具饱和传染率的脉冲免疫接种SIRS模型

J.Sys.Sci.&Math.Scis.27(4)(2007,8),563–572∗SIRS (116024;537000)(116024)SIRSSIRSMR(2000)34D05,34D201M.G.Roberts,R.R.Kao SI[1];B.Shulgin,L.Stone SIR[2];Alberto d’Onofrio SIRSEIR[3,4];SIRS[5].SIRS*(10471117,10526015),(0728249),(200607LX138)(2007YJZD08)2005-10-09,2006-04-10.56427SIRS⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩d Sd t=−βSI(1+αS)+ωR+µ(1−S),d Id t=βSI(1+αS)−λI−µI,d Rd t=λI−ωR−µR,⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭t=kτ,S(t+)=(1−θ)S(t),I(t+)=I(t),R(t+)=R(t)+θS(t),⎫⎪⎪⎪⎬⎪⎪⎪⎭t=kτ.(1)N S,I R NN=S+I+R=1.β(1+αS)λωµθτk∈Z+,α,β,ω,λ,µ,θ(1)Ω={(S,I,R)∈R3|0≤S,I,R≤1, S+I+R=1}.S+I+R=1,⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩d Sd t=−βSI(1+αS)−AS−ωI+A,d Id t=βSI(1+αS)−λI−µI,⎫⎪⎪⎪⎬⎪⎪⎪⎭t=kτ,S(t+)=(1−θ)S(t),I(t+)=I,(t)t=kτ.(2)A=ω+µ.(2)D={(S,I)∈R2|0≤S,I≤1,S+I≤1}. 21⎧⎨⎩d wd t=a−bw,t=kτ,w(t+)=(1−θ)w(t),t=kτ,(3)a,b>0,0<θ<1.(3)τw(t)=ab1−θexp(−b(t−kτ))1−(1−θ)exp(−bτ),kτ<t≤(k+1)τ.(3)w(t)=ab−(ab−w(kτ))exp(−b(t−kτ)),kτ<t≤(k+1)τ,4SIRS565w (kτ)wkt =kτ(3)F ,w ((k +1)τ)=F (w (kτ))=(1−θ)a b−a b−w (kτ) exp(−bτ),w ∗=a b (1−θ)(1−exp(−bτ))1−(1−θ)exp(−bτ).w ∗>w >0w ∗>F (w )>w ;w >w ∗w >F (w )>w ∗.w ∗(3)τw (t )=a b 1−θexp(−b (t −kτ))1−(1−θ)exp(−bτ),kτ<t ≤(k +1)τ.(2),I =0⎧⎨⎩d S d t =A (1−S )t =kτS (t +)=(1−θ)S (t )t =kτ.(4)1,(4)S ∗=(1−θ)(1−exp(−Aτ))1−(1−θ)exp(−Aτ),(5)(4)τS(t )=1−θexp(−A (t −kτ))1−(1−θ)exp(−Aτ),kτ<t ≤(k +1)τ.(6)(2)τ( S(t ),0)2R <1(2)τ( S(t ),0)R =βλ+µ1−exp(−Aτ)(1+α)(1−exp(−Aτ))+θexp(−Aτ).x (t )=S (t )− S(t ),y (t )=I (t ),t =kτ(2)τ( S(t ),0)⎛⎜⎜⎝d xd t d y d t⎞⎟⎟⎠=⎛⎜⎜⎜⎜⎝−A −β S(t )1+α S (t )−ω0β S(t )1+α S (t )−λ−µ⎞⎟⎟⎟⎟⎠x y.(7)Φ(t )(7)Φ(t )d Φ(t )d t =⎛⎜⎜⎜⎜⎝−A −β S(t )1+α S (t )−ω0β S(t )1+α S(t )−λ−µ⎞⎟⎟⎟⎟⎠Φ(t ).56627 t=kτ(2)x(t+)y(t+)=1−θ001x(t)y(t).(7)M=1−θ001Φ(τ)=⎛⎜⎜⎜⎜⎝(1−θ)exp(−Aτ)(1−θ)expτ(−β S(t)1+α S(t)−ω)d t)0expτ(β S(t)1+α S(t)−λ−µ)d t⎞⎟⎟⎟⎟⎠.(7)Floquet(M)λ1=(1−θ)exp(−Aτ)<1,λ2=expτ(β S(t)1+α S(t)−λ−µ)d t.R =βλ+µ1−exp(−Aτ)(1+α)(1−exp(−Aτ))+θexp(−Aτ)<1λ2<1.Floquet[6],τ( S(t),0)1R0<1(2)τ( S(t),0)R0=βS∗(λ+µ)(1−θ).(2)⎧⎪⎨⎪⎩d Sd t≤A(1−S),t=kτ, S(t+)=(1−θ)S(t),t=kτ.⎧⎪⎨⎪⎩d xd t=A(1−x),t=kτ, x(t+)=(1−θ)x(t),t=kτ.1(6)x(t)= S(t)=1−θexp(−A(t−kτ))1−(1−θ)exp(−Aτ),kτ<t≤(k+1)τ.[7],ε,N1∈Z+,k≥N1S(t)≤x(t)< S(t)+ε,kτ<t≤(k+1)τ,(8) (5)(6)S(t)<1−exp(−Aτ)1−(1−θ)exp(−Aτ)+ε=S∗1−θ+ε.4SIRS567(2)d I d t ≤βS∗1−θ+ε−λ−µI,d y d t =βS∗1−θ+ε−λ−µy,R0=βS∗(λ+µ)(1−θ)<1ε,β(S∗1−θ+ε)<λ+µ,limt→∞y(t)=0,lim t→∞I(t)=0,ε,N2≥N1,k≥N2I(t)<ε.(2)d Sd t≥(A−ωε)−(A+βε)S,⎧⎨⎩d ud t=(A−ωε)−(A+βε)u,t=kτ,u(t+)=(1−θ)u(t),t=kτ,1,kτ<t≤(k+1)τu(t)=A−ωεA+βε1−θexp(−(A+βε)(t−kτ))1−(1−θ)exp(−(A+βε)τ).[7],ε>0,N3≥N2,k≥N3S(t)≥u(t)> u(t)−ε,kτ<t≤(k+1)τ(9) (8)(9) u(t)−ε<S(t)< S(t)+ε.εlimt→∞S(t)= S(t).R0<1(2)τ( S(t),0)2,R <1( S(t),0) R <R0,R0<1( S(t),0)32R =β(λ+µ)(1+α)(1−θ)(1−exp(−Aτ))(1−(1−θ)exp(−Aτ))>1,(2)(S(t),I(t))D(2)t>0, S(t)≤1,I(t)≤1.m S,m I,t S(t)≥m S,I(t)≥m I.m S,t S(t)≥m S.I(t)≤1,(2)d Sd t≥µ−BS,B=β+ω+µ.⎧⎪⎨⎪⎩d Vd t=µ−BV,t=kτ,V(t+)=(1−θ)V(t),t=kτ,(10)56827 1[7],ε,k1∈Z+,k≥k1S(t)≥V(t)> V(t)−ε>µB(1−θ)(1−exp(−Bτ))1−(1−θ)exp(−Bτ)−ε>µBV∗−ε∆m S>0.V∗=(1−θ)(1−exp(−Bτ))1−(1−θ)exp(−Bτ)(10) V(t)=µB(1−θexp(−B(t−kτ))1−(1−θ)exp(−Bτ))(10)kτ<t≤(k+1)ττm I>0,t I(t)≥m I.1R =β(λ+µ)(1+α)(1−θ)(1−exp(−Aτ))(1−(1−θ)exp(−Aτ))>1,m I>0,ε>0m I<ω+µβ+ω<1,p=βξ1+α−λ−µ>0,ξ=(A−βm I−ωm I)A(1−θ)(1−exp(−Aτ))(1−(1−θ)exp(−Aτ))−ε,t1∈(0,∞)I(t1)≥m I.t∈(0,∞),I(t)<m I.(2)d Sd t≥(A−βm I−ωm I)−AS∆q−AS,⎧⎪⎨⎪⎩d zd t=q−Az,t=kτ,z(t+)=(1−θ)z(t),t=kτ.(11)1,(11)z∗=q A(1−θ)(1−exp(−Aτ))[1−(1−θ)exp(−Aτ)],kτ<t≤(k+1)ττz(t)=qA (1−θexp(−A(t−kτ))1−(1−θ)exp(−Aτ)),kτ<t≤(k+1)τ.[7],ε,k ∈Z+,k≥kS(t)≥z(t)> z(t)−ε≥(A−βm I−ωm I)A(1−θ)(1−exp(−Aτ))(1−(1−θ)exp(−Aτ))−ε∆ξ.(2),d I d t ≥β( z−ε)1+α−λ−µI≥βξ1+α−λ−µI∆pI,[kτ,(k+1)τ],k≥k I((k+1)τ)≥I(kτ)exp(pτ).k→∞I((k+k )τ)≥I(k τ)exp(kpτ)→∞,2t≥t1,I(t)≥m I,t>t1I(t)<m I.t∗=inft>t1{I(t)<m I},t∗∈(n1τ,(n1+1)τ],n1∈Z+.t∈[t1,t∗)I(t)≥m I.t∗I(t)I(t∗)=m I.t∗n0∈Z+ t∗=n0τ,t∗∗=t∗−ε∗,ε∗I(t∗∗)≥m I.t∗t∗t∗∗n2,n3∈Z+n2τ>1Aln1+z∗ε,exp(−(λ+µ)(n2+1)τ)exp(n3pτ)>1,4SIRS569[(n 1+1)τ,(n 1+1)τ+n 2τ+n 3τ]t ,I (t )≥m I .t ∈[(n 1+1)τ,(n 1+1)τ+n 2τ+n 3τ],I (t )<m I .(11),t ∈[kτ,(k +1)τ),n 1+1≤k ≤n 1+1+n 2+n 3.|z (t )− z (t )|= q A 1−(1−A qz ((n 1+1)τ))exp(−A (t −(n 1+1)τ)) −qA (1−θexp(−A (t −(n 1+1)τ))1−(1−θ)exp(−Aτ))= z ((n 1+1)τ)−q A (1−θ)(1−exp(−Aτ))[1−(1−θ)exp(−Aτ)] exp(−A (t −(n 1+1)τ))≤ 1+q A (1−θ)(1−exp(−Aτ))[1−(1−θ)exp(−Aτ)]exp(−A (t −(n 1+1)τ))=(1+z ∗)exp(−A (t −(n 1+1)τ)).(n 1+1+n 2)τ≤t ≤(n 1+1)τ+n 2τ+n 3τ|z (t )− z (t )|<ε,z (t )−ε<z (t )<z (t )+ε.1,I ((n 1+1+n 2+n 3)τ)≥I ((n 1+1+n 2)τ)exp(n 3pτ).(2)d Id t≥−(λ+µ)I ,[t ∗,(n 1+1+n 2)τ]I ((n 1+1+n 2)τ)≥I (t ∗)exp[−(λ+µ)((n 1+1+n 2)τ−t ∗)]≥m I exp(−(λ+µ)(n 2+1)τ).I ((n 1+1+n 2+n 3)τ)≥m I exp(−(λ+µ)(n 2+1)τ)exp(n 3pτ)>mI ,t =inf t ≥t∗{I (t )≥m I }.I (t )≥m I .t ∈[t ∗,t ),(2)d Id t≥−(λ+µ)I ,I (t )≥I (t ∗)exp(−(λ+µ)(t −t ∗))≥m I exp(−(λ+µ)(1+n 2+n 3)τ)∆m I .t >t ,I (t )≥m I ,2.m I >0,t ≥t 1I (t )≥m I .41,(2)( S(t ),0).τ[8]( S(t ),0)x 1(t )=S (t ),x 2(t )=I (t ),(2)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩x 1(t )=−βx 1(t )x 2(t )1+αx 1(t )−Ax 1(t )−ωx 2(t )+A ∆F 1(x 1(t ),x 2(t )),x 2(t )=βx 1(t )x 2(t )1+αx 1(t )−λx 2(t )−µx 2(t )∆F 2(x 1(t ),x 2(t )),⎫⎪⎪⎪⎬⎪⎪⎪⎭t =kτ,x 1(t +)=(1−θ)x 1(t )∆θ1(x 1(t ),x 2(t )),x 2(t +)=x 2(t )∆θ2(x 1(t ),x 2(t )),⎫⎬⎭t =kτ.57027ζ(t )=( S (t ),0)=( x 1(t ),0),∂Φ1(τ0,x 0)∂x 1=exp τ00∂F 1(ζ(r ))∂x 1d r ,∂Φ2(τ0,x 0)∂x 2=exp τ00∂F 2(ζ(r ))∂x 2d r ,∂Φ1(τ0,x 0)∂x 2= τ00 expτ0u∂F 1(ζ(r ))∂x 1d r ∂F 1(ζ(u ))∂x 2exp u 0∂F 2(ζ(r ))∂x 2d r d u,d 0=1−∂θ2∂x 2∂Φ2∂x 2 (τ0,x 0)=1−exp τ00β x 1(r )1+α x 1(r )−λ−µ d r ,(τ0d 0=0)a 0=1−∂θ1∂x 1∂Φ1∂x 1 (τ0,x 0)=1−(1−θ)exp(−Aτ0)>0,b 0=− ∂θ1∂x 1∂Φ1∂x 2+∂θ1∂x 2∂Φ2∂x 2(τ0,x 0)=−(1−θ) τ00exp(−A (τ0−u )) −β x 1(u )1+α x 1(u )−ω exp u 0β x 1(r )1+α x 1(r )d r d u >0,∂2Φ2(τ0,x 0)∂x 1∂x 2= τ00exp τ0u∂F 2(ζ(r ))∂x 2d r ∂2F 2(ζ(u ))∂x 1∂x 2exp u 0∂F 2(ζ(r ))∂x 2d rd u= τ00exp τ0u ∂F 2(ζ(r ))∂x 2d r β(1+α x 1(u ))2exp u 0∂F 2(ζ(r ))∂x 2d r d u >0,∂2Φ2(τ0,x 0)∂x 22= τ00exp τ0u∂F 2(ζ(r ))∂x 2d r ∂2F 2(ζ(u ))∂x 22exp u 0∂F 2(ζ(r ))∂x 2d rd u +τ00 exp τ0u∂F 2(ζ(r ))∂x 2d r ∂2F 2(ζ(u ))∂x 2∂x 1 u 0(exp u p ∂F 1(ζ(r ))∂x 1d r∂F 1(ζ(u ))∂x 2exp p 0∂F 2(ζ(r ))∂x 2d r ) d p d u= τ00 expτ0u∂F 2(ζ(r ))∂x 2d r β(1+α x 1(u ))2 u 0 expu p ∂F 1(ζ(r ))∂x 1d r −β x 1(u )1+α x (u )−ω exp p 0∂F 2(ζ(r ))∂x 2d r d p d u <0,4SIRS571∂2Φ2(τ0,x0)∂ τ∂x2=∂F2(ζ(τ0))∂x2expτ0∂F2(ζ(r))∂x2d r=β x1(τ0)1+α x(τ0)−λ−µexpτ0∂F2(ζ(r))∂x2d r,∂Φ1(τ0,x0)∂ τ=˙ x1(τ0)=Aθexp(−Aτ0)1−(1−θ)exp(−Aτ0)>0,B=−∂2θ2∂x1∂x2∂Φ1(τ0,x0)∂ τ+∂Φ1(τ0,x0)∂x11a 0∂θ1∂x1∂Φ1(τ0,x0)∂ τ∂Φ2(τ0,x0)∂x2−∂θ2∂x2∂2Φ2(τ0,x0)∂ τ∂x2+∂2Φ2(τ0,x0)∂x1∂x21a 0∂θ1∂x1∂Φ1(τ0,x0)∂ τ=−β x1(τ0)1+α x1(τ0)−λ−µexpτ0∂F2(ζ(r))∂x2d r+∂2Φ2(τ0,x0)∂x1∂x21a 0(1−θ)∂Φ1(τ0,x0)∂ τ,C=−2∂2θ2∂x1∂x2−ba 0∂Φ1(τ0,x0)∂x1+∂Φ1(τ0,x0)∂x2∂Φ2(τ0,x0)∂x2−∂2θ2∂x22∂Φ2(τ0,x0)∂x22+2∂θ2∂x2b 0a 0∂2Φ2(τ0,x0)∂x2∂x1−∂θ2∂x2∂2Φ2(τ0,x0)∂x22=2b 0a 0∂2Φ2(τ0,x0)∂x2∂x1−∂2Φ2(τ0,x0)∂x22>0.Bf(t)=β1−θexp(−At)1−(1−θ)exp(−Aτ)1+α1−θexp(−At)1−(1−θ)exp(−Aτ)−λ−µ,σ=θexp(−At)1−(1−θ)exp(−Aτ),f (t)=βAσ[1+α(1−σ)]2>0,f(t)τ0d 0=0τf(t)d t=f(η)τ0=0,η∈(0,τ0),f(τ0)>0,B<0(BC<0).3τ0(2)τ>τ0τ0(2) 5(1),θ> βλ+µ−1(exp(Aτ)−1),τ<1A ln1+θ(λ+µ)β−λ−µθ<β(λ+µ)(1+α)−1(exp(Aτ)−1)1+β(λ+µ)(1+α)(exp(Aτ)−1)57227[1]Roberts M G and Kao R R.The dynamics of an infectious disease in a population with birth pules.Math.Biosci.,1998,149:23–36.[2]Stone L,Shulgin B and Agur Z.Theoretical examination of the pulse vaccination policy in the SIRepidemic puter Modeling,2000,31:207–215.[3]d’Onofrio A.Pulse vaccination strategy in the SIR epidemic model:Global asymptotic stableeradication in presence of vaccine put.Modelling,2002,36:473–489.[4]d’Onofrio A.Stability properties of pulse vaccination strategy in SEIR epidemic model.Math.Biosci.,2002,179:57–72.[5]SIRS2003,24(4):235–243.[6]Bainov D D and Simeonov P S.Impulsive Differential Equations:Periodic Solutions and Applica-tions.NewYork,John Wiley&Sons,1993.[7]Lakshmikantham V,Bainov D D and Simeonov P S.Theory of Impulsive Differential Equations.Singapore,World Scientific,1989.[8]Lakmeche A and Arino O.Bifurcation of nontrivial periodic solutions of impulsive differentialequations arising chemotherapeutic treatment.Dyn.Cont.Discr.Impul.Syst.,2000,7:265–287.THE SIRS EPIDEMIC MODEL WITH SATURATED CONTACT RATE AND PULSE VACCINATIONPang Guoping(Department of Applied Mathematics Dalian University of Technology Dalian116024;Department of Methematics and Computer Science Yulin Teachers’College Yulin537000)Chen Lansun(Department of Applied Mathematics Dalian University of Technology Dalian116024)Abstract In this paper,uniform persistence and periodic solution of the SIRS epidemic model with saturated contact rate and pulse vaccination are discussed.Sufficient conditions for global asymptotic stability of the infection-free periodic solution and uniform persistence of this model are ing bifurcation theory the bifurcation parameter for existence of the positive periodic solution is given.Key words Pulse vaccination,SIRS model,uniform persistence,periodic solution.。