GLOBAL STABILITY OF TWO TUBERCULOSIS MODELS WITH TREATMENT AND SELF-CURE

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ROCKYMOUNTAINJOURNALOFMATHEMATICSVolume42,Number4,2012

GLOBALSTABILITYOFTWOTUBERCULOSISMODELSWITHTREATMENTANDSELF-CURE

YALIYANG,JIANQUANLIANDYICANGZHOUABSTRACT.Twotuberculosis(TB)modelswithtreat-mentandself-cureareinvestigated.ItisassumedthatthetreatedindividualsmayreentereitherthelatentcompartmentduetotheremainderofMycobacteriumtuberculosisortheinfectiouscompartmentduetothetreatmentfailure.Ontheotherhand,infectiousindividualswithmildsymptomsmayreenterthelatentcompartmentduetoself-cure.Afterinves-tigatingasimplemodelwithonelatentcompartment,aTBmodelwithtwolatentcompartmentsisstudiedtodescribetheslowandfastprogressionoftheexposedindividuals.Theba-sicreproductionnumbersofthosetwomodelsaredefined,andtheirepidemiologicalinterpretationisgiven.Similarthresh-olddynamicsisobtainedforthosetwomodels:thediseasediesoutultimatelywhenthebasicreproductionnumberislessthanorequaltoone;theuniqueendemicequilibriumisgloballystablewhenthebasicreproductionnumberisgreaterthanone.Theinfluenceoftreatmentfailureandself-cureonthebasicreproductionnumberisalsodiscussed.

1.Introduction.Tuberculosis(TB)isabacterialdiseasecausedbyMycobacteriumtuberculosis,whichisusuallyacquiredthroughair-borneinfectionfromactiveTBcases.Itisoneofthemostcommoninfectiousdiseasesandhasbeenaleadingcauseofdeathintheworldforcenturies.Aboutone-thirdoftheworld’spopulationisinfectedwiththeTBbacillus[1,14].Incomparisonwithotherinfectivedis-eases,followingprimaryinfection,mostoftheindividualsinfectedbyTB(approximately90percent)stayinalatentstage,andonlyasmallproportionofindividualsdeveloptheactiveTB.Therefore,introducingthelatentcompartmentisnecessaryformodelingthetransmissionofTB.

Keywordsandphrases.Tuberculosis,treatment,self-cure,equilibrium,globalstability.ThisworkwaspartiallysupportedbytheNatureScienceFoundationofShaanxi

ProvinceofChina(No.2012JQ1019)andtheNatureScienceFoundationofChina(No.11071256).ReceivedbytheeditorsonAugust4,2009,andinrevisedformonJanuary10,

2010.

DOI:10.1216/RMJ-2012-42-4-1367Copyrightc󰀁2012RockyMountainMathematicsConsortium13671368YALIYANG,JIANQUANLIANDYICANGZHOU

TherearevariousmathematicalmodelsonTBtransmission.In[4],thetreatmentTBmodelswithone-strainandtwo-strainareinvesti-gated.In[9],atwo-strainTBmodelwithageofinfectionisestab-lished,andtheefficiencyofthevariableperiodoflatencyondiseasedynamicsisanalyzed.In[8],theexogenousreinfectionisincorporatedinthetransmissionmodelofTB,andcomplexdynamics(backwardbi-furcation)isfoundundercertainconditions.In[13],aTBmodelwithcasedetectionandtwotreatmentstagesisdiscussed,andtheglobaldynamicsofthemodelisobtained.

In[4,13],itisassumedthatthetreatedindividualhasapartialimmunity,thatis,thetreatedindividualmaybeinfectedagain.Inthispaper,weassumethatthetreatedindividualsareisolated,andtheyarenotabletoinfectothersandcannotbeinfected.Afteraperiodoftreatment(usually612months)thetreatedoneleavestheisolatedenvironment.Sincethetuberculosisbacillusmayhidefromtheimmunesysteminitshost’sfatcells,thispathogenisprotectedagainsteventhemostpowerfulantibioticsinthesecells,inwhichitmayremaindormantforyears[15,16].MosttreatmentofinfectiousTBpatientsisincomplete.AfterbeingtreatedtheTBsymptomsofapatientmaydisappear,andtheremaystillbeafewtuberclebacillusinthebodyofthepatient;then,theseindividualsmaystillbeTBcarriersandbecomelatent,ormayreentertheinfectiouscompartmentduetothefailureoftreatment[2].

Inthispaper,incompletetreatmentandself-cureareincorporatedintwomodels,whichincludeoneandtwolatentcompartments,respec-tively.Self-curemeansthatinfectiousindividualsreverttothelatentcompartmentwithoutbeingtreated[3].ThebasicTBmodelwithtreatmentandself-cureisformulatedanddiscussedinSection2.SinceTBhasalonglatentperiodbetweenmonthsanddecades,andsomelatentTBcasesmayoccuralongwithotherdiseases(suchasinanHIVcarrier)whichcanweakenaperson’simmunityandshortenthelatentperiodofTB[7,17],wedividethelatentindividualsintotwocom-partments:slowandfastlatenttodescribethesloworfastprogressoflatentTBtoactiveTB.AgeneralTBmodelwithtreatment,self-cureandslowandfastprogressisformulatedandinvestigatedinSection3.Thedisease-freeandendemicequilibriaofthosetwomodelsarestudiedandthresholddynamicsisobtainedonthebasisofthebasicreproduc-