballyasymptoticallystable.
However, whenRp< R0 0,b > 0. When < 0with ||
1,0islocallyasymptoticallystable,andthereexistsapositiveunstableequilibrium;when0
<
1,0isunstableandthereexistsanegativeandlocallyasymptoticallystableequilibrium;ii.
a 0, U < 0,x> 0, U > 0,x< 0, Sa < 0,b < 0 < 0,
U > 0, S < 0,x> 0, S > 0,x< 0, Ua < 0,b > 0
< 0, S > 0, U < 0,x< 0, U > 0,x> 0, Sa > 0,b
< 0 < 0, U > 0, S < 0,x< 0, S > 0,x> 0,
URemark 1. The requirement that w is nonnegative in Theorem 4.1 is
not
necessary.Whensomecomponentsinwarenegative,westillcanapplythistheorem,butonehastocomparewwithactual
theequilibriumbecausethegeneral
parameterizationofthecentermanifoldbeforethecoordinatechangeisWc=
{x0 +c(t)w +h(c, ) : v h(c, ) = 0, |c| c0, c(0) =
0}providedthatx0isanonnegativeequilibriumofinterest(Usuallyx0isthedisease-freeequilibrium).
Hence,x02ba> 0requiresthatw(j) > 0wheneverx0(j) = 0.Ifx0(j)
> 0,thenw(j)neednotbepositive.Corollary4.1. Whena>0andb>0,
thebifurcationat
=0issubcritical(backward).ApplyingCorollary4.1tomodel (2023),
wegivearigorousproof thatmodel(2023)undergoesabackwardbifurcation.
Wedoitinarathersimplecase= 1thoughwhen =1theargumentsareidentical.
Let=cbethebifurcation376 C. CASTILLO-CHAVEZANDB. SONGparameter.
Introducingx1= S +T,x2= E,x3= I,theSystem(2023)becomesdx1dt= x1x3x1
+x2 +x3x1:= f1, (32)dx2dt= x1x3x1 +x2 +x3px2x3x1 +x2 +x3( +k)x2:=
f2, (33)dx3dt= px2x3x1 +x2 +x3+kx2( +r +d)x3:= f3, (34)with R0= 1
corresponding to = =(k+)(+r+d)k. The disease-free equilibriumis
[x1=, x2= 0, x3= 0]. The linearization matrix of system (3234)
around thedisease-free-equilibriumwhen = isDxf=__ 0 0 (k +) 0 k (
+r +d)__.It is clear that 0 is a simple eigenvalue of Dxf. A right
eigenvector associated with0 eigenvalue is w = [+kk,1k,1+r+d], and
the left eigenvector vsatisfying v w = 1isv=
[0,k(+r+d)(k+)+(+r+d),(k+)(+r+d)(k+)+(+r+d)].
Algebraiccalculationsshowthat2f2x2x3= (1 +p)x1,2f2x23= 2x1,2f3x2x3=
px1,2f2x1=
1.Therestofthesecondderivativesappearingintheformulaforain(26)andbin(27)areallzero.
Hence,a =x1k((k +) + ( +r +d))_p k_1 +k +r +d__,b = v2w3=k(k +) + (
+r +d)> 0.Wecollecttheseresultsinthetheorembelow:Theorem4.2. If
p >p0=k_1 +k+r+d_, the directionof the
bifurcationofsystem(3234)(orsystem(2023))at R0=
1isbackward.Model(2023)exhibitstotallydierentbehaviorasitsupportsmultiplestable-steadystatesviaabackwardbifurcation(subcriticalbifurcation).
Theresultsarecounter-intuitive.
thatis,whenthebasicreproductivenumber R0=c+r+dk+k nL, Q0 p1np1+n+(1
p)KKnwhich levels oat the value p1+(1 p)). Hence,an increase in n
translates an increase in TBtransmissionbuttheincreaseisnon-linear.
Initially, thisincreaseisalmostlinearbut as n becomes larger the
rate of increase decreases, because the time spent by aninfectious
individual per contact cancels out increases in cluster size.
Hence,
R0(n)isboundedbyaconstantvalue,andthisboundlimitsthesizeofTBprevalence.Case2.
Thesecondcaseassumes(n)tobeinverselyproportional ton(thatis,(n) =1n
). Itcanbeseenfrom45that R0(n)isthesumofcontributionsfromboth the
within and out of cluster new secondary cases of infection. The
ratio E(n)ofwithintobetweenclustercontributionsisgivenbyE(n) =1pK(1
p)n(K n)(p1 +n).Function E(n) increases with n and reaches its
maximum value at n=K1+_1+Kp1.Hence,nheredenestheoptimal
clustersize;thatis,thevaluemaximizeswithinclustertransmission.Singular
perturbationtheoryandmultipletimescales techniques
areusedtostudytheglobaldynamicsoftheclustermodels[71].
Sincetheaverageinfectiousperiod(about34months)ismuchshorterthantheaveragelatentperiod,
whichhas the same order as the host population, two dierent time
scales can be identied.Time is measured using the average latency
period 1/kas the unit of time (that is,=kt).
VariablesS2andE2arere-scaledby, thetotalasymptoticpopulationsize (
=); N1 could be re-scaled by instead, it is re-scaled by the
balance factork+. The rescaling formulae are x1=S2 , x2=E2, y1=+kS1
, y2=+kE1, andy3=+kI.
Thesenewre-scaledvariablesandparametersarenon-dimensional.380 C.
CASTILLO-CHAVEZANDB. SONGThere-scaledmodelequationsaregivenbydx1d=
B(1 x1) + (1 m)y1nx1x2x1 +x2, (46)dx2d= (1 m)y2(1 +B)x2nx22x1 +x2,
(47)dy1d= y1 +nx1x2x1 +x2, (48)dy2d= my1(1 m)y2 +nx22x1 +x2,
(49)dy3d= x2(1 m)y3, (50)where =k+,m =+< 1andB=k.
Thetermsinright-handsideofsystem(4647)all
havethesameorderofmagnitudewhenever 1. Therefore, y1,
y2,andy3arefastvariablesandx1andx2areslowvariables.
Hoppensteadtsearlytheorem[44]helpustoshowtheglobalstabilityoftheendemicequilibriumwhenissmall.
ALiapunovfunctionandaDulacfunctionareselectedtoestablishtheglobalstabilityofthedisease-freeequilibrium.
Hence,aglobalforwardbifurcation(seeFig.7)characterizesthedynamicsoftheclustermodels.R0I*1global
transcriticalbifurcationFigure7.
GlobalforwardbifurcationdiagramfortheTBmodels.5.
Modelswithdensitydependentdemography. Demographyplaysanim-portant
role inthe transmissiondynamics of TBsince the average rate of
pro-gressionfrominfected(non-infectious)toactive(infectious)TBisveryslow.
Infact, theaveragelatentperiodhasthesameorderof
magnitudeasthelife-spanof thehostpopulation.
Twodistinctdemographicscenariosarestudied. Intherst,
exponentialgrowthisobservedoveralongtimescale,
andinthesecondex-ponential
growthisobservedoverashorttimescale(quasi-exponential
growth).DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS
381Consequently,threedierentrecruitmentratesareusedinthestudyofdynamicalmodelsforTBtransmissionwithdemography:
constantrecruitmentrate()[15],linear recruitment rate (rN) [74],
and logistic recruitment rate (rN(1N/K))
[74].ThegeneralmodelisgivenbydSdt= B(N) cSIN S, (51)dEdt= cSIN ( +k
+r1)E +cTIN , (52)dIdt= kE ( +d +r2)I, (53)dTdt= r1E +r2I cTIN T,
(54)N= S +E +I +T,where the recruitment rate B(N) includes the
three demographies described above,namely, rN, rN(1N/K), and . The
basic epidemiological reproductive numberisgivenbyR0=_k +r1 +k__c
+r2
+d_(55)However,thisnon-dimensionalnumberisnotenoughtocharacterizethedynamicsofmodel(51-54).5.1.
Linearrecruitmentrate. Currently,
mostdeathscausedbyTBrepresentasmall proportionof
thedeathsformostpopulations, inotherwords, disofteninsignicant.
Therefore,
alinearrecruitmentrateB(N)=rNwithreasonablervaluesislikelytosupportexponential
growthonaTB-infectedpopulation.
TheuseofalogisticrecruitmentrateB(N) = rN(1
N/K)tomodelthedemographyin general is also likely to result in
logistic growth for the total population Nin
thepresenceofTB.Tosimplifyouranalysis,wefurtherassumethattheinfectedandreinfectedproportionsareequal
=. Theuseofthevariables, N, EandI, isenough.
Hence,model(5154)reducesto:dNdt= B(N) N dI, (56)dEdt= c(N E I)IN (
+k +r1)E, (57)dIdt= kE ( +d +r2)I.
(58)Weshallconsistentlyusethefollowingcompressednotationmr= r +r2
+d, nr=r +r1+k, m= +r2+d, n= +r1+k, and = c to simplify the
discussions.First, westudythedynamicsof model (5658)withB(N)=rN.
Thatis, itisassumedthatthetotal populationexhibitsexponential
growthintheabsenceof TB(thenet growthrateof thepopulation,
intheabsenceof thedisease, isr ). Total population size increases
exponentially if r > , and remains constantif r =. Thecasewherer
if d is large enough; that is, technically, a fatal disease can
impactpopulation growth (see also May and Anderson,[57];Busenberg
and Hadeler [11]).Realistic examples of situations where a disease
has or is likely to to have an
impactonthedemographicgrowthcanbefoundintheworkonmyxomatosis[51]andon382
C. CASTILLO-CHAVEZANDB. SONGHIV[1,56].
Threenon-dimensionalthresholdparameters R0, R1,and R2provideafull
characterizationofthepossibledynamical
regimesofsystem(5658)underthevariousdemographicregimes.
ThebasicreproductivenumberR0=_ +r2 +d__k +r1 +k_,
(59)givestheaveragenumberofsecondarycasesproducedbyatypicalinfectiousindi-vidual
during his/her entire life in a population of mostly susceptibles.
It is impliedfrom R0<
1thattheinfectedpopulationsgoestozero,while R0> 1impliesthatthe
infectedpopulations grows (initially) exponentially(together
withthe totalpopulation N). In this last case, there are two
possibilities: Ngrows faster than I,orNdoesnotgrowfasterthanI.
Intherstcase,thefractionu
=INapproacheszeroastimeincreases,andtheadditionalthresholdparameterR1=_r
+r2 +d__kr +r1 +k_(60)plays a role;that is, R1discriminates between
the last two possibilities. If R1< 1,then limtu=0, while
R1>1implies that limtu>u>0. Our assumptionr >
impliesthat R0> R1istrue.
Iftheinfectious(I)populationchangesfasterthanthetotal
population(N)then(afatal)diseasecandrivethepopulationtoextinction(evenwhen
R1>1).
ThethresholdparameterthatdecidesthislastsituationisgivenbyR2=r du,
(61)where uis a positive constant (independent of (see (67)); that
is, R2determineswhetherornotthetotal
populationsizegrowsexponentially. Infact, populationsize
woulddecrease exponentially(fromTB) onlyif R2, the trivial
equilibrium(0, 0) isgloballyasymptoticallystableif R1 1.
Furthermorethereexistsauniquepositiveequilibriumthatisgloballyasymptoticallystableif
R1> 1.The standard classication of planar quadratic dierential
systems rules out theexistenceof closedorbitsorlimitcycles.
(Otherapproachescanbeusedtodrawthesameconclusion, forexample,
seeBusenbergandvandenDriessche[12]; Linand Hethcote [53]. The full
structure of system (5658) is provided in Theorem
5.2below:Theorem5.2. Considersystem(5658)andassumethatr > :i. If
R0< 1,then(, 0, 0)isgloballyasymptoticallystable.ii. If R1< 1
< R0, then (, , ) is globally asymptotically stable and
limtIN=0, limtEN= 0.iii. If1 < R1< R0,then(a). (0, 0,
0)isgloballyasymptoticallystableand limtIN=u, limtEN=vwhen R2<
1,(b). (, , )isgloballyasymptoticallystableand limtIN=u,
limtEN=vwhen R2> 1,whereu= [d(mr +nr) (mr +k)] +2d( d)(k mrnr),
(67)v=mr(a2 +a3 + 1/2) 2a2du22a2k,= [d(mr +nr) (mr +k)]2+ 4d( d)(k
mrnr), = (a2 +a3)2+ 4a1a2> 0.Hence, whenever R0< 1 the
disease dies out while the total population increasesexponentially.
Althoughthediseasespreads(thatis,thenumberofinfectedgrowsin total
number) when R1< 1 < R0, the proportionsINandENapproach zero.
Onecanobservefromthecase1< R1<
R0thatdisease-inducedmortalitycanleadtotheextinctionof
apopulationthatwouldotherwiseincreaseexponentially(afatal disease
can indeed regulate a population). Note that R2is positive since
uispositiveandindependentof. Wealsohaveestablishedthatwhenr<
,(0, 0, 0)384 C. CASTILLO-CHAVEZANDB.
SONGisgloballyasymptoticallystableeventhough limtIN= uand limtEN=
vwhenR1>1, and R1 . Theorem 5.2 provides a complete
characterization of the dynamicstructureof model (5658). Theglobal
dynamicsareshowninFig. 8. Here,
weprovidetheproofthatthedisease-freeequilibriumisaglobalattractor.R1R0I
I/N 0I 0I/N 0I/N u*N 0r={N 0 (R21)11Figure8.
Bifurcationdiagramforlinearrecruitmentrate.Proof. The disease-free
equilibrium is (rr, 0, 0). It is straightforward to show
thattheendemicequilibriumisuniquewhenever R0>1and
R2>1andthedisease-freeequilibriumislocallystablewhenever R0 1.
Here,weonlyneedtoestablishtheglobal stabilityof
thedisease-freeequilibriumundertheassumption R0 1.Letf(t)=E(t) +
2I(t), where=_(mn)2+ 4k + m n. Itsucestoshowlimtf(t) = 0.df(t)dt=
dE(t)dt+ 2dI(t)dt (I(t) nE(t)) + 2(kE(t) mI(t))= (2k n)E(t) + (
2m)I(t)= (n +2k)E(t) + (2 m)2I(t)= (E(t) + 2I(t))__(mn)2+ 4k (m
+n)2_= (1 R0)_(mn)2+ 4k +m
+nf(t).DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS
385Thisactuallyproducesadierentialinequalityonthefunctionf(t);thatis,df(t)dt<
(1 R0)_(mn)2+ 4k +m +nf(t). (68)It follows that limtf(t)
=0from(1R0)(mn)2+4k+m+n>0and R0
0.Hence,f(t)isadecreasingfunctionandf(t) f(0)e1_t0I(t)dt. (69)If
liminftI(t) > 0, thenlimtf(t) = 0 by (69), which yields limtI(t)
= 0 from thedenitionoff(t). Hence, liminftI(t)=0.
ItfollowsthatliminftE(t)=0from the uctuation lemma of Hirsch et al.
[43] and Proposition 2.2 by Thieme [77].Consequently, liminftf(t) =
0 and limtf(t) = 0 because f(t) is decreasing.5.2.
Logisticrecruitmentrate. When logistic recruitment rate B(N) =
rN(1N/K) is considered in model (5154), the dynamics become
determined by R0andR2=r +dk+d+r2+kR01R0.Forsystem(5154), if R0 1,
thedisease-freeequilibriumisaglobal attractor;if R0>1and
R2>1,
thereexistsauniqueendemicequilibriumthatisgloballyasymptotically
stable under some assumptions. Hence, a global forward
bifurcationdescribes thedynamics if R2>1(seeFig. 7).
Toshowtheglobal stabilityoftheendemicequilibrium,
anequivalentmonotonesystemtotheoriginal
onewasidentiedandastrongversionofthePoincare-Bendixontheoremapplied.
Furtherwhen the disease dies out, the decay is of exponential form
with a rate proportionalto R0 1. Particularly, interestingdynamics
are observedwhen1 k=0.00256.(One has seen this in Ferebees
assumptions in section 3.)Model (9194) allows onetoexploretheroleof
treatingearlylatent-TBcases. Resultsfromthistwo-stage392 C.
CASTILLO-CHAVEZANDB. SONGlatent TB model show that treatment of 25%
of early latent-TB cases together
withtreatmentof80%ofactive-TBcasesmayresultintheeliminationofTB.8.3.
EliminationofTBintheU.S.A.. A comprehensive and executable
modelthat leads to TB elimination in the U.S.A. is proposed by Song
[72, 73].
CollectingpublicTBanddemographicdatafortheU.S.A.forthepasthalfcentury,amodelofnon-autonomoussystemsofordinarydierentialequationsisusedtotU.S.A.tuberculosisincidenceoverthepastvedecades.
ItisshownthattuberculosisintheU.S.A.
maybecontrolledtothepointofmeetingCDCscriterionofonecasepermillion,
butonlybytheyear2020. Thisgoal maybeaccomplishedonlyifatleast 20%
of latently-infected individuals are treated. The eect of HIV/AIDS
after1983isincludedintheanalysisviaavariationof ourmodel
thatincorporatesafunctionthatacceleratesTBprogressionoverawindowoftime.
ItisshownthatTBscaseratemaybecontrolleddespiteincreasesintherateof
TBprogressionduetoHIV. Fromthecensusandprojectiondata, thetotal
populationsizeN(t)is includedinthemodel as anexternal input.
Twolatently-infectedclasses areintroduced, primary latent/exposed
class (L1) and a permanently latent class
(L2).Anon-autonomousODEmodel
withtwolatentclassesandtheincorporationofHIVstandsdL1dt= (N(t)
L1L2I)IN(t) ((t) +k +r1 +p +A(t))L1, (95)dL2dt= pL1((t) +r2
+A(t))L2, (96)dIdt= kL1 +A(t)(L1 +L2) ((t) +d(t) +r3)I,
(97)whereA(t)hastheform:A(t) =_1(t 1983)2e3(t1983)4if1983 < t,0
otherwise.Hereiisconstanttobedeterminedviasimulations. N(t),
nowassumedtobeindependent of the disease, is aknownexternalinput
tothe epidemiologicalmodel. Thevaluesof
N(t)areinfactinputfromextrapolatedpublishedU.S.A.demographicdata.
Thetransmissionrateisassumedtobeaconstant; k,
TBsactivationrate,isalsoassumedtobeconstant;ri(i = 1, 2,
3),thetreatmentratesdenedbefore, arealsoassumedtobeconstant; pis
therateat whichprimarylatent-TB cases become permanent latent-TB
cases; and (t) and d(t) are functionsoftime.
EstimatesforsomeoftheseparametersarelistedinTable5.
Fig.9showsTable5.
EstimatedparametersofTBtransmissionfortheU.S.A..parameter c k
r1r2r3pestimation 0.22 10 0.01 0.05 0.05 0.65
0.1thatfortheselectedparameterranges,
theprogressionratefunctionA(t)tsthedataverywell.
Thatis,thetsuccessfullycapturesthepasthistoryofTBintheU.S.A..
Thevaluesofr1=r2=0.05meansthatinthepastonly5%oflatently-infectedindividuals
got treatedper year. The treatment of 100%of active-TBcases per
unit time (r3= 1, instead of 0.65) is insucient to reach CDCs goal
(seeDYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS 393Fig. 10).
However, treatment of morelatently-infectedindividuals,
forinstance,raisingr1andr2to20%peryear, wouldhelpreachCDCstargetof
1/1,000,000inamorereasonableperiodoftime(seeFig.s10and11). Fig.
12illustratestheeectofHIVonthecontrolofTB.ItisclearthatHIVdelaystheachievementofCDCsgoalbuthasnopermanentimpactonthelong-termpersistenceofTB.However,prolongingTBssurvivalenhancesthe
likelihoodof its evolution, asituationthat is not exploredhere. We
haveintroduced the impact of HIV/AIDS on TB progression during the
past two decadesviaatemporaryperturbationonthedistributionof
TBprogressiontimes. Ourselection of this perturbation is driven by
our desire to t active-TB data since ourprimarygoal
istolookatnotthecoevolutionof
co-infectionsbutatHIV/AIDSco-infections on the ability of the
U.S.A. to meet CDCs target by 2010. Our
modelsuggeststhatifemphasisisplacedontreatingatleast20%ofthelatently-infectedindividuals
then CDCs target may be met by 2020. Our model also shows that
re-emergence of diseases that compromise the immune system (or
recurrent
outbreaks)wouldmakeitverydiculttocontrolTBunlesstreatmentemphasisisputontheearlier(non-detectable)stagesofTBdisease.TheoreticallysucientconditionsforTBextinctionandpersistencearederivedintermsof
upperlimitsandlowerlimitsof
themortalityfunctionsforthenon-autonomousmodel.
Weplacethismathematicalresultinsubsection8.58.4.
Regressionapproach.
Topartiallybackupourconclusionsfromastatis-tical viewpoint,
weuseregressiontostudythetrendof
newTBcaseseachyear.Weletthenumberofnewcasesbetheresponsevariable,denotedbyY
,andtimebethepredictor, denotedbyX. AscanbeseenfromFig. 1,
thescatterplotofannual newcasesY versusyearXappearstobeexponential.
Intuitively,
aloga-rithmictransformationistakenontheresponsevariable.
Thequadratic regression1950 1960 1970 1980 1990 2000123456789x
104yearnew casespredicted valuesdataFigure9.
NewcasesofTBanddata.394 C. CASTILLO-CHAVEZANDB. SONG2010 2015 2020
2025 2030 2035 2040 2045 205005101520253035r1=r2=5%yearcase rate
per millionpredicted valuesCDCs goalFigure10. r1=r2=5%.
CDCsTBeliminationcannotbeachievedby2020.2010 2015 2020 2025
2030012345678910 r1=r2=20%yearcase rate per millionpredicted
valuesCDCs goalFigure 11. r1=r2=20%. CDCs
TBeliminationcanbeachievedby2020.isturnedouttobethebestt.
Theregressionequationlog Y= 11.3970 0.0597X + 0.0006X2, (98)is
thebest t. Fig. 13shows thettedcurve, 90%condencebands,
and90%predictionbands.DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS
3951950 1960 1970 1980 1990 2000 2010 20200123456789x 104yearnew
casesFigure12. Impactof HIV. ThelowercurverepresentsnoHIVeect;
theuppercurverepresentsthecaseratewhenHIVisin-cluded;botharethesamebefore1983.
Dotsrepresentrealdata.0 10 20 30 40
509.69.81010.210.410.610.81111.211.411.6yearlogcasesregression95%
CI95% PIFigure13. Quadraticregressionlog Y vs. X.
Dotsaretherealdata.
Condencebandsandpredictionbandsareincluded.Thequadraticregressionmodelturnsouttobeappropriateaftertheregressionassumptionsareveried.
Consequently,
wecanuseEquation(98)topredictthenumberofnewcasesinthenearfuture.
Theresultsshowsthatthepredictedcase396 C. CASTILLO-CHAVEZANDB.
SONGrate in 2010 is 7.1659/1, 000, 000. The same rate from our
deterministic model (9597)is5.2952/1, 000, 000.
Bothguresdrawthesameconclusion; thatis,
CDCsgoalsforTBeliminationareunrealisticwithintheproposeedtimehorizon.8.5.
Asymptotic behavior of thenon-autonomous model.
Anasymptoticanalysisof model
(9597)iscarriedoutandtheresultsarediscussed[72]
inthissectionsincetheyplayaroleintheparameterizationof themodel.
Theanalysishelpsestablishacriterionfordiseasepersistence(thatis,
athresholdcondition)whichmustbemetbytheparameterizedmodel.
Thelong-termbehaviorof
oursystemisdeterminedbytheasymptoticpropertyofthefunctionsN(t),
(t), andd(t).
Thefollowingtheoremcharacterizessuchbehavior.Theorem8.1. Assume
that liminft(t) = , liminftd(t) =d, andlimsupt(t) = ,limsuptd(t) =
d.i. If R=_kk++r1+p__+d+r3_ 1 then limtL1(t) = L2(t) =limtI(t)
=0.ii. IfR=_kk++r1+p__+d+r3_ 1thenlimsuptL1(t) > 0,limsuptL2(t)
> 0,andlimsuptI(t) > 0.Proof.
Wewillusethefollowingequalities,whicharestraightforwardinrealanal-ysis.
WheneverlimAexists,thefollowinglimitequalitieshold:limsup(A+B) =
limA+ limsupB,liminf(A+B) = limA+ liminfB,limsup(AB) =
limAlimsupB,liminf(AB) = limAliminfB.The proof is basedonthe
uctuationlemma[43] andits extensionbyThieme(seeTheorem2.3in[77]).
ApplyingTheorem2.3from[77]toEquations(95)and(97) directly, one
obtains that 0 I(+k+r1+p)L1andkL1(+ d+ r3)I.
ItfollowsthatI+k+r1+pk(+ d+ r3)I; thatis,I_kk++r1+p+d+r31_= I(R 1)
0. SinceR< 1andI(t)isbounded,itfollowsthatI= 0.
AsimilarargumentresultsinL= 0. Therstpartofthetheoremisproved.It is
not dicult to show that limsuptL1(t) = 0 if and only iflimsuptI(t)
=0. For instance, the fact that limsuptI(t) =0 implieslimsuptL1(t)
= 0isveriedbelow. FromEquation(95),weobtaindL1dt I ((t) +k +r1
+p)L1.ItfollowsfromthecomparisonprinciplethatL1(t) L1(0)
+_t0I(s)e_s0 (()+k+r1+p)ddse_t0(()+k+r1+p)d.Hence,limsuptL1(t)
limsuptI(t)e_t0(()+k+r1+p)d((t) +k +r2)e_t0(()+k+r1+p)d=
limsuptI(t)(t) +k +r1 +p=
0.DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS 397The same
argument can be used to show that limsuptL1(t) = 0 im-plies
limsuptI(t) =0. It is also clear that limsuptL1(t) =0
implieslimsuptL2(t) = 0fromEquation(96).
Twocasearehandledseparately:Case 1: If kL1(t) ((t)+k+r1+p)I(t)
holds for all t > 0, thendIdt> 0 directlyimplieslimsuptI(t)
> 0;Case2: IfkL1(t) < ((t) +k +r1 +p)I(t)holdsforallt >
0,thenlimsuptI(t) > 0. Thecase2isprovedbycontradiction.
SupposelimsuptI(t) =0,thendL1dt I(t)N(t)_N(t) (t) +d(t) +r3kI
L2I_((t) +d(t) +r3)((t) +k +r1 +p)kI=_ ((t) +d(t) +r3)((t) +k +r1
+p)k_I +o(I)_ (+d) +r3)(+k +r1 +p)k_I +o(I)=(+d) +r3)(+k +r1
+p)k(R1)I +o(I) > 0, fort >> 1.This implies that
limsuptL1(t) > 0, which contradicts the assumptionlimsuptI(t)
=0. Trajectories of system(9597) cannot intercept kL1(t) =((t)
+d(t) +r3)I(t)innitelymanytimesifI(t) 0ast ,becausedL1dt>
0whenkL1(t) = ((t) +d(t) +r3)I(t)wheneverR> 1.
Therefore,limsuptI(t) = 0implies that either kL1(t) > ((t) +d(t)
+r3)I(t) or kL1(t) ((t) +d(t) +r3)I(t)holdseventually.
Hence,limsuptI(t) > 0.When (t) and d(t) d (both constant), R= R=
R0gives the classicalbasicreproductivenumberR0= _1 +d +r3__kk + +r1
+p_whereistheeectivecontactrate;1r3++distheeectiveinfectiousperiod;andkr1+p++kis
theproportionof primarily-infectedindividuals whomakeit
totheactivestage.Thetheoremprovides conditions for dierentiationof
thetwoimportant bio-logical states:
diseaseeliminationorpersistenceforthisnon-autonomoussystem.Thesethresholdsarehelpful
notonlyinverifyingthereasonablenessofpublishedparametersbutalsointheselectionofreasonablerangesofunknownparameters.Ourmodel
generalizestheresultsestablishedforrelatedautonomoussystemsbyFengetal.[31]andSongetal.[74].9.
TBtransmittedbypublictransportation. In Argentina, the TB
incidencerateinthe1990swas42/100,000ofthepopulation,butthisvaluewasmisleadingsinceintheinnercityofBuenosAiresitwas160/100,000(fourtimeshigherthanthenationalaverage).
BuenosAireshas12millionpeople,and9.2millionpassen-gersarecarriedbythebussystem,accountingfor82%ofthemovementinpublictransportation[19].
Aspecicmodel targetingthepopulation(inBuenosAires)398 C.
CASTILLO-CHAVEZANDB.
SONGthatincorporatestheimpactofpublictransportation(buses)hasbeendevelopedbyCastillo-Chavezetal.
[19].ThecityisdividedintoNneighborhoods.
Eachneighborhoodisfurthersub-dividedaccordingtowhetherornotanindividual
frequentlytakesabus.
TypeIindividualsarethosewhoseldomtakebusesordonottakethematallwhiletypeIIindividualsarethosewhofrequentlytakebuses.
Anindividualofanytypefallsintooneof fourepidemiological
groupsatanytimesusceptible(S),
infectedbutnotinfectious(E),infectious(I),andtreated(T).Type I and
type II individuals have dierent levels of activity, which are
modeledbythecontactratesC1iandC2i , whereiindexestheneighborhood.
Themixingstructureof the populationis drivenbythe bus system,
whichdepends ontheaveragetimespentonthebusbythetypeImembersof
eachneighborhood. Todescribethemodel,letQji=
Sji+Eji+Iji+Tjibethetotalnumberofindividualsof type-jin the
ithneighborhood,j= 1, 2. The followingparameters are
required:ai=per-capitaaveragecontactrateoftypeIindividualsinneighborhoodi;bi=per-capitaaveragecontactrateoftypeIIindividualsinneighborhoodi;i=
per-capita rate of getting o the bus by type II individuals in
neighborhood
i;i=per-capitarateofboardingabusbytypeIIindividualsinneighborhoodi.Then,1i=averagetimeonabusbyatypeIIperson;ii+i=probabilityofstayinginthebus(typeIIperson);ii+i=probabilityofstayingothebus(typeIIperson).Proportionate
mixing is assumed [13]. The mixing probabilities are calculated
usingtheabovedenitions.P11i=aiQ1iaiQ1i+biii+iQ2iisthemixingprobabilityof
typeIindividualswithinthesameneighborhoodi;P12i=biii+iQ2iaiQ1i+biii+iQ2iisthemixingprobabilitybetweentypeIandtypeIIindi-vidualswithinthesameneighborhoodi;P21i=aiQ1iaiQ1i+biii+iQ2i_ii+i_isthemixingprobabilitybetweentypeIIandtypeIindividualswithinthesameneighborhoodi;P22i=biii+iQ2iaiQ1i+biii+iQ2i_ii+i_is
the mixing probability between type II
individualsintheithneighborhood;P22ij=bjjj+jQ2jN
i=1_biii +iQ2i__ii+i_ the mixing probability between type II
individ-ualsintheithneighborhoodandtypeIIindividualsinthejthneighborhood.ThesemixingprobabilitiessatisfyP11i+P12i=
1andP21i+P22i+N
j=1P22ij= 1.The recruitment rate jivaryacross neighborhoods
andtypes. The
naturalmortalityrateisassumedtobeidenticalforallneighborhoodsandallepidemi-ologicalgroups.
Treatmentratesri, progressionrateki,
andthelossofimmunityrate(treatedpersonbecomessusceptibleagain)
idependontheneighborhoodDYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS
399butnotonthetypes. ThemodelequationsaredSjidt= ji Bji(t) Sji+iTji
, (99)dEjidt= Bji(t) ( +ki)Eji, (100)dIjidt= kiEji ( +ri +d)Eji,
(101)dTjidt= riEji ( +i)Tji , (102)Qji= Sji+Eji+Iji+Tji ,wherei=1,
2, 3, . . . , N, andj=1, 2.
Thesuperscriptsrefertothetypeandthesubscriptsindexneighborhoods.
TheequationsoftypeIandtypeIIpopulationsandthepopulationacrossneighborhoodsarecoupledbytheincidenceratesB1i
(t)andB2i (t),whereB1i (t) = iC1i
Si_P11iI1iQ1i+Q2iii+i+P12iI2iii+iQ1i+Q2iii+i_,B2i (t) = iC2i
S2i__P21iI1iQ1i+Q2iii+i+P22iI2iii+iQ1i+Q2iii+i+N
j=1P22ijI2jjj+jQ2jjj+j__.This researchfoundthat the larger the
dierence of prevalence
betweenneigh-borhoods,thelargerthebasicreproductivenumber.
Afterestimatingtherelevantparameters, it was found that on average
100 people enter and leave the bus hourly,and that one TB infection
per 1,000 travelers was generated per hour of travel. Us-ing
another model they found bus travel could be responsible for about
30% of newcasesofTB[9,19].
TheyalsofoundthatvariationsinTBtransmissionweremostsensitivetotransmissionwithinthetransportationsystem.10.
Questionsandconclusions.10.1. Anoldprediction. In the context of TB
control, the importance of R0wasestablishedin 1937,more
thantwodecades before the introductionof rst dynam-ical model for
TB [36]. W. H. Frost, an epidemiologist at John Hopkins
University,addressedthefundamentalroleofthereproductivenumberinthefollowingway:However,
fortheeventualeradicationoftuberculosisitisnotnecessarythat
transmission be immediately and completely prevented. It is
necessaryonlythattherateoftransmissionbeheldpermanentlybelowthelevelatwhichagivennumberofinfectionspreading(i.e.,open)casessucceedinestablishing
an equivalent number to carry on the succession. If in succes-sive
periods of time, the number of infectious hosts is continuously
reduced,the end-result of this diminishing ratio, if continued long
enough, must beexterminationof tuberclebacillus..... This means
that under presentconditionsof
humanresistanceandenvironmentthetuberclebacillusislosingground,
andthattheeventual eradicationof
tuberculosisrequiresonlythatthepresentbalanceagainstitbemaintained.[36]400
C. CASTILLO-CHAVEZANDB. SONGTBis aslowdisease, andtherefore,
thebasicreproductivenumberR0plays afundamental role in the study
its dynamics and control. Its role (transcritical bifur-cation)
goes well with the observed downward trend of TB mortality and
incidencerates. In fact, it suggests that R0is moving downward as
parameters (naturally)change. Mathematically,
theresultsarenotsurprising, becausethemodelsusedaremodicationsof
theframeworksdevelopedbyKermackandMcKendrick[47]andRoss[66].10.2.
Challengingquestions. We have collected a number of dynamical
models,theresults, andinsightsthathavegeneratedinthestudyof
TBdynamics. Thehistorical evolution of dynamical models of TB
follows a common pattern in biologyfromlineartononlinear,
fromonestraintomultiplestrains,
fromhomogenoustoheterogeneous,fromdeterministictostochastic,fromempiricaltotheoretical(andviceversa).
Themodelsaregivenbysystemof dierenceequations, dierentialequations
(ODEs andPDEs), integro-dierential equations,
andMarkovchains.Althoughwehaveseenaremarkableprogressinthedevelopmentofatheoreticalframework
for the study of the dynamics of TB and other epidemiological
processes,theremanyinterestingandchallengingtopicsandquestionsremain.
Apartiallistincludesi.
ImmigrationAllmodelsessentiallyassumeclosedpopulations,ignoringtheeectsofim-migration.
Immigration is probably the critical factor in the generation of
newTBcases.
IntheU.S.A.alone,over40%oftotalnewcaseshavebeenamongimmigrantsinthepastfewyears.ii.
RaceandethnicityThereisevidenceshowingthatcaseratesofTBaredierentamongdierentgroupsofpeople.
Thesechangesmayberelatedtovariationsinsusceptibilitytothe tubercle
bacilli. We have seenacomplexMarkovchainmodel
thattakesthisintoaccount. However, moreworkisrequiredif
wearetobetterunderstandtheroleofraceandethnicityindiseasedynamics.iii.
GeneticsThemajorreductiononTBmortalityrateswasachievedlongbeforethein-troductionof
antibiotics. Canthisreductionbeexplained,
atleastpartially,bytheevolutionofhumansusceptibility?
RecentworkbyAparicioetal. [4]provideagoodstart.
TheworkonHIVandgeneticsbyHsuSchmitz[69,70]suggestsvaluableapproaches.iv.
SanitariumThesanitariumwaxedandwanedhistorically. Itplayedacritical
roleiniso-latingandcuringactive-TBcaseswhenantibioticswerenotavailable.
ModelsthatincorporatetheroleofisolationonTBcontrolarerare.
Frostconcludedthatitcoulddelaythenumberofcases[36].v. Global
dynamicsTheoretically, we characterized the global dynamics of a
few models. For mostmodels, the characterization of their global
dynamics remains an open question.Multiple strain models, like
(1014) and models with fast and slow
progression,like(35)shouldbefurtheranalyzed.vi. TimedependenceThe
case of time-dependent coecients is not only more realistic but
often nec-essary[73].
Time-dependentparametersleadtothestudyofnon-autonomousDYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS
401models.
Tostudythelong-termbehaviorofmodelswithtime-dependentco-ecients,
thresholdvalues (like the basic reproductive number)
needtobefurther developed. Obviously,the classic approach for
computing the basic re-productive number is not helpful [26]. We
found upper and lower limits for
thepersistenceanderadicationofTB,butitseemshardtogetthesharpthresh-oldexplicitly.
Themethodsof averagesbyMaetal. [55] maybehelpful inthestudyof this
problem. Time-dependent models provideauseful
wayofconnectingparameterstodata. Theworkof Aparicioetal. [4]
showsthisconclusively.vii. MeanlatentperiodThe distribution of the
latent period is unknown as well as its mean.
Knowledgeoftheshapeofthisdistributionseemscriticalforcontrol.
TheresultsofFengetal.[31]haveshownthatitmaynothaveanimportantqualitativerole,butitcertainlyplaysacriticalquantitativerole.REFERENCES[1]
R. M. Anderson, R. M. May and A. R. Mclean (1988). Possible
demographic impact of AIDSindevelopingcountries,
Nature,332:228234.[2] J. P. Aparicio, A. F. Capurro, andC.
Castillo-Chavez(2000).
Transmissionanddynamicsoftuberculosisongeneralizedhouseholds, J.
Theor. Biol.,206:327341.[3] J. P. Aparicio, A. F. CapurroandC.
Castillo-Chavez(2002).
Frequencydependentriskofinfectionandthespreadof infectiousdiseases,
inMathematical Approaches forEmergingand Reemerging Infectious
Diseases: An Introduction, C. Castillo-Chavez with S. M. Blower,P
van den Driessche, D. Kirschner, and A. A. Yakubu (Eds.), IMA,
Vol.125, Springer-Verlag,NewYork,341350.[4] J. P. Aparicio, A. F.
Capurro, C. Castillo-Chavez (2002a). Long-term dynamics and
reemer-gence of tuberculosis, in:
MathematicalApproachesforEmergingandReemergingInfectiousDiseases:
AnIntroduction,C.Castillo-ChavezwithS.M.Blower,P.vandenDriessche,D.Kirschner,A.A.Yakubu(Eds.),IMA,Vol.125,Springer-Verlag,NewYork,pp.351360.[5]
B. R. Bloom and C. J. L. Murray (1992). Tuberculosis: Commentary on
a reemergent killer,Science,257:10551064.[6] S.M.Blower,
A.R.McLean, T.C.Porco, P.M.Small, P.C.Hopwell, M.A.Sanchez,andA. R.
Moss(1995). Theintrinsictransmissiondynamicsof
tuberculosisepidemics, NatureMedicine,1(8):815821.[7] S. M. Blower,
P. M. Small, andP. C. Hopwell (1996). Control
strategiesfortuberculosisepidemics: Newmodelsforoldproblems,
Science,273:497500.[8] S. M. BlowerandJ. L. Gerberding(1998).
Understanding, predictingandcontrollingtheemergenceof
drug-resistant tuberculosis: Atheoretical framework, J.Mod.
Med.,76:624636.[9] D. Brand (1999). Tuberculosis spreads via
crowded city buses, biomathemtician
nds,CornellChronicle,Vol.30,Number20,February4,1999.[10] S. Brogger
(1967). Systems analysis intuberculosis control: Amodel, Amer. Rev.
Resp.Dis.,95:419434.[11] S. Busenberg and K. P. Hadeler (1990).
Demography and epidemics,Math.Biosci., 101:4162.[12] S.Busenberg
andP.van
denDriessche(1990).Nonexistenceofperiodicsolutionsforaclassof
epidemiological models, inDierential EquationsModelsinBiology,
EpidemiologyandEcology, S. Busenberg and M. Martelli (Eds.),
Lecture Notes in Biology 92,
Springer-Verlag,Berlin-Heidelberg-NewYork,7179.[13] S.
BusenbergandC. Castillo-Chavez(1991). Ageneral solutionof
theproblemof mixingsub-populations, andits applicationto
risk-andage-structuredepidemic models for thespreadofAIDS,IMAJ. of
MathematicsAppliedinMed. andBiol.,8:129.[14] J.Carr(1981),
ApplicationsCentreManifoldTheory,Springer-Verlag,NewYork.[15] C.
Castillo-ChavezandZ. Feng(1997), Totreatornottotreat: Thecaseof
tuberculosis,J. Math. Biol.,35:629656.402 C. CASTILLO-CHAVEZANDB.
SONG[16] C. Castillo-ChavezandZ. Feng(1998). Global stabilityof
anage-structuremodel
forTBanditsapplicationtooptimalvaccinationstrategies, Math.
Biosci.,151:135154.[17] C. Castillo-ChavezandZ. Feng(1998).
Mathematical modelsforthediseasedynamicsoftuberculosis, inAdvances
in Mathematical Population Dynamics: Molecules, Cells
andMan,O.Arino,D.Axelrod,andM.Kimmel(Eds.),WorldScientic,629656.[18]
C. Castillo-ChavezandW. Huang(2002). Age-structurecoregroupmodel
anditsimpactonSTDdynamics, in:. Mathematical
ApproachesforEmergingandReemergingInfectiousDiseases: Models,
MethodsandTheory, C. Castillo-ChavezwithS. M. Blower, P.
vandenDriessche, D. Kirschner, andA. A. Yakubu(Eds.), IMA, Vol.
126, Springer-Verlag, NewYork,261274.[19] C. Castillo-Chavez, A. F.
Capurro, M. Zellner, and J. X. Velasco-Hernandez (1998). El
trans-porte pblico y la dinmica de la tuberculosis a nivel
poblacional.AportacionesMatemaqticas,SerieComunicaciones,22:209225.[20]
A.J.E.Cave(1939). BritishJournal of Tuberculosis,33:142.[21] CDC
(1989). A strategic plan for the elimination of tuberculosis in the
UnitedStates,MMWR,38: (suppl.No.S-3)125.[22] CDC (1999).
Surveillance reports reported tuberculosis in the United States,
1999.http://www.cdc.gov/nchstp/tb/surv/surv99/surv99.htm.[23] A.
Cockburn and E. Cockburn (1980).
Mummies,disease,andancientCultures, Cambridge.[24] T. M. Daniel
(2000). Captainof death: Thestoryof tuberculosis, Universityof
RochesterPress,Rochester,NewYork.[25]
A.M.Debanne,R.A.Bielefeld,G.M.Cauthen,T.M.Daniel,andD.Y.Rowland(2000).Multivariate
Markovian modeling of tuberculosis: Forecasts for the United
States,EmergingInfectiousDiseases,6(2):148157.[26] O. Diekmann, J.
A. P. HeesterbeekandJ. A. J. Metz(1990).
Onthedenitionandcom-putationof
thebasicreproductiveratioinmodelsforinfectiousdiseasesinheterogeneouspopulation,
J. Math. Biol.,28:365382.[27] P. van den Driessche and J. Watmough
(2000). A simple SIS epidemic model with a bickwardbifurcation, J.
Math. Biol.,40:525540.[28] P. vandenDriesscheandJ. Watmough(2001).
Reproductivenumbersandsub-thresholdendemic equilibria for
compartment models of disease trsmission,Math.Biosci.,
180:2948.[29] R. DubosandJ. Dubos(1952). TheWhitePlague:
tuberculosis, man, andsociety, LittleandBrown,Boston.[30]
J.Dusho,W.Huang,andC.Castillo-Chavez(1998).Backwardbifurcationsandcatastro-pheinsimplemodelsoffataldiseases,J.
Math. Biol.,36:227248.[31] Z. Feng, W. Huang, andC.
Castillo-Chavez(2001). Ontheroleof
variablelatentperiodsinmathematicalmodelsfortuberculosis, Journal
of DynamicsandDierential Equations,13(2):425452.[32] Z. Feng, C.
Castillo-Chavez,andA. F. Capurro(2000). Amodel
fortuberculosiswithex-ogenousreinfection, Theor. Pop.
Biol.,57:235247.[33] Z. Feng, M. Iannelli, and F. Milner (2002). A
two-strain TB model with age-structure,SIAMJ. Appl.
Math.,62(5):16341656.[34] S. H. Ferebee(1967). Anepidemiological
model oftuberculosisintheUnitedStates, NTABulletin,January,47.[35]
J. A. Frimodt-M oller(1960). TuberculosisstudyinasouthIndiarural
populations, 19501955, Bull. WorldHealthOrgan.,22:413.[36]
W.H.Frost(1937).Howmuchcontroloftuberculosis, AmericanJournal of
PublicHealthandtheNationsHealth,27(8):759766.[37]
J.GukerhamerandP.Homes(1983).Nonlinearoscillations,dynamical
systems,andbifur-cationsof vectorelds,Springer-Verlag,NewYork.[38]
J.Guckenheimer(1996).Towardsaglobaltheoryofsingularlyperturbedsystems,ProgressinNonlinearDierential
EquationsandTheirApplications,19:214225.[39] K. P. Hadeler and K.
Ngoma (1990). Homogeneous models for sexually transmitted
diseases,RockyMountain, J. Math.,20:967986.[40] K. P. Hadeler
(1992). Periodic solutions of homogeneous equations, J. Dierential
Equations,95:183202[41]
K.P.HadelerandC.Castillo-Chavez(1995).Acoregroupmodelfordiseasetransmission,J.
Math.
Biosci.128:4155DYNAMICALMODELSOFTUBERCULOSISANDTHEIRAPPLICATIONS
403[42]
C.Herrera,S.Lima,R.Munoz,G.Ramos,A.Rodriguez,andC.Salzberg(1996).Amodeldescribingtheresponseof
immunesystemtoMycobacteriumtuberculosis,
DepartmentofBiometricsTechnicalReportSeries#BU-1364-M,BiometricsDepartment,CornellUniver-sity.[43]
W.M.Hirsch, H.Hanisch, andJ.P.Gabriel(1985).
Dierentialequationmodelsforsomeparasitic infections: methods for
the study of asymptotic behavor, Comm.PureAppl.Math.,38:733753.[44]
F. Hoppensteadt(1974).
Asymptoticstabilityinsingularperturbationproblems. II: Prob-lems
having matched asymptotic expansion
solutions,J.DierentialEquations, 15:510521.[45] W. Huang, K. L.
Cook,and C. Castillo-Chavez (1992). Stability and bifurcation for
amultiple-groupmodel forthedynamicsofHIV/AIDStransmission, SIAMJ.
Appl. Math.,52(3):835854.[46]
T.A.Kenyon,S.E.Valway,W.W.Ihle,etal.(1996).Transmissionofmultidrug-resistantMycobacterium
tuberculosis during a long airplane ight, NewEnglandJournal of
Medicine,334:933938.[47]
W.O.KermackandA.G.McKendrick(1927).Acontributiontothemathematicaltheoryofepidemics,
Proc. R. Soc.,A,115:700721.[48]
D.Kirschner(1999).Dynamicsofco-infectionwithM.tuberculosisandHIV-1,TheoreticalPopulationBiology,55:94109.[49]
G. Kolata(1995). Firstdocumentedcasesof
TBpassedonairlinerisreportedbyU.S.A.,TheNewYorkTimes,March3.[50] C.
M. Krisb-ZeletaandJ. X. Velasco-Hernandez(2001).
Asimplevaccinationmodel withmultipleendemicstates,Math.
Biosci.,164(2):183201.[51]
S.A.LevinandD.Pimentel(1981).Selectionofintermediateratesofincreaseinparasite-hostsystems,Am.
Nat.,117:308315.[52] T. LietmanandS. M. Blowe(2000). Potential
impactoftuberculosisvaccinesasepidemiccontrolagents, Clinical
InfectiousDiseases,30(Suppl. 3): S316322.[53] X. Lin, H. W.
Hethcote, andP. vandenDriessche(1993). Anepidemiological
modelsforHIV/AIDSwithproportionalrecruitment, Math.
Biosci.,118:181195.[54] A. M. Lowell, L. B. Edwards, andC. E.
Palme(1969). Tuberculosis, HarvardUniversityPress,Cambridge,MA.[55]
Z. Ma, B. Song, andT. G. Hallam(1989). The thresholdof survival for
systems inauctuatingenvironment, Bull. Math.
Biol.,51(3):311323.[56] R. M. MayandR. M. Anderson(1989).
Possibledemographicconsequenceof HIV/AIDSepidemics: II,
assumingHIVinfectiondoesnotnecessarilyleadtoAIDS,
inMathematicalApproaches toProblems inResourceManagement
andEpidemiology, C. Castillo-Chavez,S. A. Levin, andC. A.
Shoemaker(Eds.), LectureNotesinBiomathematics81,
Springer-Verlag,Berlin,220248.[57]
R.M.MayandR.M.Anderson(1985).Endemicinfectionsingrowingpopulations,
Math.Biosci.,77:141156.[58] J. W. McGrath(1988). Social networkof
disease spreadinthe lower Illinios valley: asimulationapproach, Am.
J. Phys. Anthropol.,77:483496.[59]
NewsinBrief(1999).WHOwarningforairpassengers, Lancet,353:305.[60]
T. C. PorcoandS. M. Blower(1998).
Quantifyingtheintrinsictransmissiondynamicsoftuberculosis,
Theoretical PopulationBiology,54:117132.[61] J. Raalli, K. A.
Sepkowitz, andD. Armstrong(1996)Community-basedoutbreaksof
tu-berculosis, Arch. Intern. Med.,156:10531060.[62] L. B.
ReichmanandJ. H. Tanne(2002). Timebomb: The global epidemic of
multi-drugresistanttuberculosis,McGraw-Hill,NewYork.[63] C. S.
ReVelle,W. R. Lynn, and F. Feldmann (1967). Mathematical models for
the economicallocationof tuberculosis control activities
indevelopingnations, Am. Rev. Respir. Dis.,96:893909.[64] C. S.
ReVelle (1967). The economics allocation of tuberculosis control
activities in developingnations,CornellUniversity.[65]
Reuters(1993). AgencycitiesurgentneedtoghtincreaseinTB.
TheNewYorkTimes,November16,C8.[66] R.Ross(1911). Thepreventionof
malaria,Murry,London.[67] A.A.SalyersandD.D.Whitt(1994). Bacterial
pathogenesis,WashingtonD.C.404 C. CASTILLO-CHAVEZANDB. SONG[68] P.
A. Selwyn, D. Hartel, andV. A. Lewisetal. (1989).
Aprospectivestudyof
theriskoftuberculosisamongintravenousdruguserswithhumanimmunodeciencyvirusinfection.N.
Engl. J. Med.,320:545550.[69] S-F. Shu Schmitz (2000a). A
mathematical model of HIV transimission in homosexuals
withgeneticheterogenity, Journal of Theoretical
Medicine,bf2:285296.[70] S-F. ShuSchmitz(2000b).
TreatmentandvaccinationagainstHIV/AIDSinhomosexualswithgeneticheterogenity,
Math. Biosci.,167:118.[71] B. Song, J. P. Aparicio, andC.
Castillo-Chavez(2002). Tuberculosismodelswithfastandslowdynamics:
Theroleofcloseandcasualcontacts, Math. Biosci.,180:187205.[72]
B.Song(2002).Dynamicalepidemicalmodelsandtheirapplications,CornellUniversity.[73]
B. Song andC. Castillo-Chavez (2001). Tuberculosis control inthe
U.S.A.: Astrategytomeet CDCs goal, Department of Biometrics
Technical Report Series
#BU-1562-M,BiometricsDepartment,CornellUniversity.[74] B. Song, C.
Castillo-Chavez, andJ. P. Aparicio(2002). Global dynamics of
tuberculosismodels with density dependent demography,
inMathematicalApproachesforEmergingandReemergingInfectiousDiseases:
Models,Methods,andTheory.,C.Castillo-ChavezwithS.M. Blower, P.
vandenDriessche, D. Kirschner, andA. A. Yakubu(Eds.), IMA, Vol.
126,Springer-Verlag,NewYork,275294.[75]
K.Styblo(1985).Therelationshipbetweentheriskoftuberculosisinfectionandtheriskofdeveloping
infectious
tuberculosis,BulletinoftheInternationalUnionagainstTuberculosis,60:117119.[76]
K. Styblo(1991). Epidemiologyof tuberculosis, SelectedPapers, 24,
Royal NetherlandsTuberculosisAssociation,TheHague.[77] H. R.
Thieme(1993).
Persistenceunderrelaxedpoint-dissipativity(withapplicationtoanendemicmodel),
SIAMJ. Math. Anal.,24(2): 407435.[78] U.S. Census Bureau(1975).
Historical statistics of the UnitedStates: Colonial times
to1970,GovernmentPrintingOceWashington,D.C..[79]
U.S.CensusBureau(1999). Statistical Abstractsof
theUnitedStates,119thedition,Wash-ington,D.C.[80] E. Vynnycky and
P. E. M. Fine (1997). The natural history of tuberculosis: The
implicationsof age-dependent risks of disease and the role of
reinfection, Epidemiol.Infect., 119:183201.[81] H. T. Waaler, A.
Gese, andS. Anderson(1962) Theuseof mathematical models
inthestudyoftheepidemiologyoftuberculosis, Am. J. Publ.
Health,52:10021013.[82] H. T. WaalerandM. A. Piot(1970). Useof
anepidemiological model
forestimatingtheeectivenessoftuberculosiscontrolmeasures, Bull.
WorldHealth. Org.,43:116.[83] W. F. Wells(1995). Aerodynamicsof
droplet nuclei, airbornecontagion,
andairhygiene,HarvardUniversityPress,Cambridge.[84] WHO(2001).
Global TuberculosisControl, WHOreport2001,Geneva.[85] WHO(2000).
Global TuberculosisControl, WHOreport2000,Geneva.[86] S. Wiggins
(1990). Introduction to Applied Nonlinear Dynamical Systems and
Chaos,Springer-Verlag,Berlin.[87] Y. Ye(Ed.) (1986) Theoryof limit
cycles, Translations of mathematical monographs
ByAmericanMathematicalSociety,66: 245260.[88]
E.Ziv,C.L.Daley,andS.M.Blower(2001)Earlytherapyforlatenttuberculosisinfection,Am.
J. Epidemiol.,153(4):381385.ReceivedonFeb. 27,2004.
RevisedonJune20,2004.E-mail address: [email protected]
address: [email protected]
