ModellingoftheHFMDwiththeCarrierPopulation

RuzhangZhao1,LijunYang1*1DepartmentofMathematicalScience,TsinghuaUniversity,ChinaRuzhangZhao:zrz15@mails.tsinghua.edu.cn*LijunYang:lyang@math.tsinghua.edu.cn

Abstract: Thispaperfocusesonthemodellingofthehand,footandmouthdisease(HFMD)inpeoplecarryingthepathogen.TheHFMDhasbecomewidespreadinChina,andtheChinesepublichealthhasbeenseriouslythreatened.ThemodellingoftheHFMDhasalreadytakenseveralaspectsintoconsiderationbutinthetraditionalmodellingofinfectiousdisease,thepeoplewhoarenotinfectedbutcarrythepathogenareignored.Inthispaper,the“carrier”isincludedinthemodellingoftheHFMDandthenewlydevelopedmodel(SIRC)isprovedtoplayakeyroleinthemodellingandstabilityanalysisoftheHFMD. Keywords:HFMD,Carrierpopulation,SIRCmodel,Stabilityanalysis.

1. IntroductionThe existing solutions tomethods for the development of infectionmodels have beenwidely taken intoconsideration for a long time. Additionally, the application of modeling to control the spread of someinfectiousdisease isalsowidelytaken intoaccount.Differentkindsof infectiousdiseaseswillsharemanyimportantcharacteristics,suchaswhetherthepeoplewhorecoverfromthemwillbecomeimmunetothem.However,thespecificcharacteristicofsomeinfectiousdiseaseswillbelargelydifferentfortheirspread.Theprevious models for the spread of infectious diseases do not really work for the hand, foot andmouthdisease(HFMD),duetoitsspecificcharacteristic. The most significant feature of the HFMD is that people older than 10 years of age have lifelong

immunization and its pathogen can be spread via interpersonal communication (Centers for DiseaseControlandPrevention,2017).IfweignorethisfeatureandsimplyapplytheexistingmodelstothespreadoftheHFMD,thecriterionfortheexistenceofthedisease-freeequilibriummaybesatisfied.However,ifweapplytheSIRCmodeltothemodellingprocess,wemayobtainaresultthatiscontradictorytothepreviousresult,which canbe clearly seen in the following simulation section. The result appeals to us because ifpoliciesaremadeunder theprevious result, theexistenceof theendemicequilibrium is ignoredand thecontrolof theHFMDwillbelowerthanrequired,which isdangerous forthepublic.Thematerialalreadypresentwillbringtheproblemintothediseasetransmissioncoefficientatagiventime(RoyNetal.,2012).However, the changeof thedisease transmission coefficient isnoteasytoevaluate.Now,wecanprovideanother solution to theproblem.We can extend themodels further by including the carrier population,peoplewhoarenotinfectedbutcarrythepathogen,inthemodellingprocess,whichhasneverbeentriedinpreviousmodellingstudies.ThenewmodeliscalledtheSIRCmodel,whichisthemaintopicofthispaper.TheHFMDiscloselyconnectedwiththekindofpeoplementionedabove,becausetheadultscarryingthe

pathogenwillbeanimportantreasonforchildrencatchingHFMD.Thatmakesthemodeldescribedinthispaperreasonable. TherestofthepaperconsistsofthreesectionsafterthedevelopmentoftheHFMDmodelincludingthe

carrierpopulation.Insection2,wediscussthedisease-freeequilibrium.Section3providesthediscussionon endemic equilibrium.Section 4 comprises the overall discussion of thepaperand section 5provides

somesimulationsusingtheSIRCmodel.

2. AssumptionsandnotationsofthemodellingThis section gives the assumptions made in the modelling and lists the basic notations used in the paper.

AssumptionsofthemodelInthemodelling,thefollowingassumptionsarealwaysmade:Thepopulationisdividedintofourgroups:susceptible(S),infected(I),removed(R)andcarrier(C).The

removedpopulationgroupcomprises thepeople immuneto theHFMDandthe carrierpopulationgroupcomprises the people carrying the pathogen but who are not infected. The total number of people isdenoted as N, which equals to the sum of people included in the S, I, R and C population groups. Thevariableschangewithtime,whichmeansthattheyareafunctionoftime. Itisassumedthattheindividualswhoareolderthan5yearsofagehavelifelongimmunization,because

theHFMDisadiseasecausedbyagroupofvirusesthatmainlyinfectpeopleyoungerthan5yearsofage.AnditisassumedthatallthenewbornchildrenaresusceptibletoHFMD.ItisassumedthatthepeoplewhorecoverfromtheHFMDhavelifelongimmunization.The population is growing at a low speed, whichmeans that the natural birth rate is larger than the

naturalmortality rateandwedonot consider that thepeoplemove inandout the city.Additionally, it isassumedthatthenaturalbirthrateislargerthanthemortalityrateduetoHFMD.Thestandardincidenceratioisused.

NotationsofthepaperInthispaper,thefollowingnotationsareused. Susceptible,S;infected,I;removed,R;carrier,C;totalnumberofpeople,N;ratioofStoN,s;ratioofItoN,

i;ratioofRtoN,r;ratioofCtoN,c.Naturalbirthrate,b;naturalmortalityrate,d;contactcoefficientbetweenthesusceptibleandinfected

population, ;contactcoefficientbetweensusceptiblepopulationandcarrierpopulation, ;ratioofpeoplewhogetlifelongimmunizationatanageabove10yearsold(CentersforDiseaseControlandPrevention,2017), ;mortalityrateduetoillness, ;recoverycoefficientoftheinfectedpeople, ;transferredcoefficientfromthesusceptiblepopulationtothecarrierpopulation, ;exclusivecoefficientofthecarrierpopulation, .Thestandardincidencebetweenthesusceptibleandinfectedpopulation, ;thestandard

incidencebetweenthesusceptibleandinfectedpopulation, .

3. ModellingoftheHFMDwiththecarriers ThefollowingflowchartillustratesthepropagationofHFMDasobtainedbythemodel.

Where denotes the number of new-borns and is the number of new-borns deaths due tonatural causes. denotes the fraction of susceptible individuals that are infected by

infectedandcarrier individuals. and denote thenumberof individualswhoare removed

and the fraction of carrier individuals respectively. denotes the infected population who are

killedbyHFMDandnaturalcausesofdeath,and isthefractionoftheinfectedpopulationwhorecoversfromthedisease. and alsodenotethenumberofnaturaldeathsof theremoved individualsand

β β1

σ α γ

ζρ

βSI / N

β1SC / N

bN dSβSI / N + β1SC / N

σS ζSI / N(d +α )I

γ IdR dC

carrier individuals respectively. is the fraction of the removed populationwho become carrier

individualsand isthefractionofthecarrierindividualswhoaremovedinto‘R’populationgroup.

Fig.1.ModellingoftheHFMDwiththecarrierpopulation.

Fromtheflowchartdepictedabove,thefollowingsystemofdifferentialequationscanbederived.

(1)

The sum of the four differential equations is . Using the transformation

, the followingsystemofdifferentialequations,whichhasbeennormalized,can

beobtained.

(2)

Then,eliminatingthevariabler,thesystemwiththreevariablesisobtained.

ζ RI / NρC

dSdt

= bN − (d +σ )S − (β +ζ ) SIN

− β1SCN

dIdt

= β SIN

+ β1SCN

− (d +α + γ )I

dRdt

=σS + γ I + ρC − dR −ζ RIN

dCdt

= ζ RIN

+ζ SIN

− (ρ + d)C

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

dN / dt = (b − d)N −α I

S, I ,R,C( ) = N s,i,r,c( )

dsdt

= b − (b +σ )s − (β +ζ −α )si − β1sc

didt

= βsi + β1sc − (b +α + γ )i +αi2

drdt

=σ s + γ i + ρc − br + (α −ζ )ri

dcdt

= ζ ri +ζ si − (ρ + b)c +αci

1= s + i + r + c

⎧

⎨

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

(3)

4. Basicreproductionnumber Thebasicreproductionnumber(BRN)inthenormalizedsystemneedstobemodified.Whenthereareonlysusceptible individuals in the system, the sum of the ratio of the individuals infected by one infectedindividual tothetotalpopulationandtheratioofthe individuals infectedbyonecarrier individual tothetotalpopulationiswhatweredefinedastheBRN.

Letting the right-handsideof the system(3)beequal to zero,weobtain thedisease-freeequilibrium,whichmeansthatwhenthereareonlysusceptibleindividualsandtheratioofthemis

,thesystemisinequilibrium.

When only the recovery is considered, the average morbidity cycle is , and when the natural

mortalityandthemortalityduetoillnessarebothadded,theaveragemorbidityperiodis

Likewise,theaveragecarryingperiodofthecarrierindividualsis .

According the definition of the BRN, we assume that there are only susceptible individuals. The

individuals infectedbyone infectedperson is .Thecarrier individual isequal to ,so

theindividualsinfectedbythecarrierindividualis .

Thus,BRNisequaltothesumofthetwokindsofindividuals.

(4)

5. Disease-freeequilibriumThe disease-free equilibrium was obtained above, and the local stability and global asymptotic stability of the disease-free equilibrium will be discussed next.

Localstabilityofdisease-freeequilibriumTheorem4.1:When ,thedisease-freeequilibriumislocallyasymptoticallystable.When ,thedisease-freeequilibriumisunstable.

Proof:Thedefinitiondomainis .

Denotethedisease-freeequilibriumas ,thenthefollowingJacobianmatrixcanbeobtained.

(5)

Letusdenotethreeeigenvaluesas , and .Itistheneasytogetthethreefollowingequations.

dsdt

= b − (b +σ )s − (β +ζ −α )si − β1sc

didt

= βsi + β1sc − (b +α + γ )i +αi2

dcdt

= ζ i −ζ i2 − (ρ + b)c + (α −ζ )ci

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

U0 (b / (b +σ ),0,0)b / (b +σ )

1/γ1/ (b +α + γ )

1/ (ρ + b)

βs0 / (b +α + γ ) ζ

ζβ1s0 / (ρ + b)(b +α + γ )

R0 =βs0

(b +α + γ )+ ζβ1s0

(ρ + b)(b +α + γ )= bβ

(b +σ )(b +α + γ )+ bζβ1

(b +σ )(ρ + b)(b +α + γ )

U0 (b / (b +σ ),0,0)

R0 < 1R0 > 1

(s, i,c) 0 ≤ s ≤ 1,0 ≤ i ≤ 1,0 ≤ c ≤ 1{ }(s0, i0,c0 )

−(b +σ ) −(β +ζ −α )s0 −β1s0

0 βs0 − (b+α +γ ) β1s0

0 ζ −(ρ + b)

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ξ1 ξ2 ξ3

(6)

Using the condition “ ”, we can get , , , which can also be

transformed into , and . Thus, the disease-free equilibrium is locallyasymptoticallystable.

Using the condition“ ”,we canget ,wheretheremustbeoneeigenvalue larger thanzero.Thus,thedisease-freeequilibrium isunstable.

Globalasymptoticstabilityofdisease-freeequilibrium

Theorem4.2: .When , thedisease-freeequilibrium is

globallyasymptoticallystable.Proof:Tonarrowthedefinitiondomainof“s”,thefollowingdifferentialinequalitywillbeapplied.

(7)

Thedomainof“s”canbeeasilyobtained .Usingthemethodofundeterminedcoefficients,theLiapunovfunction canbeconstructed.

(8)

(9)

(10)Letthecoefficientatthefrontof“c”beequaltozero.Wecanthenget .We can also assumed that , . Also, using the condition ,

canbeobtained.

(11)

As ,wecanget .

Then,let ,thesolutiontotheequationis .

Inconclusion, , and iff .Also, istheglobalattractivesetunderthe

condition .Thus,thetheoremhasbeenproofed.

6. Endemicequilibrium

ξ1 = −(b +σ )ξ2 + ξ3 = βs0 − (b +α + γ )− (ρ + b)ξ2ξ3 = [−βs0 + (b +α + γ )](ρ + b)−ζβ1s0 = (b +α + γ )(ρ + b)(1− R0 )

⎧⎨⎪

⎩⎪

R0 <1 ξ1 < 0 ξ2 + ξ3 < 0 ξ2ξ3 > 0ξ1 < 0 ξ2 < 0 ξ3 < 0 U0

R0 >1 ξ2ξ3 < 0U0

R0! = β

(b +α + γ )+ ζβ1

(ρ + b)(b +α + γ ) R0! ≤1 U0

dsdt

= b − (b +σ )s − (β +ζ −α )si − β1sc ≤ b − (b +σ )s

−s0 ≤ (s − s0 ) ≤ 0L

L = k1i+ k2c

dLdt

= k1didt+ k2

dcdt

= k1[βsi + β1(s − s0 )c + β1s0c − (b +α + γ )i +αi2 ]+k2[ζ i −ζ i2 − (ρ + b)c+ (α −ζ )ci]

dLdt

= k1β1(s − s0 )i + c[k1β1s0 − k2(ρ + b)]+

i[k1βs + (k1α − k2ζ )i + k2(α −ζ )c+ k2ζ − k1(b+α +γ )]

k1β1s0 − k2 (ρ + b) = 0k1 =1 k2 = β1s0 / (ρ + b) s + i + c ≤1

dLdt

≤ i[βs + (α − k2ζ )i + k2(α −ζ )c+ k2ζ − (b+α +γ )]

≤ i[max{β,(α − k2ζ ),k2(α −ζ )}+ k2ζ − (b+α +γ )]

R0! =

β(b +α + γ )

+ζβ1

(ρ + b)(b +α + γ )≤ 1 dL / dt ≤ 0

dL / dt = 0 i i = 0{ }L ≥ 0 ′L ≤ 0 ′L = 0 i = 0 i i = 0{ }

R0! ≤1

Beforetheanalysis,weneedtomakethefollowingadditionalassumptions. and

ExistenceanduniquenessofendemicequilibriumTheorem5.1:When ,theendemicequilibriumexistsandisunique.

Proof: Let us denote the endemic equilibrium as and , , can be solved with thefollowingequations.

(12)

Thefeasibleregionoftheequationsis .

Andapplying toexpress and ,thefollowingcanbeobtained

, . (13)

And satisfiesthefollowingequation.= 0

(14)Denotingequation(14)as andanalyzingtheexistenceof intheendemicequilibrium.

canbewrittenas . (15) (16)

(17)Then,

, , (18)

(19)

Thus,thereexists intheequation .Regardingtheuniquenessof ,letusperformthefollowinganalysis.

(20)

Accordingly, the cubic function hasa negative coefficient of cubic term. And, as ,

wecanobtaintheimageof andthey-axisintersect,atmost,attwopoints,whichmeansthatthe

function has,atmost,twoextremepoints.Andas , , ,theimage

of andthey-axisintersectatonlyonepoint,whichmeansthattheequation onlyhasonesolutionatthefeasibleregionof .

Meanwhile, it iseasytoget , , , .Andsince is

unique, and arealsounique.Inconclusion,theendemicequilibrium existencehasbeenprovedanditisunique.

Localstabilityofendemicequilibrium

α <ζ ,σ ζβ1 < (ζ −α )β

R0 >1U* s*, i*,c*( ) s* i* c*

b− (b+σ )s* − (β +ζ −α )s*i* −β1s*c* = 0βs*i* +β1s*c* − (b+α +γ )i* +α (i*)2 = 0ζ i* −ζ (i*)2 − (ρ +b)c* + (α −ζ )c*i* = 0

⎧

⎨⎪⎪

⎩⎪⎪

Ω = (s*,i*,c*) 0 ≤ s*,i*,c* ≤1{ }i* s* c*

s* = b − (b +α + γ )i* +α (i*)2

(b +σ )− (α −ζ )i*c* = ζ i* −ζ (i*)2

(ρ + b)− (α −ζ )i*

i*

[(ρ + b)− (α −ζ )i*]b[(b +σ )− (α −ζ )i*]− [(ρ + b)− (α −ζ )i*](b +σ )[b − (b +α + γ )i* +α (i*)2 ]−[(ρ + b)− (α −ζ )i*](β +ζ −α )[b − (b +α + γ )i* +α (i*)2 ]i* − β1[b − (b +α + γ )i* +α (i*)2 ](ζ i* −ζ (i*)2 )

W (i* ) i*

W (i* ) W (i* ) = g1(i*)4 + g2 (i*)3 + g3(i*)2 + g4 (i*)

g1 = ζαβ1 + (α −ζ )(β +ζ −α )αg4 = −(ρ + b)b(α −ζ )+ (ρ + b)(b +σ )(b +α + γ )− (ρ + b)(β +ζ −α )b − β1bζ

W (0) = 0 W 1( ) = (ρ + b −α +ζ )[b(b +σ −α +ζ )+ γ (b +σ + β +ζ −α )]> 0W ′ 0( ) = g4 = −(ρ + b)b(α −ζ )+ (ρ + b)(b +σ )(b +α + γ )− (ρ + b)(β +ζ −α )b − β1bζ= (ρ + b)(b +σ )(b +α + γ )(1− R0 ) < 0

i* W (i* ) = 0

i*

g1 =α[ζβ1 − (ζ −α )β − (ζ −α )2 ]< 0′W (i*) W ′ 0( ) < 0

′W (i*)′W (i*) W (0) = 0 W 1( ) > 0 W ′ 0( ) < 0

′W (i*) W (i* ) = 0

i*

s* > 0 c* > 0 s* ≤ b +αb +σ

≤1 c* ≤ ζ i*

ζ i* + (ρ + b −α )≤1 i*

s* c*

U* s*, i*,c*( )

Theorem5.2:When and �theendemicequilibriumislocally

asymptoticallystable.Proof:Let ,andcalculatelinearapproximationsystem ,whereAis:

(21)

Applying the Routh-Hurwitz criterion to prove the local stability of endemic equilibrium.Where theRouth-Hurwitzcriterionstatesthat ifall theorderprincipalminordeterminantsaremorethanzero, thematrix islocallyasymptoticallystable.Denoting asthe thorderprincipalminordeterminant.As forA,where

(22)

(23)

(24)

When ,itiseasytoget .

When ,thefollowingtransformationcanbeobtained.

(25)

Accordingtotheassumption, . (26)Thus,onlythefollowinginequality(27)needstobeproveninordertoprove .

(27)Applyingthecondition ,theinequality(27)istrue.Thus, .In conclusion, according to the Routh-Hurwitz criterion, the endemic equilibrium is locally

asymptoticallystable.

7. DiscussionConsideringtheproblemthatiscausedbythespecificcharacteristicoftheHFMD,theexistingmodelsfordescribing the spreadof infectiousdiseasesarenot suitable for theHFMD.Theaimof this studywas to

R0 >1 R1 =bβ1ζ

(ρ + b)(b +α + γ )(b +σ )<1

y = (s, i , c) = (s − s0,i,c) ′y = Ay

−(b +σ )− (β +ζ −α )i* − β1c* −(β +ζ −α )s* −β1s

*

βi* + β1c* βs* − (b +α + γ )+ 2αi* β1s

*

0 ζ − 2ζ i* + (α −ζ )c* −(ρ + b)+ (α −ζ )i*

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

∆ i i

∆ 1 = −(b +σ )− (β +ζ −α )i* − β1c* < 0

Δ2 = [(b +σ )+ (β +ζ −α )i* + β1c*](−βs* + (b +α + γ )− 2αi*)

+(β +ζ −α )s*(βi* + β1c*)

= [(b +σ )+ (β +ζ −α )i* + β1c*]((b +α + γ )− 2αi*)

−(b +σ )βs* + (ζ −α )β1s*c*

> 1i*

[(b +σ )+ (β +ζ −α )i* + β1c*](β1s

*c* −α (i*)2 )

= [(b +σ )+ (β +ζ −α )i* + β1c*][(b +α + γ )− βs* − 2αi*]> 0

∆ 3 = [−(ρ + b)+ (α −ζ )i*]∆ 2+ (ζ − 2ζ i* + (α −ζ )c*){[(b +σ )+ (β +ζ −α )i* + β1c

*]β1s* − β1s

*(βi* + β1c*)}

(ζ − 2ζ i* + (α −ζ )c*) ≤ 0 Δ3 < 0(ζ − 2ζ i* + (α −ζ )c*) > 0

Δ3 = [(b +σ )+ (β +ζ −α )i*]{[−(ρ + b)+ (α −ζ )i*]((b +α + γ )− 2αi*)+ (ζ − 2ζ i* + (α −ζ )c*)β1s

*}+[(ρ + b)+ (ζ −α )i*](b +σ )βs* + [−(ρ + b)+ (α −ζ )i*]β1c

*[(b +α + γ )− 2αi* + (ζ −α )s*]− (ζ − 2ζ i* + (α −ζ )c*)β1s*βi*

(b +α + γ )− 2αi* + (ζ −α )s* > 0Δ3 < 0

[−(ρ + b)+ (α −ζ )i*]((b +α + γ )− 2αi* − βs*)[(b +σ )− (α −ζ )i*]+ (ζ − 2ζ i* + (α −ζ )c*)β1[b − (b +α + γ )i* +α (i*)2 ]< 0

R1 <1Δ3 < 0

provideasolutiontothisproblem.TheexistingstudiesfocusonchangingtheexistingmodelsandproduceresultsthatarenoteasytoanalyzeandcannotreflectthecharacteristicsofthespreadoftheHFMD. ThemostsignificantfactorsfortheHFMDcannotbeadequatelydescribedbyexistingmodels.ThereasonforthedevelopmentofthemodelistoprovideamoresuitablemodelforthespreadoftheHFMD. ThemostinnovativepartfortheSIRCmodelwithcarriersistheintroductionofthenewindividuals,whichis alsoanovel approach to improve thepreviousmodels according to the specific characteristics of theHFMD. In thepreviousmodels, the spreadprocess of the infectiousmodel includesmany parts, such asdifferentenvironmentforspread,theactivitiesamongpeople,etc.RegardingtheHFMD,itisadiseasethatis spread from person to person through coughing and sneezing, or contact with an infected person(Centers forDiseaseControlandPrevention,2017).Forexample, ifakidwhohasnooutdoorsactivitiessuffersfromtheHFMD,thekid’sparentsmaybethereasonforhisinfectionwiththevirusthatcausesthisdisease.Fromapathologicalperspective,thepeoplearoundthekidsplayanimportantpartinthespreadofthe disease because the pathogens of the HFMD, such as the Coxsackie virus, are highly contagious(Graham,2012),thusitiseasytospreadbytheindividualsthatcarrytheseviruses.Ifweignorethisfact,thespreadmodelwillnotadequatelydescribetherealsituationoftheHFMDandcanevenbewrong.

Oncewe include the carrier individuals in themodelling, the spread part can beanalyzed instead ofbeing regarded as a normal part of the spread process. The interrelationships between the carrierindividualsandother individualsareanalogoustothosewiththeexisting individuals.Oncetheexistenceandcharacteristicsofthedisease-freeequilibriumandtheendemicequilibriumareproved,theSIRCmodelcanbeenprovedtoberational.Suchworkwascompletedinsection5andsection6.TheadvantagesoftheSIRCmodelovertheevaluationofthechangeofthecoefficientswillbeclear,because,insteadofignoringcarrierindividuals,wewillnotonlyincludethetheminthespreadsystembutwealsowillfindaneasierwaytodeterminethecriterionforthedisease.Makingthecarrieranindividualinsteadofcombiningallthespreadmediatogethercanbenefittheanalysisoftheinfluenceofthecarrierindividuals.

Fortheadditionalapplicationof theSIRCmodel,ascanbeseenfromthemodellingprocess, themostimportantconsiderationoftheSIRCmodelisthecarrierindividual.Manysimilarinfectiousdiseasessharethesamecharacteristics,forexample,varicella.Thus,iftheSIRCmodelcanalsobeappliedtothemodellingofsuchdiseases,thespreadpartwillbemoreclearlyanalyzed.Moreover,thecriterionfortheexistenceofdisease-free equilibrium and endemic equilibrium are easy to operate in ways that are suitable for themedicalevaluation.

Inthefollowingsimulationsection,wecanobtaindifferentresultswiththeSIRmodelandSIRCmodel,fortheexistingofcarriers.

Encouraged by the consideration of this study, we can also change the spread of some infectiousdiseases by introducinga new group of individuals,which is a novel idea for analyzing the influence ofsomeparts.

Inconclusion,inthispaper,wemodifiedthespreadprocessoftheHFMDbyintroducinganewcarrierindividualgroup.Additionally, therationalityof themodelwasprovedandthedifferentresultsobtainedcomparedwiththepreviousmodelscanbeusedtogeneratedifferentpoliciesfordiseasecontrol.

8. SimulationsTosimulatetheprevalenceofHFMDunderdifferentconditions,letussettwogroupsofparametersandrepresentthesituationswiththefollowingfigures.

Parameterssettingswithdisease-freeequilibriumWemaysettheparametersasshowninTable1.Also, and .

AccordingtoTheorem5.2,thesystemhasadisease-freeequilibriumwithglobalasymptoticstability.R0 = 0.1626<1 R0

! = 0.9882<1

Table1.Parametersettings

Parameters Number

Letusset the initialconditionsas andtheresultspresented inFigure2areobtained.

Fig.2.Simulationwithdisease-freeequilibrium.

TheresultsshowninFigure2reveal that theproportionof threepartswillbecomestableandthe linewill become almost horizontal. Additionally, i and c both go down to zero. That is just the disease-freeequilibrium.

ParameterssettingswithendemicequilibriumWemaysettheparametersassowninTable2.Also, and .

However, as we mentioned in section 7, we will provide different consequences under the existingmodelwhichwerefertohereastheSIRmodelandtheSIRCmodel.

The BRN for the SIR model is (Xiao Y, 2002). According thedefinition of theBRN, under thegivenparameters settings, the SIRmodelwillproduce the disease-freeresultandtheSIRCmodelwillgeneratetheendemicresult,whicharecontradictory

AccordingtotheTheorem5.2,thesystemhasanendemicequilibriumwithlocalasymptoticstability.

Table2.Parametersettings

Parameters Number

Fig.3.Simulationwithendemicequilibrium.

b β β1 σ α γ ζ ρ0.013 0.11 0.05 0.066 0.003 0.117 0.02 0.1

s0,i0,c0( ) = (0.75,0.1,0.15)

R0 = 2.1653>1 R1 = 0.8087<1

R0 = βb / [γ (b +σ )]= 0.1547 <1

b β β1 σ α γ ζ ρ0.013 1 0.4 0.066 0.003 0.117 0.3 0.2

TheresultspresentedinFigure3revealthat,afterchaos,theproportionofthreepartswillbecomestableandthelinewillbecomealmosthorizontal.Allthethreepartshaveapositiveproportion.Thatisjusttheendemicequilibrium.

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