A mathematical analysis of the effects of control

AmathematicalanalysisoftheeffectsofcontrolstrategiesonthetransmissiondynamicsofmalariaC.Chiyakaa,*,J.M.Tchuencheb,1,W.Gariraa,S.DubecaDepartmentofAppliedMathematics,NationalUniversityofScienceandTechnology,P.O.BoxAC939Ascot,Bulawayo,ZimbabwebMathematicsDepartment,UniversityofTexasatArlington,Box19408,Arlington,TX76019,USAcDepartmentofAppliedBiology/Biochemistry,NationalUniversityofScienceandTechnology,P.O.BoxAC939Ascot,Bulawayo,ZimbabweAbstractWeformulateadeterministicmodelwithtwolatentperiodsinthenon-constanthostandvectorpopulations,inordertotheoreticallyassessthepotentialimpactofpersonalprotection,treatmentandpossiblevaccinationstrategiesonthetrans-missiondynamicsofmalaria.Thethresholdsandequilibriaforthemodelaredetermined.Themodelisanalysedqualita-tivelytodeterminecriteriaforcontrolofamalariaepidemicandisusedtocomputethethresholdvaccinationandtreatmentratesnecessaryforcommunity-widecontrolofmalaria.Inadditiontohavingadisease-freeequilibrium,whichislocallyasymptoticallystablewhenthebasicreproductivenumberislessthanunity,themodelexhibitsthephenomenonofbackwardbifurcationwhereastabledisease-freeequilibriumcoexistswithastableendemicequilibriumforacertainrangeofassociatedreproductivenumberlessthanone.Fromtheanalysiswededucethatpersonalprotectionhasapositiveimpactondiseasecontrolbuttoeradicatethediseaseintheabsenceofanyothercontrolmeasurestheefficacyandcom-plianceshouldbeveryhigh.Ourresultsshowthatvaccinationandpersonalprotectioncansuppressthetransmissionratesoftheparasitefromhumantovectorandvice-versa.Ifthetreatedpopulationsareinfectiousthencertainconditionsshouldbesatisfiedfortreatmenttoreducethespreadofmalariainacommunity.Amongtheinterestingdynamicalbehav-ioursofthemodel,numericalsimulationsshowabackwardbifurcationwhichgivesachallengetothedesigningofeffectivecontrolmeasures.2007ElsevierInc.Allrightsreserved.Keywords:Treatment;Vaccination;Malaria;Reproductivenumbers;Backwardbifurcation1.IntroductionPlasmodiumfalciparummalariaisstillamajorcauseofmortalityandmorbidityinthetropicalandsubtrop-icalareasoftheglobe,wherearound200millionpersonsareatconstantriskofinfection,withsomepartsof0096-3003/$-seefrontmatter2007ElsevierInc.Allrightsreserved.doi:10.1016/j.amc.2007.05.016*Correspondingauthor.E-mailaddress:[e-mail](C.Chiyaka).1Permanentaddress:DepartmentofMathematics,UniversityofDaresSalaam,P.O.Box35062,DaresSalaam,Tanzania.Availableonlineat(2008)641–662ffected[1].WHOrevealedthatmalariakillsatleastonemillionpeopleannuallyinsub-SaharanAfrica[2]withthepotentialtosignificantlyincreaseinresponsetoclimatechange(duetotheroleoftemperatureandrainfallinthepopulationdynamicsofitsmosquitovector)[3,4].MalariacasesarealsobeingexacerbatedbythehighlevelsofHIVinfection,thatweakentheimmunesystemrenderingpeoplewithHIVmoresusceptibletocontractingthedisease[5]anditalsoenhancesmortalityinadvancedHIVpatientsbyafactorofabout25%innon-stablemalariaareas[6].Sincemalariaincreasesmorbidityandmor-tality,itcontinuestoinflictmajorpublichealthandsocio-economicburdensindevelopingcountries.ItslowseconomicgrowthinAfricabyupto1.3%eachyear[7].Itsdevastatingimpacthasincreasedresearcheffortstofindaneffectivevaccinethatwouldstoptheprogressionandtransmissionofmalaria.Althoughthereisoptimismaboutdevelopingamalariavaccinesincetheyarebeingproducedandtested[8,9],malariacontrolcurrentlyreliesheavilyonpersonalprotectionandchemotherapy.Personalprotectionmeasuresarethefirstlineofdefenseagainstmosquito-bornediseases.Oneofthemeth-odsofpersonalprotectionistheuseofmosquitorepellents.Thesearesubstancesappliedtoexposedskinortoclothingtopreventhuman–mosquitocontact.Theseonlyrepelbutdonotkillmosquitoes.Theuseofinsec-ticide-treatedbednets(ITNs)forindividualsagainstmalariahasbeenshowntoreducemorbidityofchild-hoodmalaria(belowfiveyearsofage)by50%andglobalchildmortalityby20–30%[10,11].WhenusedonalargescaleITNsareconsideredtorepresentefficienttoolsformalariavectorcontrol.Thereishoweveralimitingfactorofresistanceintheinsecticidesusedforimpregnatednets.ResistanceofthemostimportantAfricanmalariavectorAnophelesgambiaes.l.topyrethroidisalreadywidespreadinseveralWestAfricancountries[12,13].TheeffortsofreducingthespreadofmalariathroughchemotherapyarenowbecomingmildlysuccessfulsincetheP.falciparumisbecomingresistanttocheapandavailabledrugssuchaschloroquine[14].Resistancetonewersecondandthird-linedrugscontinuestogrow.Unfortunatelymanyofthesenewdrugsarenotonlyexpensiveandhaveserioussideeffects,butmostwilleventuallyberenderedineffectivebythemalariaorgan-ism’scomplexepidemiologyandfacilityforrapidmutation.Massadministrationofantimalarialdrugsreducesprevalenceofmalariainfection.Suchinterventionsmaynotsustainablyreducetransmissionstrategies[15,16].Althoughchemotherapymayhavelittleeffectonthetransmissionofmalariawhereendemicityisintense[15–17]studieshavedemonstratedthatevenasub-optimaltherapeuticregimenreducesthedurationofinfectiousnesssufficientlytoterminateseasonalmalariatransmissionduringalarg