declinenaning

RESEARCHOpenAccessThebasicreproductiveratioofBarbour'stwo-hostschistosomiasismodelwithseasonalfluctuationsShu-JingGao1,2,Hua-HuaCao2,Yu-YingHe2,Yu-JiangLiu2,Xiang-YuZhang2,Guo-JingYang3,4andXiao-NongZhou1*AbstractBackground:MotivatedbythefirstmathematicalmodelforschistosomiasisproposedbyMacdonaldandBarbour'sclassicalschistosomiasismodeltrackingthedynamicsofinfectedhumanpopulationandinfectedsnailhostsinacommunity,inourpreviousstudy,weincorporatedseasonalfluctuationsintoBarbour'smodel,butignoredtheeffectofbovinereservoirhostinthetransmissionofschistosomiasis.
Inspiredbythefindingsfromourpreviouswork,themodelwasfurtherimprovedbyintegratingtwodefinitivehosts(humanandbovine)andseasonalfluctuations,soastounderstandthetransmissiondynamicsofschistosomiasisjaponicaandevaluatetheongoingcontrolmeasuresinLiaonanvillage,XingziCounty,JiangxiProvince.
Methods:ThebasicreproductiveratioR0anditscomputationformulaewerederivedbyusingtheoperatortheoryinfunctionalanalysisandthemonodromymatrixtheory.
ThemathematicalmethodsforglobaldynamicsofperiodicsystemswereusedinordertoshowthatR0servesasathresholdvaluethatdetermineswhethertherewasdiseaseoutbreakornot.
TheparameterfittingandtheratiocalculationwereperformedwithsurveillancedataobtainedfromthevillageofLiaonanusingnumericalsimulation.
SensitivityanalysiswascarriedoutinordertounderstandtheimpactofR0onseasonalfluctuationsandsnailhostcontrol.
Themodifiedbasicreproductiveratioswerecomparedwithknownresultstoillustratetheinfectionrisk.
Results:TheBarbour'stwo-hostmodelwithseasonalfluctuationswasproposed.
TheimplicitexpressionofR0forthemodelwasgivenbythespectralradiusofnextinfectionoperator.
TheR0sforthemodelrangedbetween1.
030and1.
097from2003to2010inthevillageofLiaonan,XingziCounty,China,with1.
097recordedasthemaximumvaluein2005butdeclineddramaticallyafterwards.
Inaddition,weprovedthatthediseasegoesintoextinctionwhenR0islessthanoneandpersistswhenR0isgreaterthanone.
Comparisonsofthedifferentimprovedmodelswerealsomade.
Conclusions:Basedonthemechanismandcharacteristicsofschistosomiasistransmission,Barbour'smodelwasimprovedbyconsideringseasonality.
TheimplicitformulaofR0forthemodelanditscalculationweregiven.
TheoreticalresultsshowedthatR0gaveasharpthresholdthatdetermineswhetherthediseasediesoutornot.
Simulationsconcludedthat:(i)ignoringseasonalitywouldoverestimatethetransmissionriskofschistosomiasis,and(ii)mollusicidingisaneffectivecontrolmeasuretocurtailschistosomiasistransmissioninXingziCountywhentheremovalrateofinfectedsnailsissmall.
Keywords:Schistosomiasisjaponica,Mathematicalmodel,Seasonalfluctuation,Basicreproductiveratio,Parameterestimation,Barbour'stwo-hostmodel*Correspondence:zhouxn1@chinacdc.
cn1NationalInstituteofParasiticDiseases,ChineseCenterforDiseaseControlandPrevention;KeyLaboratoryofParasiteandVectorBiology,MOH,WHOCollaboratingCenterforTropicalDiseases,Shanghai200025,ChinaFulllistofauthorinformationisavailableattheendofthearticleTheAuthor(s).
2017OpenAccessThisarticleisdistributedunderthetermsoftheCreativeCommonsAttribution4.
0InternationalLicense(http://creativecommons.
org/licenses/by/4.
0/),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedyougiveappropriatecredittotheoriginalauthor(s)andthesource,providealinktotheCreativeCommonslicense,andindicateifchangesweremade.
TheCreativeCommonsPublicDomainDedicationwaiver(http://creativecommons.
org/publicdomain/zero/1.
0/)appliestothedatamadeavailableinthisarticle,unlessotherwisestated.
Gaoetal.
Parasites&Vectors(2017)10:42DOI10.
1186/s13071-017-1983-1BackgroundSchistosomiasisjaponica,aparasiticdiseasecausedbySchistosomajaponicum,hasbeeninexistenceinthePeople'sRepublicofChina(P.
R.
China)forover2000yearswithconsiderablepublic-healthandeconomicsignificance[1,2].
Alarge-scalenationalschistosomiasiscontrolprogrammewasinitiatedinthemid-1950s[2,3],whenChina'spopulationwasapproximately600million.
Anestimated11.
8millionpeoplewereinfectedwithS.
japonicum[4].
Sustainedcontroleffortshavecontributedsignificantlytothedramaticreductionofbothtransmis-sionintensityandschistosomiasisdistributioninChinainthepastsixdecades[5–9].
Recentdatahaveshownthatprogresstowardschistosomiasiseliminationencountereddifficultiesandsetbackduetohighre-infectionrates,particularlyinlakeregionoftheendemicareas[10].
Forinstance,schistosomiasisre-emergedshortlyaftertheter-minationofTheWorldBankLoanProject(WBLP)attheendof2001[1,5,11].
InordertoachievethegoalofschistosomiasiseliminationinP.
R.
China,Chinesegov-ernmentstrengthenedthenationalschistosomiasiscontrolprogrammein2004.
Thismadeschistosomiasiscontroltoppriorityalongwiththelistofothercommunicabledis-easessuchasHIV/AIDS,tuberculosis,andhepatitisBinChina[4,12].
Moreover,arevisedstrategytoeffectivelycontrolschistosomiasisbyusingintegratedmeasuresinthenationalcontrolprogrammehasbeenimplementedsince2005[11].
Over40differentspeciesofwildanddomesticanimalshavebeenidentifiedasdefinitivehostsofS.
japonicum[13].
BovinesarethemajorreservoirsforS.
japonicuminthelakeandmarshlandregionsofsouthernChina[14,15].
Alargenumberofschistosome-infectedbovinesaredistributedintheseregionsandtheyexcretelargequantitiesofS.
japonicumeggs,themajorityofwhicharedepositednearorinthelake[14].
Thedailyfaecaloutputfromawaterbuffalo(~25kg)hasbeenestimatedtobeatleast100timesmorethanthat(~250g)fromanindividualhuman[16,17].
Itwasreportedthattheover-allprevalenceofS.
japonicumwas9.
6and7.
2%inwaterbuffaloandcattle,respectively,in1995[18].
Highpreva-lenceofS.
japonicuminfectioninbovinereservoirhostsisbelievedtobethemajorfactormaintainingactivetransmissionincertainareas.
Aclusterrandomizedinterventiontrialwhichwasdesignedtocomparethecontrol(humantreatment)andintervention(humanandbovinetreatment)insomevillagesconcludedthattheincidenceofhumanS.
japonicuminfectionisre-ducedwithadeclineintheinfectionratesofwaterbuf-faloes[19].
Inordertoremovebovinesasasourceofinfection,severaleffectivemeasures,includingreplacingcattlewithfarmmachinery,isolatingmarshland,andprohibitinggrazinginsusceptibleareas,havebeenimplemented.
Mathematicalmodellingisapowerfultooltostudythetransmissiondynamicsofschistosomiasis[20].
ItwasfirstproposedbyMacdonald[21].
Followingthispioneeringwork,manymathematicalmodelshavebeendeveloped,allofwhichshowgreatpotentialinaidingourunderstandingoftheinterplayofbiology,transmis-siondynamicsandcontrolofschistosomiasis[22–27].
In1996,Macdonald'smodelwasimprovedbyBarbour[28].
Ittracksdynamicsofbothinfectedhumanandsnailsinacommunity.
Barbour'smodelhasplayedanimportantroleinevaluatingpossiblecontrolstrategies[29].
Traditionally,schistosomiasismodelsassumethatallpa-rametersareconstant.
Arealworldenvironmentisobvi-ouslynon-stationary,andshouldincludeseasonalvariationsinsnailpopulation.
Infectionratesvarysea-sonallyduetonaturalfactors(i.
e.
changesinmoistureandtemperature)andsocialfactors(i.
e.
changesincon-tactrates).
Therefore,itismorerealistictoassumethattheinfectionratesareperiodicratherthanconstant.
Inourpreviousstudy[30],weconstructedtheBarbour'ssingle-hostmodelwithseasonalfluctuations(BSHSFmodel)andcalculatedthebasicreproductiveratiotoas-sesstheeffectofintegratedcontrolmeasuresagainstschistosomiasisinLiaonanvillage,XingziCounty,JiangxiProvince.
However,theimpactofthebovineres-ervoirhostonthetransmissionofschistosomiasisshouldnotbeignored.
ThisstudyaimstofurthermodifytheBarbour'stwo-hostmodelwithseasonalfluctuations(BTHSFmodel)andgiveanimplicitexpressionofthebasicreproductiveratioofschistosomiasisandcomputationmethod.
Also,wewillfurtherestablishsomeindexesthatpredictprevalencevariationcharacteristicsofschistosomiasis,andevaluatethepreventionandcontrolstrategiesforschistosomiasis.
MethodsMathematicalformulationTheBarbour'ssingle-hostmodelwithseasonalfluctua-tions(BSHSFmodel)inGaoetal.
[30]wasgivenbythefollowingequations:dPdtatΔy1PgP;dydtbtPΔP1yμy;8>:1AllvariablesofEq.
(1)aredescribedinTable1.
Thein-fectionratea(t)representstherateatwhichasinglede-finitivehostbecomesinfectedatunitdensityofinfectedsnailsattimet.
Theinfectionrateb(t)istherateatwhichsnailsbecomeinfectedattimet.
Thetwoinfec-tionratescanbecalculatedasfollows(see[28]):Gaoetal.
Parasites&Vectors(2017)10:42Page2of9aa1a2a3;bβ1β2β3β4;Here,α1denotestherateatwhichahosthascontactwithcontaminatedwaterperunittime,α2denotesthedensityofcercariae,α3denotestheprobabilitythatanencounterwithcercarialeadstotheestablishmentofanadultparasite,β1istherateofegg-laying,β2istheprob-abilityofaneggdevelopingintoamiracidium,β3istheprobabilitythatamiracidiumpenetratesasnail,β4istheprobabilitythatmiracidialpenetrationintoanuninfectedsnaildevelopsintoinfection.
Seasonalitywastakenintoconsiderationbyassumingthattheinfectionratesaandbinmodel(1)weretime-varyingandperiodic.
a(t),b(t)weretakenastheabsolutevalueofsinefunctionswith365daysperiod,whiletheaveragevaluesoffunctionsa(t)andb(t)peryear(365days)wereassumedtobeequaltothevaluesofpa-rametersaandbin[31],respectively.
Furthermore,weconsideredtheimpactsofbovineres-ervoirhostonthetransmissionofschistosomiasis.
TheBSHSFmodel(1)wasmodifiedandusedtocreatetheBTHSFmodel.
Thedynamicsofthemodelaregovernedbythefollowingdifferentialequations:dP1dta1tΔy1p1g1P1;dP2dta2tΔy1p2g2P2dydtb1tP1P1b2tP2P2Δ1yμy:8>>>>>>>>>:2AllvariablesofEq.
(2)aredescribedinTable2,wherea1(t)denotestherateofincidenceforonehumanhostatunitdensityinfectedbyoneinfectioussnailperunittime(oneday)attimet,a2(t)denotestherateofinci-denceforonebovinehostatunitdensityinfectedbyoneinfectioussnailperunittime(1day)attimet,b1(t)istherateofincidenceforonesusceptiblesnailatunitdensityinfectedbyoneinfectedhumanperunittime(1day)attimet,b2(t)istherateofincidenceforonesusceptiblesnailatunitdensityinfectedbyoneinfectedbovineperunittime(1day)attimet.
Inthisstudy,weassumethatb1isproportionaltob1,i.
e.
b2=kb1,wherekisthecoefficientratio.
AccordingtoZhaoetal.
[32]:kb2b1A2B2A1B1;whereAi(i=1,2)denotefaecesoutputperhumanandbovine,Bi(i=1,2)denotetheepgofhumanandbovine,respectively,hereepgisthenumberofeggspergramoffaeces.
Thenumberofeggsperbovineperdayismorethan105[33],andthenumberofeggsperhumanperdayisabout~2800–3000[34].
Furthermore,manyeffectivemeasures,suchasim-provedsanitationandhealtheducationareimple-mentedtocontrolthespreadofschistosomiasisamonghumanbeings.
Inthispaper,wechoosek=50.
Wealsotaketheinfectionratesas:a1ta10sinπt=365jj;a2ta20sinπt=365jj;b1tb10sinπt=365jj;b2ta20sinπt=365jj:8>>>:Sinceinfectionratesaiandbi(i=1,2)denotetheaver-agevaluesofai(t)andbi(t)peryear,respectively,wehavea11365Z3650a1tdt;a21365Z3650a2tdt;andb11365Z3650b1tdt:Thuswecanderivea10π2a1;a20π2a2;b10π2b1;b20π2kb1andTable1Interpretationofthemodel(1)ParameterInterpretationa(t)therateofincidenceforasingledefinitivehostatunitdensityofinfectedsnailsattimetb(t)therateatwhichaninfecteddefinitivehostcausessnailinfectionsattimetgtherecoveryratefordefinitivehostinfectionsΔthedensityofsnailsΣthedensityofdefinitivehostsμpercapitaremovalrateofinfectedsnailsAbbreviations:P,theprevalenceofinfectioninthedefinitivehostpopulation;y,theproportionofinfectedsnailsTable2Interpretationofthemodel(2)ParameterInterpretationa1(t)theinfectionratefromsnailtohumanattimeta2(t)theinfectionratefromsnailtobovineattimetb1(t)theinfectionratefromhumantosnailattimetb2(t)theinfectionratefrombovinetosnailattimetg1therecoveryrateforhumanhostinfectionsg2therecoveryrateforbovinehostinfectionsΔthedensityofsnailsΣ1thedensityofhumanhostsΣ2thedensityofbovinehostsμpercapitaremovalrateofinfectedsnailsAbbreviations:P1,prevalenceofinfectioninthehumanhostpopulation;P2,theprevalenceofinfectioninthebovinehostpopulation;y,theproportionofinfectedsnailsGaoetal.
Parasites&Vectors(2017)10:42Page3of9a1tπ2a1sinπt=365jj;a2tπ2a2sinπt=365jj;b1tπ2b1sinπt=365jj;b2tπ2kb2sinπt=365jj:8>>>>>>>>>>>>>:Thebasicreproductiveratio(R0)Forepidemiologicalmodels,thebasicreproductiveratio,whichisoftenintroducedasathresholdparameter,isdefinedastheexpectednumberofsecondaryinfectionsproducedbyasingleinfectiveindividualinacompletelysusceptiblepopulationduringitsentireinfectiousperiod[35].
R0hasoftenbeenusedtopredictthetrendofthediseasetransmissionandalsotoassesstheeffectsofcontrolmeasures.
Here,wegivetheimplicitformulaR0ofmodel(2).
ItisexpectedthattheinfectionwillpersistifR0isgreaterthanone.
AsR0increases,thespreadingrateofthediseasealsoincreases.
Thisimpliesthatmoreinterventioneffortsorimprovedcontrolmeasuresshouldbeimplemented.
WehavealsoshownthatR0isasharpthresholdvaluewhichdetermineswhetherthedis-easediesoutornotinAdditionalfile1:AppendixB.
Thatis,ifR01,thediseasewillbeendemic.
WithreferencetoBacar[36],abiologicallymeaning-fulthresholdvalueR0ofBTHSFmodel(2)wasderivedbyusingoperatortheoryinfunctionalanalysisandthemonodromymatrixoflinearperiodicsystemtheory.
Thenumericalcomputationofthebasicreproductivera-tiowascarriedoutusingthemathematicalprogramminglanguageMATLAB7.
1.
Also,sensitivityanalysiswascar-riedoutinordertoassesstheimpactofseasonalfluctu-ationsonR0StudyareaandestimationofmodelparametersLiaonanvillage,XingziCounty,JiangxiProvincewasse-lectedasthestudyarea.
Thebasicreproductiveratiowascalculatedbasedontheparametersinmodel(2).
Determinationofaccuratemodelparameterswastheprioritywhenthisthresholdwasapplied.
Thedetailsofthetechniqueusedinestimatingtheparametershavebeendescribedinourpreviousstudy[30].
Inthispaper,partoftheparametersinmodel(2)weredeterminedac-cordingtotheirbiologicalsignificance.
Inaddition,theremainingparameterswereestimatedbasedonthean-nualreportsurveillancedataofXingziCountyfrom2003to2010.
Themeaningsofparametersgi,Σi(i=1,2),μ,ωareconsistentwiththosereportedin[30](seeTa-bles1and2).
Theparametersarestatedasfollows;g1=0.
00093perday,μ=0.
0055perday,andg2=0.
00183perday(thedataaregivenbyWu[31]).
Thedensityofsnails,Δ(/m2),andthedensityofdefinitivehosts,Σ1,Σ2(/m2),wereestimatedfromannualreportsurveillancedata(seeTable3).
Itisassumedthatthecoefficientsofmodel(2)areconstant.
Bysettingtheright-handsideofequationsinmodel(2)tozero,thesteady-state(equilibrium)valuesfortheprevalenceofinfectioninhumans,bovinesandsnailswereobtained[28]11yt1MSy1=t1SMt2MSy1=t2SM;3Piyy1=tiSM;4wheretMS(i)=bi∑i/(μΔ)andtSM(i)=aiΔ/gi(i=1,2)arethetransmissionfactorsfordefinitivehosti.
Bysolving(3)and(4),weobtaintheprevalenceofinfectioninhumans,bovinesandsnailsfrom2003to2010.
ResultsCalculationofR0FollowingtheresultsofBacar[36]andWang&Zhao[37],weshowthatR0isequaltothevalueofλ,whereλisthepositiverootoftheequationρW365;0;λ1;5HereW365;0;λexp"Z3650g10b1tP1=λΔ0g2b2tP2=λΔa1tΔ=λa2tΔ=λμ!
dt#;λ>0:andρ(W(365,0,λ))isthespectralradiusofmatrixW(365,0,λ).
ThemathematicaldetailsthatwereusedtoderivetheexpressionforthebasicreproductiveratiocanbefoundinAdditionalfile1:AppendicesA,B(DerivationofR0andproofofmainresults).
NumericalsimulationBasedontheannualreportdataforXingziCountyfrom2003to2010,theequilibriumvaluesP1;;P2;y(preva-lenceofinfectioninhumanhost,bovinereservoirhostandsnail)areshowninTable3.
Inviewoftheparame-ters'valuesgivenabove(seeTable3),theinfectionratesa1,a2,b1andb2ofBTHmodelwasderivedfromEqs.
(3)and(4),andareshowninTable4.
InAdditionalfile1:AppendixA,weillustratethatR0isthepositiverootoftheequationρ(W(365,0,λ))=1.
UsingthedatainTables3and4,thevaluesofR0forGaoetal.
Parasites&Vectors(2017)10:42Page4of9BTHSFmodelwerecalculatedandshowninTable5.
LetR0′bethebasicreproductiveratioforBSHSFmodel.
Itwascalculatedusingestablishedmethods(seeTable6).
Barbour[28],proposedthefollowingmodelwithoutseasonality(BTHmodel):dP1dta1Δy1P1g1P1;dP2dta2Δy1P2g2P2;dydtb1P1P1b2P2P2Δ1yμy;8>>>>>>>>>:andobtainedthebasicreproductiveratioR0a1b1P1g1μa2b2P2g2μ:UsingthedatainTables3and4,weobtainthevaluesofR0inthevillageofLiaonanfrom2003to2010.
R0forBTHSFmodelisshowninTable3,anditclearlyshowsthatthevaluesrangebetween1.
030and1.
097from2003to2010inthevillageofLiaonan.
Moreover,R0forBTHmodelisalwaysgreaterthanthatofBTHSFmodelatthesametime.
Furthermore,thevariationten-dencyofR0forBTHSFmodelisshowninFig.
1.
Theprevalenceoftwohostsandvectorhosthasstrongef-fectonR0forBTHSFmodel.
Exceptin2003and2010,thevaluesofR0forBTHSFmodelaregreaterthanthoseforBSHSFmodelintheotheryears(seeTable4).
Thepreva-lenceofinfectedbovinesin2003and2010arelessthantheotheryears(seeTable2).
ThisphenomenonillustratesthatiftheprevalenceratioofinfectedbovinehostsandinfectedhumanhostsP2=P1issmall,R0forBTHSFmodelmaybesmallerthanforBSHSFmodel.
Moreover,wenotethatR0andtheprevalenceofinfectioninbovinesP2forBTHSFmodelarehigherin2004and2005amongtheyearsunderstudyasshownin(Tables1and3).
SensitivityanalysisInmodel(2),infectionratesa1(t)andb1(t)(i=1,2)weretakenassinefunctionsof365daysperiod,withtheforma1tπ2a1sinπt=365jjandb1tπ2b1sinπt=365jj.
Thus,πai/2andπbi/2reflecttheamplitudeofinfectionratesai(t)andbi(t),respectively.
Wefixg1=0.
00093,g2=0.
00183,Δ=91.
4602,Σ1=0.
0475,g1=0.
00093,Σ2=0.
0020,μ=0.
0055,a2=0.
0426,b2=0.
1224,andvarya1andb1in[0,0.
04]in(5),withotherparametersun-changedasabove,bynumericalsimulationsweobtainthecurveofR0withrespecttoa1andb1(seeFig.
2).
Itshowsthatthelargera1andb1arethehigherR0is.
Whena1andb1arenearto0,R0islessthan1,whichindicatesthatthecontrolstrategyisveryeffectiveanddiseasetransmissionwillbeinterrupted.
ThisimpliesthatR0isverysensitivetothechangesina1andb1whentheirvaluesarehigherthan0.
03.
Next,ifwefixa1=0.
0068,b1=0.
0024,andleta2andb2vary.
Agraphindi-catingtherelationshipbetweenR0anda2andb2wasobtained(Fig.
3).
ThisgraphshowsthatR0increaseswithanincreaseintheamplitudeofa2andb2.
Finally,asμvariesin[0,0.
012]withotherparametersunchangedasabove,numericalsimulationsprovidetherelationshipbetweenR0andμ(Fig.
4).
Figure4showsthatR0ismoresensitivetochangesinμwhenμislessthan0.
002.
DiscussionThispaperproposedBarbour'stwo-hostmodelwithsea-sonality(BTHSFmodel),whichisanimprovementonTable3ThedataforΔ,Σ1,Σ2,P1,P2andyfromannualreportdatainthevillageofLiaonan,XingziCounty,JiangxiProvince,ChinaParameter20032004200520062007200820092010Δ27.
99733.
923091.
4602133.
523841.
061517.
496011.
31212.
4183Σ10.
03000.
01990.
04750.
01670.
03090.
01780.
03090.
0176Σ20.
00130.
00070.
00200.
00060.
00100.
00060.
00130.
0008P10.
06710.
05330.
06310.
04540.
04540.
04540.
045300.
0457P20.
02830.
13790.
17560.
11440.
10840.
06760.
06390.
0466y0.
00020.
000050.
00010.
00010.
00020.
00050.
00080.
0015Table4Thecalculatedvaluesofcompositeparametersa1,a2,b1andb2Parameters20032004200520062007200820092010a10.
01160.
03090.
00510.
00240.
00610.
00500.
00510.
0120a20.
00930.
17260.
03200.
01280.
03050.
01510.
01440.
0241b10.
00820.
00150.
00320.
02600.
00570.
01710.
00850.
0075b20.
40840.
07540.
16061.
29960.
28710.
85650.
42680.
3725Gaoetal.
Parasites&Vectors(2017)10:42Page5of9themodelsBTHandBSHSF.
FollowingtheideaofWang&Zhao[37],theimplicitformulaofR0forBTHSFwasgiven.
Bycomputation,theannualchangingtrendofR0sinLiaonanvillagefrom2003to2010wasgiven(seeFig.
1).
ItshowedthatR0peakedin2005at1.
09anddeclineddramaticallyuntil2010.
Therearethreepossiblefactorsresponsibleforthisdecline.
First,in2001,theWorldBankLoanProject(WBLP)forschis-tosomiasiscontrol(1992–2001)wasterminated.
Thisledtotheresurgenceofschistosomiasistransmissionafter2001[38,39].
Secondly,theimplementationofinte-gratedcontrolstrategieswithemphasisoninfectionsourcecontrolcommencedin2005.
Finally,thehighprevalenceofS.
japonicuminbovinescontributestothehighvalueofR0.
Bynumericalsimulation,werealizethatthebasicre-productivenumberforBarbour'smodelwithconstantinfectionratesisalwayshigherthanthatofperiodicin-fectionrates.
Thisillustratesthatignoringseasonalfluc-tuationswouldleadtooverestimatingschistosomiasistransmissionrisk.
Chemotherapy,healtheducation,provisionofwater,sanitationandhygiene,bovinecontrolandsnailhostcontrolarethemainmeasuresforschistosomiasiscon-trol.
Weknowthattheimplementationofchemotherapystrategyisbeneficialtotherecoveryandwell-beingofpatientsandcattle.
Thiswouldresultinincreasingtheparametersg1andg2(recoveryrates)inmodel(2).
Ac-cordingly,theimplementationofcontrolstrategies,suchashealtheducation,provisionofwater,sanitationandhygiene,bovinecontrolwouldleadtodecreasein;thehost'scontactratewithcontaminatedwater,theprob-abilitythatanencounterwithcercariaeresultsinthede-velopmentofanadultparasiteandtherateatwhichschistosomeeggsarelaid.
Therefore,theinfectionrates(a1(t),a2(t),b1(t),b2(t))inmodel(2)woulddecrease.
Moreso,theimplementationofsnailcontrolstrategywillincreasetheremovalrateofthesnail(μ).
DuetotheimplementationofpreventivechemotherapyinXingziCounty,weassessedthesensitivityanalysisoftheaver-ageinfectionratesandremovalrate.
NumericalresultshaveshownthatR0isamonotoneincreasingfunctionofthefourinfectionratesandamonotonedecreasingfunctionoftheremovalrate.
Thisimpliesthatasthein-fectionratesincreasesortheremovalratedecreases,R0increases.
Ourresultsalsoindicatethatmollusicidingisaneffectivemeansofcontrollingschistosomiasistrans-missioninXingziCountywhentheremovalrateofsnailsissmall.
Finally,inlinewiththetheoreticalconceptofthispaper,weintroduceseveralindexesthatwereexploredextensively.
Similartothemethodsestablishedin[28],R0sareestimatedfromprevalencedataandassumedtobeatasteadyequilibriumstate,sotheyarealwaysmorethan1.
Inreality,R0maybelessthan1.
However,wecanstudythetransmissionintensitybyusingthevaluesofR0.
Forexample,ifwedefinetherelativebasicrepro-ductiveratioas:RjiR0jR0iwhereR0kdenotesthebasicreproductiveratioofyeark.
IfRij>1,thetransmissionstatusofschistosomiasisinyearjismoreseriousthanyeariandviceversa.
Next,wedefinetherelativemeanbasicreproductiveratioasmR0mR0Here,R01ji1XjkiR0km>1showsthatthebasicreproductiveratioinyearmisgreaterthantheaverageratiofromyearitoj,andviceversa.
TakingLiaonanvil-lageastheexample,therelativebasicreproductiveratioin2005and2006isdefinedas:Table5TheresultofthebasicreproductiveratiofromannualreportdatainthevillageofLiaonan,XingziCounty,JiangxiProvince,ChinaThebasicreproductiveratios20032004200520062007200820092010R0(BTHSFmodel)1.
0311.
0741.
0971.
0601.
0571.
0381.
0371.
0300(BTHmodel)1.
0521.
1421.
1921.
1141.
1071.
0661.
0641.
050Table6TheresultofthebasicreproductiveratiofromannualreportdatainthevillageofLiaonan,XingziCounty,JiangxiProvince,ChinaThebasicreproductiveratios20032004200520062007200820092010R0(BTHSFmodel)1.
0311.
0741.
0971.
0601.
0571.
0381.
0371.
030R0'(BSHSFmodel)1.
0431.
0351.
0401.
0311.
0311.
0311.
0311.
032Gaoetal.
Parasites&Vectors(2017)10:42Page6of9R20062005R02006R020051:121:200:9Naning:NanjingUniversityPress;1998.
19.
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