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1、? ? ? ? ?112(1“!“#“$“%“88?88A8B8C8D8E8F8G8H8I8J8K8L8M8N8OSIRP8Q8R8S8T8UWV8X8G8HYZM/_abUWcd=eRfghijklmFnopqrMst.uvw:xyzW|zWzW/zW Z?O175:A.A delay SIR epidemic model with with vaccination and constant inputLIU Shao-ying1CAO Xin-jie1PEI Yong-zhen2(1School of Mathematics and Information Science, Pingdin
2、gshan College, Pingdingshan, 467000, China)(2Shool of Science, Tianjin Polytechnic University, Tianjin, 300160, China)Abstract: This paper investigates a delay SIR periodic model with vaccination and constant input. Byusing the comparison theorem in impulsive differential equation, we obtain some su
3、fficient conditions for globalattractivity of the infection-free periodic solution and for uniformly persistent of the disease.Keywords: Impulsive; impulsive; periodic solution; delay; globally attractive; Permanence.1PQRSTg;.PoMRST tNMNQRMCf(S,I) = S(t)I(t)1,S(t)tNMN/QMC, I(t)tNMN/QMC.,pmPQMPQRST,f
4、(S,I) = Sq(t)Ip(t)3, Satsuma288Mf(S,I) = Sq(t)I(t)(0 ABCDEFGHKLMNOSIRST: S0(t) = A eSq(t )I(t ) S(t),I0(t) = eSq(t )I(t ) I(t) I(t),R0(t) = R(t),t 6= k,k Z+S(t+) = (1 )S(t),I(t+) = I(t),R(t+) = R(t) + S(t), t = k,k Z+(2.1)N(t) = S(t) + I(t) + R(t)tNMC, A 0BCDE, N/QRQMC, 0aMN,N/QRRPd, !“#$N% x(t) 0Lt
5、 ,0,ACMN:(i)Oa b,Glimt+x(t) = +.3QPRST2CU;ijJKaMfghS(t) =A (1 1 (1 )ee(tk),k 0,:ce?A1 e 1 (1 )e+ ?q0cS0(t) A S(t),M/_no x0(t) = A x(t), t 6= k,k Z+x(t+) = (1 )x(t), t = k,k Z+(3.3)1lb10h,no(3.3)HIijJKaMfghx(t) =A (1 1 (1 )ee(tk),k 0M8h, x(t)n8o(3.3)bvcv8v98s8tx(0+) = S0Mh,1GHYZM/_ab,Hk1 0,:cS(t) x(t
6、) k1,wk k1N,AS(t) x,1no(2.2)Mtyz(3.4)0c,L,AMt k + , k k1,AI0(t) e(SM)qI(t ) ( + )I(t)/_y0(t) = e(SM)qy(t ) ( + )y(t),(3.5)31(3.2)he(SM)q k1,:cL,AMt k2 + ,AI(t) ,cS(t) (A e(A)q) S(t),GAz(t) S(t), w 0N,Az(t) S(t).z(t)MI-fgh, z0(t) = (A e(A)q) z(t), t 6= k,z(t+) = (1 )z(t), t = k,z(0+) = S(0+),(3.7)1(3
7、.7)0c,Lk 0,H-Ck3 K2,:cS(t) z(t) ,k k3,(3.8)ZsM,1(3.4)F(3.8)0climtS(t) = S(t).(3.9)|,1(3.6)F(3.9)0cno(2.2)MeRfgh(S(t),0)ijklM.C; 1,Gno(2.2)MRqrM.lmW(S(t),I(t)no(2.2)Mqr-h,Lt 0a0YCV (t) = I(t) + e(S)qZttI(s)ds.LV (t)(2.2)Mh ,cV0(t)=I0(t) + e(S)qI(t) e(S)qI(t )=e(Sq(t ) (S)q)I(t ) + e(S)q ( + )I(t)=e(S
8、q(t ) (S)q)I(t )=e?Sq1(t ) + Sq2(t )S+ + S(t )(S)q2+ (S)q1?(S(t ) S)I(t )41,GI 0, (0 csM0 0,:cSMijJKaM-fgh.HT 0,L,AMt t0+ T =t1,bcS(t) u(t) 0A e(A)qI(1 )(1 e) 1 (1 )e 0,(3.12)1(3.10)F(3.12)0c,Lt t1+ =T1,AV0(t)=e?Sq1(t ) + Sq2(t )S+ + S(t )(S)q2+ (S)q1?(S(t ) S)I(t )qe(S)q1( S)I(t ).Il=min tT1,T1+I(t
9、)8vDEL,A8Mt T1, I(t) Il.vG,H88v8v8B8CT2,:8cvLt T1,T1+ + T2, I(t) Il, I(T1+ + T2) = IlI0(T1+ + T2) 0.*1no(2.2)Mtyz(3.12)0hI0(T1+ + T2)eSq(T1+ T2) ( + )Il=( + )e+Sq(T1+ T2) 1Il( + )( S)q 1Il 0,I0(T1+ + T2) 0,0c,L,AMt T1, I(t) Il 0.,L,AMt T1+ ,AV0(t) qe(S)q1( S)Il 0,GAwt +N, V (t) +.V (t) A (1 + e(S)q)
10、(tc).cD.fg62L:t,L,AZMt, I(t) I;ty, tZN, I(t)I.LtL,abMMN*M.56tyL.UDEL,AMtc,AI(t) m,m = minI2,Ie(+).5t1pA GH,0cI(t)pI 2.vt2 t1 ,G8AI(t) Ie(+), t1 0,:8cI(t) Ie(+), t1 t t1+ + T4, I(t1+ + T4) = Ie(+)I0(t1+ + T4) 0,wt1cN,S(t) Lt1( + )( S)q 1Ie(+) 0I0(t1+ + T4) 0. ,Lt t1,t2, I(t) m.1t1,t2qrM,tyL,LcMt,AI(t
11、) m.fg|MN, mMopno(2.2)M-h LcMtbcI(t) m.abD.4QST9:N=?ABCDEFGHKLMNOSIRPQRST,cd2xcdw 1N,RqrM,NBLRUR.1 V.Capasso, Mathematical Structure of Epidemic ModelsM, Spinger-Verlag, Berlin, New York,1993.1974.2 J.Satsuma,R. Willox, A. Ramani, A.S. Carstea, B. Grammaticos, Extending the SIR epidemic model, Physica AJ336, 2004, 369-375.3 D. BAINOV, P. SIMEONOV, Impulsive differential equations: peri
