一类媒介-宿主传染病模型的稳定性分析(英文).docx

一类媒介-宿主传染病模型的稳定性分析(英文)
Introduction
Infectiousdiseasesareamajorconcernforpublichealtharoundtheworld,anditisimportanttounderstandhowinfectiousdiseasesaretransmittedbetweenthehostandtheenvironment.Oneimportantaspectofdiseasetransmissionistherelationshipbetweenthehostandtheenvironment,whichisoftenmediatedbyavectorsuchasamosquitoortick.Inthispaper,wewillanalyzethestabilityofatypeofinfectiousdiseasemodelknownasthehost-vectormodel,whichdescribesthetransmissionofadiseasebetweenahost(suchasahuman)andavector(suchasamosquito).
ModelDescription
Thehost-vectormodelcanbedescribedbyasetofdifferentialequationsthatquantifythechangesinthenumberofinfectedanduninfectedindividualsovertime.Themodelconsistsoftwocompartments:oneforthehostpopulation(H)andoneforthevectorpopulation(V).Eachcompartmentisfurthersubdividedintocompartmentsforthesusceptible(S),infected(I)andrecovered(R)individuals.Themodelcanbewrittenasfollows:
dS/dt=rN-βSI-μS
dI/dt=βSI-γI-μI
dR/dt=γI-μR
dV/dt=a-cv-βVSI
dW/dt=βVSI-δW
Inthismodel,Nisthetotalpopulationsize,risthebirthrate,μisthedeathrate,βisthetransmissionratefromthevectortothehost,γistherecoveryrate,aistherecruitmentrateofthevectorpopulation,cisthedeathrateofthevectorpopulation,δistheincubationrateofthevectorpopulation,andWisthenumberofvectorsthatarelatentwiththedisease.
StabilityAnalysis
Toanalyzethestabilityofthehost-vectormodel,wecanusethebasicreproductionnumber(R0),whichisdefinedasthenumberofsecondarycasesthataregeneratedbyeachprimarycase.R0canbecalculatedas:
R0=βSV/(μγ)
IfR0<1,thediseasewilldieouteventuallyandthedisease-freeequilibrium(DFE)isstable.IfR0>1,thediseasewillpersistandtheendemicequilibrium(EE)willbestable.IfR0=1,thesystemisontheboundarybetweenthetwoequilibria.
Next,wecancalculatetheJacobianmatrixofthesystemevaluatedattheequilibriumpoints.FortheDFE,theJacobianmatrixis:
J=[-βNμ0βNμ0
βSμ-γ-μ00
0γ-μ0
000-cv-βSI]
FortheEE,theJacobianmatrixis:
J=[-βNμ0βNμ0βSμ0
0-γ-μ000
0γ-μ00
000-cv-βVIδβS]
UsingtheeigenvaluesoftheJacobianmatrix,wecandeterminethestability