一类具有接种疫苗和时滞的传染病模型的行波解稳定性研究.pdf

AMATHEMATICA APPLICATA2024,37(2):423-438ak?D/?.?1)-5?,(-O?,-400067)?ak?.?1)?-5,?.XkN5,?Uy?.1)?-5,6I3x=+k.c?.;-5;N1;?UaO29AMS(2010)Ka35K57;92D30zIA?1001-9847(2024)02-0423-161.:5?D/,;.?Y?5D/.kD/5r,k?p?A:,-?,J?,Y,P&,$9).L?/?Y D,D,(,)y)?/c-.Qg-.61,E?k?.-.)|(WHO)?O?zc?k130?400?,k?p21000143000.?.?31)uX-.Codeco1gY?l?SIR?.SIRBY?5D/?.Mukandavire?2Jz?SIRB?.52008-2009c9n,5u)5xY?m?D.3NXn-5n?:,WANG?4y?2)u1A*?.?-5.DUAN?4?UYWANG?3,?2)?u1T?.3:?-5.LIAO?5?5?.?1)35.WANG?3?#?.e5?.?/:?5pD/561?K,(JrN?pD/5?-59m5?p.Cc5,?m5D/?.1)?,Jku?)|?D/.1)-51n?(J?K,k-?nd.-?1)-53Nz?2?,|?U(?n?,710.,?,uN?,?,?1)-5?(Jk.MEI?1113kJE?UA*?1-5.ZHANG?14|E?U?A*D/?.?1)-5.LI?15?N*SIRD/?.1vF2023-03-3178-:c&8(cstc2020jcyj-msxmX0394),x,:).424A2024)?-55.YANG?16HUANG?17A?E?U)?kN5?.?1)-5.WU?18|E?U?n,?NeDD/?.?1)?-5.|E?UFp“C,ZHANG19?k?A*?N1)?-5,y?k?.1)3e?-?,?.1)3e?-?.a;.?Y?5D/kD,)?DY?m?m?D.?)D/1,?k?.?1)-5.?8c,vk?N?D/?.?1)-5?1,c5?Y.?I?n?E?U,k?O.2.?.3“3puX-.4k?19,u61u)g,/?p+,19l?-:?)xJ?.320?:,?A*?.Xe:u1(t,x)t=(wu4(t,x)+hu3(t ,x)u1(t,x)(+)u1(t,x)+d1u1(t,x),(2.1)u2(t,x)t=u1(t,x)(wu4(t,x)+hu3(t ,x)u2(t,x)u2(t,x)+d2u2(t,x),(2.2)u3(t,x)t=w(u1(t,x)+u2(t,x)u4(t,x)+h(u1(t,x)+u2(t,x)u3(t ,x)(+)u3(t,x)+d3u3(t,x),(2.3)u4(t,x)t=u3(t,x)u4(t,x)+d4u4(t,x),(2.4)u5(t,x)t=u3(t,x)u5(t,x)+d5u5(t,x),(2.5),u1(t,x),u2(t,x),u3(t,x)u5(t,x)La,?,a/O3tx?,u4(t,x)LY?3tx?.?0,1.=0L?k?,=1L?.a?,P,?P,ag,)/k?,k?,whOLm?D 0(i=1 5)?*X.,h(u1(t,x)+u2(t,x)u3(t ,x)“LLd?ta/;?.D/6Xe:5?.(2.1)-(2.4)u5(t,x)k5,?3?Y?.|(2.1)-(2.4)=.?.(2.1)-(2.4)?lim(u1,u2,u3,u4)=(+,(+),0,0),lim(u1,u2,u3,u4)=(k1,k2,k3,k4).1 2?ak?D/?.?1)-54252.16w,?.|(2.1)-(2.4)k:E0=(+,(+),0,0),9/:E=(u1,u2,u3,u4)=(k1,k2,k3,k4),E?k1=?wk3+hk3?+,k2=?wk3+hk3?+?wk3+hk3?+?,k3=u3,k4=k3.?.(2.1)-(2.4)?2)20OR0=?+(+)?(h+w)(+).(2.6)3.n(=x+ct,c 0,uj(t,x)=j(),j=1,2,3,4“X(2.1)-(2.4),?A?1Xe:001()c01()=1()(w4()+h3(,t )(+)1(),(3.1)002()c02()=1()2()(w4()+h3(,t )2(),(3.2)003()c03()=w4()(1()+2()+h(1()+2()3(,t )(+)3(),(3.3)004()c04()=3()4().(3.4)(1(),2(),3(),4()ve.:lim(1(),2(),3(),4()=(+,(+),0,0),lim(1(),2(),3(),4()=(k1,k2,k3,k4).?C 0,Ci 0(i=1,2,)?A?.?Im,Ae?I=R.L2(I)3I?m.Hk(I)(k 0)Sobolevm,3mI?L2f(x)?8,?/didxif(i=1,2,k)uL2(I).L2(I)L2-mD?(x)0?m,:kf(x)kL2=?ZI(x)|f(x)|2dx?12.426A2024Hk(I)(k 0)k?Sobolevmkf(x)kHk=?kXi=0ZI(x)?dif(x)dxi?2dx?12.?T 0,BBanachm.2C(0,TB)m0,T?BYm.:hh(,c)00hw(,c)M(,c)=0w(+(+)0,M(,c)?A?Xe:2(,c)=(,c)=w(+(+)hh(,c)hw(,c),(3.5)hh(,c)=d32 c (+)+(+(+),hw(,c)=d42 c .3?c,e(:n11)ec c,(,c)=1,3?121 2 c,?(1,2),(,c)1.2)ec=c,?(0,c),K(,c)1 1=2 c.3)ec 1.?X(3.1)-(3.4)?1)-5,?()=e(0),R,(3.6)0?0?1.0,K()1 lim()=+,lim+()=0.b?maxc,C1,C2,C3,C4,Ci3?,XJ6v(ui0(x)i(x)C(R)H1(R),i=1,2,4,u30(s,x)3(x+cs)C(,0;C(R)H1(R)L2(,0;H1(R)limx+(ui0(x)i(x):=ui0,(x),i=1,2,4,limx+(u30(s,x)3(x+cs):=u30,(s),03.?3?uxt,C,0 0,i=1,2,3,4.1 2?ak?D/?.?1)-5427Cunif,T,0,0 T .Cunif(,T:=u(t,x)C(,T R),limx+e2tu(t,x)3u,T limx+ux(t,x)=limx+uxx(t,x)=0ut ,T.?Ui(t,),i=1,2,3,4,(3.1)-(3.4)?),Ui(t,)=ui(t,x)i(),i=1,2,3,4,Ui0(0,)=ui0(x)i(x),i=1,2,4,U30(s,)=u30(s,x)3(x+cs),s ,0,x R.?K(3.1)-(3.4)-#LU1t+cU12U12+(+)U1=P1(U1(t,),U2(t,),U3(t ,c),U4(t,),(3.7)U2t+cU22U22+U2 U1=P2(U1(t,),U2(t,),U3(t ,c),U4(t,),(3.8)U3t+cU32U32+(+)U3=P3(U1(t,),U2(t,),U3(t ,c),U4(t,),(3.9)U4t+cU42U42=U3 U4,(3.10)P1=w14+h13(c)w(U1+1)(U4+4)h(U1+1)(U3(t ,c)+3(c),P2=w24+h23(c)w(U2+2)(U4+4)h(U2+2)(U3(t ,c)+3(c),P3=P1+P2.Taylor,?.X(3.7)-(3.10)-#?n?U1t+cU12U12+(+T1)U1 h1U3(t ,c)=w1U4 Q1(t ,c),(3.11)U2t+cU22U22+(+T2)U2+w2U4+h2U3(t ,c)U1=Q2(t ,c),(3.12)U3t+cU32U32+(+)U3 T1U1 T2U2 T3U3(t ,c)T4U4=Q1(t ,c)+Q2(t ,c),(3.13)U4t+cU42U42+U4=U3,(3.14)Q1(t ,c)=wU1U4+hU1U3(t ,c),Q2(t ,c)=wU2U4+hU2U3(t ,c),T1=w4+h3(c),T2=w4+h3(c),428A2024T3=h1+h2,T4=w1+w2.)m:X(r ,r+T)=(U1,U2,U3,U4)|Ui C(r i,r+T;C(R)H1(R)L2(r i,r+T;H1(R)Cunifr i,r+T,i=1,2,3,4,1=2=4=0,3=,:M2(T)=4Xi=1?suptr,r+T(kUi(t)k2C+kUi(t)k2H1)+Zr+TrkUi(s)k2H1ds?.n1?duen2:n2 0,3?t0=t01 0,K)(U1(t,),U2(t,),U3(t,),U4(t,)X(r ,r+t0)M(t0)p2(1+)M(0).K19?,?d.eyX(3.11)-(3.13)3Cunif0,+Cunif0,+Cunif,+Cunif0,+m)Ui(t,x)?35.K2(?35)-(1(x+ct),2(x+ct),3(x+ct),4(x+ct)?1),?6(U10(),U20(),U30(s,),U40()X(,0),KX(3.11)-(3.13)?)U(t,)=(U1(t,),U2(t,),U3(t,),U4(t,)3Cunif0,Cunif0,Cunif,Cunif0,?3.y-Ui(t,)X(,t0),i=1,2,3,4,1=+,2=,3=+,4=,L?O,?Ui(t,)=eitZRGi(t,)Ui0(0,)d+Zt0ei(ts)ZRGi(,t s)Pi(U1(s,),U2(s,),U30(s ,c),U4(s,)dds,(4.1)ut 0,Gi(t,)=14ditexp?(+ct)24dit?,i=1,2,3,4.duUi0(s,)Cunif,0,i=1,2,3,4,=lim+Ui0(s,)=Ui0(s,),lim+Ui0,(s,)=lim+Ui0,(s,)=0 s ,0.1 2?ak?D/?.?1)-5429(4.1)?4,(RRGi(t s,)d=1,i=1,2,3,4,Kklim+Ui(t,)=eaitZRGi(t,)lim+Ui0(0,)d+Zt0eai(ts)ZRGi(t s,)lim+Pi?U1(s,),U2(s,),U30(s ,c),U4(s,)?dds=eaitUi0(0,)ZRGi(t,)d+Zt0eai(ts)Pi?U1(s,),U2(s,),U30(s ,c),U4(s,)?ZRGi(t s,)dds=eaitUi0(0,)+Zt0eai(ts)Pi?U1(s,),U2(s,),U30(s ,),U4(s,)?ds=gi(t),3t 0,Gi(t s,)=0,2(Gi(t s,)|=0,?lim+kUi(t,)=eitZRkGi(t,)lim+Ui0(0,)d+Zt0ei(ts)ZRkGi(,t s)lim+Pi?U1(s,),U2(s,),U30(s ,c),U4(s,)?dds=eitUi0(0,)ZRkGi(t,)d+Zt0ei(ts)Pi?U1(s,),U2(s,),U30(s ,),U4(s,)?ZRkGi(,t s)dds=0,3t 0,.d,yU3(t,)Cunif,Ui(t,)Cunif0,i=1,2,4.aqy3t n,(n+1),3?)Ui Cunif0,(n+1)Cunif0,(n+1)Cunif,(n+1)Cunif0,(n+1).y3m?)Ui Cunif0,Cunif0,Cunif,Cunif0,3.?e5,k?O,:B1()=c012(0)2+2+2T1 2 h1 w1,B2()=c012(0)2+2+2w4+2T2 2 h2 w2,B3()=c012(0)2+2+2 2 T1 T2 T3 T4 2he2(1(+c)+2(+c)(+c)(),B4()=c012(0)2+2 2 T3,C1()=12cc+2+2T1 2 T3,430A2024C2()=12cc+2+2P2 2 T4,C3()=12cc+2 2 T1 T2 3T3 2T3(e2 1)T4,C4()=12cc+2 2 T4.n3(?)-0 0OCi()=0,i=1,2,3,4?)Ci()=0,i=1,2,3,4,KBi()Ci()0,R.yd()=ec(0),0()()=c,(+c)()=ecc12cc+2+2T1 2 T3=C1(),0 12cc+2+2T2 2 T4=C2(),0 12cc+2 2 T1 T2 T3 2T3e2 T4=12cc+2 2 T1 T2 3T3 2T3(e2 1)T4=C3(),0 12cc+2 2 T4=C4(),0 Ci()0,i=1,2,3,4.y?.n4(k?O)-(U1(t,),U2(t,),U3(t,),U4(t,)Xloc(,)?.X(3.11)-(3.13)?),3?,2eC,?M(0)2,k4Xi=1?kUi(t)k2C+kUi(t)k2H1+Zt0e2(ts)kUi(s)k2H1ds?eCe2t,t 0.yk(3.11)-(3.14)?e2t()Ui(t,),?n12e2tU12ot+e2tnc2U12 U1U1o+e2tU21+e2t0U1U1+e2tnc02+T1 oU21+e2th1U1U3(t ,c)+e2tw1U1U4=e2tU1Q1(t ,c),(4.2)n12e2tU22ot+e2tnc2U22 U2U2o+e2tU22+e2t0U2U2+e2tnc02+T2 oU22+e2tw2U2U4+e2th2U2U3(t ,c)e2tU1U21 2?ak?D/?.?1)-5431=e2tU2Q2(t ,c),(4.3)n12e2tU32ot+e2tnc2U32 U3U3o+e2tU23+e2t0U3U3+e2tnc02+oU23 e2tT3U3U3(t ,c)e2tT4U3U4 e2tT1U1U3 e2tT2U2U3=e2tU3(Q1(t ,c)+Q2(t ,c),(4.4)n12e2tU42ot+e2tnc2U42 U4U4o+e2tU24+e2t0U4U4+e2tnc02+oU24=e2tU3U4.(4.5)dCauchy-Schwarz?,?|e2t0UiUi|e2tU2i+14e2t?0?U2i.“(4.2)-(4.5),?n12e2tU12ot+e2tnc2U12 U1U1o+e2tnc02+T1 14(0)2oU21+e2th1U1U3(t ,c)+e2tw1U1U4 e2tU1Q1(t ,c),(4.6)n12e2tU22ot+e2tnc2U22 U2U2o+e2tnc02+T2 14(0)2oU22+e2tw2U2U4+e2th2U2U3(t ,c)e2tU1U2 e2tU2Q2(t ,c),(4.7)n12e2tU32ot+e2tnc2U32 U3U3o+e2tnc02+14(0)2oU23 e2tT3U3U3(t ,c)e2tT4U3U4 e2tT1U1U3 e2tT2U2U3e2tU3(Q1(t ,c)+Q2(t ,c),(4.8)n12e2tU42ot+e2tnc2U42 U4U4o+e2tnc02+14(0)2oU24e2tU3U4.(4.9)(4.6)-(4.9)3R t,0ut,?e2tk U1k2L2+Zt0ZRe2sn c0+2+2+2T1 2 12(0)2oU21dds+2Zt0ZRe2sh1U1U3(s ,c)dds+2Zt0ZRe2sw1U1U4dds k U10(0)k2L22Zt0ZRe2sU1Q1(s ,c)dds,(4.10)e2tk U2k2L2+Zt0ZRe2sn c0+2+2T2 2 12(0)2oU22dds+2Zt0ZRe2sw2U2U4dds+2Zt0ZRe2sh2U2 U3(s ,c)dds 2Zt0ZRe2sU1U2dds432A2024 k U20(0)k2L22Zt0ZRe2sU2Q2(s ,c)dds,(4.11)e2tk U3k2L2+Zt0ZRe2sn c0+2+2 2 12(0)2oU23dds 2Zt0ZRe2sT3U3U3(s ,c)dds 2Zt0ZRe2sT4U3U4dds 2Zt0ZRe2sT1U1U3dds 2Zt0ZRe2sT2U2U3dds k U30(0)k2L2+2Zt0ZRe2sU3(Q1+Q2)dds,(4.12)e2tk U4k2L2+Zt0ZRe2sn c0+2 2 12(0)2oU24dds 2Zt0ZRe2sU3U4 k U40(0)k2L2.(4.13)2gCauchy-Schwarz?,?|2e2sUi(s,)Uj(s ,c)dds|e2s?U2i(s,)+U2j(s ,c)?,i=1,2,j=1,2,3.,:?2Zt0ZRe2s()i()Uj(s,)U3(s ,c)dds?Zt0ZRe2s()i()U2j(s,)dds+Zt0ZRe2s()i()U23(s ,c)dds=Zt0ZRe2s()i()U2j(s,)dds+e2ZtZRe2s(+c)i(+c)U23(s,)ddsZt0ZRe2s()i()U2j(s,)dds+e2Zt0ZRe2s(+c)i(+c)U23(s,)dds+e2Z0ZRe2s(+c)i(+c)U230(s,)dds.(4.14)(4.14)“(4.10)-(4.13),?e2tk U1k2L2+Zt0ZRe2sn c0+2+2+2T1 2 12(0)2 h1 w1oU21dds he2Zt0ZRe2s()1(+c)(+c)()U23dds Zt0ZRe2sw1U24dds k U10(0)k2L2+he2Z0ZRe2s(+c)1(+c)U230dds 2Zt0ZRe2sU1Q1(s ,c)dds,(4.15)e2tk U2k2L2+Zt0ZRe2sn c0+2+2T2 2 12(0)2 h2 w2 oU22dds he2Zt0ZRe2s()2(+c)(+c)()U22dds k U20(0)k2L22Zt0ZRe2s()U2Q2(s ,c)dds+he21 2?ak?D/?.?1)-5433Z0ZRe2s()2(+c)(+c)()U230dds+Zt0ZRe2sU21dds,(4.16)e2tk U3k2L2+Zt0ZRe2sn c0+2+2 2 12(0)2 T1 T2 T3 T4oU23dds he2Zt0ZRe2s()(1(+c)+2(+c)(+c)()U23dds k U30(0)k2L2+2Zt0ZRe2sU3(Q1(s ,c)+Q2(s ,c)dds+Zt0ZRe2s(T4U24+T1U21 T2U22)dds+he2Z0ZRe2s()(+c)()(1(+c)+2(+c)U230dds,(4.17)e2tk U4k2L2+Zt0ZRe2sn c0+2 2 12(0)2 oU24dds Zt0ZRe2sU23dds k U40(0)k2L2.(4.18)LBi Ci,(4.15)-(4.18)?4Xi=1?e2tkUik2L2+Zt0ZRCie2sU2idds?4Xi=1kUi(0)k2L2+Z0ZRe2s()?2h1(+c)+2h2(+c)?e2(+c)()U230dds+2Zt0ZRe2sU1Q1(s ,c)dds+2Zt0ZRe2sU2Q2(s ,c)dds 2Zt0ZRe2sU3(Q1(s ,c)+Q2(s ,c)dds4Xi=1kUi(0)k2L2+2supT3e2Z0ZRe2s()U230dds+2Zt0ZRe2sU1Q1(s ,c)dds+2Zt0ZRe2sU2Q2(s ,c)dds 2Zt0ZRe2sU3(Q1(s ,c)+Q2(s ,c)dds.(4.19)dTaylor?|Q1(t ,c)|=|hU1U3(t ,c)+wU1U4|h|U1(t,)|U3(t ,c)|+w|U1(t,)|U4(t,)|,|Q2(t ,c)|=|hU2U3(t ,c)+wU2U4|h|U2(t,)|U3(t ,c)|+w|U2(t,)|U4(t,)|.dM()?,|Ui(t,)|CM(),t 0,R,|Ui(t,)|CM(),t 0,R,C 0 i,d5?O2?Zt0ZRe2sU1Q1(s ,c)dds?CM()nZt0ZRe2s?|U1(s,)|2+|U4(s,)|2?dds+Z0ZRe2s|U30(s,)|2ddso,(4.20)434A20242?Zt0ZRe2sU2Q2(s ,c)dds?CM()nZt0ZRe2s?|U2(s,)|2+|U4(s,)|2?dds+Z0ZRe2s|U30(s,)|2ddso.(4.21)d,?(4.19)z4Xi=1e2tkUik2L2+Xi=1,2,4(Ci()CiM()e2sU2idds+Zt0ZRC3()e2sU23ddsC?4Xi=1kUik2L2+Z0ZRe2sU230dds?.(4.22)?2 0,?Ci CiM()Ci Ci2 0,K4Xi=1e2t?kUik2L2+Zt0e2(ts)kUi(s)k2L2ds?C?4Xi=1kUi0(0)k2L2+Z0kU30(s)k2L2ds?.(4.23)aq?3?(3.11)-(3.14)e2t()Ui(t,),?n12e2tU21ot+e2tnc2U21 U1U1o+e2tnc02+T1 14(0)2oU21+e2t?h1U1U3(t ,c)+w1U1U4?=e2tU1Q01(t ,c),(4.24)n12e2tU22ot+e2tnc2U22 U2U2o+e2tnc02+T2 14(0)2oU22+e2t?w2U2U4+h2U2U3(t ,c)U1U2?=e2tU2Q02(t ,c),(4.25)n12e2tU23ot+e2tnc2U23 U3U3o+e2tnc02+14(0)2oU23e2t?T3U3U3(t ,c)+T4U3U4+T1U1U3+T2U2U3?=e2tU3(Q1(t ,c)+Q2(t ,c),(4.26)n12e2tU24ot+e2tnc2U24 U4U4o+e2tnc02+14(0)2oU24=e2tU3U4.(4.27)Q01=T1U1+w1U4+h1U3(t ,c)+wU1U4+wU1U4+hU1U3(t ,c)+hU1U3(t ,c),Q02=T2U2+w2U4+w2U3(t ,c)+wU2U4+wU2U4+wU2U3(t ,c)+wU2U3+wU2U3(t ,c).1 2?ak?D/?.?1)-5435(4.24)-(4.27)tO3R t,0,?e2tkU1k2L2+Zt0ZRe2sn c0+2+2+2T1 2 12(0)2 h1 w1oU21dds he2Zt0ZRe2s1(+c)(+c)()U23dds Zt0ZRe2sw1()U24ddskU10(0)k2L2+he2Z0ZRe2s()1(+c)(+c)()U230dds 2Zt0ZRe2sU1Q01(s ,c)dds,(4.28)e2tkU2k2L2+Zt0ZRe2sn c0+2+2T2 2 12(0)2 h2 w2 oU22dds he2Zt0ZRe2s2(+c)(+c)()U22dds k U20(0)k2L2+Zt0ZRe2sU21dds 2Zt0ZRe2sU2(s,)Q02(s ,c)dds+he2Z0ZRe2s2(+c)(+c)()U230dds,(4.29)e2tkU3k2L2+Zt0ZRe2snc0+2+2 2 12(0)2 T1 T2 T3 T4oU23dds he2Zt0ZRe2s(1(+c)+2(+c)(+c)()U23dds k U30(0)k2L2+2Zt0ZRe2sU3(Q01(t ,c)+Q02(t ,c)dds+Zt0ZRe2s?T4U24+T1U21 T2U22?dds+he2Zt0ZRe2s(+c)()(1(+c)+2(+c)U230dds,(4.30)e2tkU4k2L2+Zt0ZRe2snc0+2 2 12(0)2 oU24dds k U40(0)k2L2+Zt0ZRe2sU23dds.(4.31)n?e?U?O:4Xi=1e2t?kUik2L2+Zt0e2(ts)kUi(s)k2L2ds?C?4Xi=1kUi0(0)k2H1+Z0kU30(s)k2H1ds?,t 0,i=1,2,3,4.(4.32)(4.23)(4.32),?Xe(:4Xi=1e2t?kUik2L2+Zt0e2(ts)kUi(s)k2L2ds?C?4Xi=1kUi0(0)k2H1+Z0kU30(s)k2H1ds?.(4.33)436A2024-Ui H1(R),i=1,2,3,4,Ui H1(R)kUikC CkUikH1,supx(,0kUi(t,)k Cet,t 0,i=1,2,3,4.TnyLaqu19,?.eyUi(t,)3=+?P5.,Ui(t,)Xloc(,),?d?3t 0,),Cunif0,),lim+Ui(t,)3.XJ 0,i=1,2,3,4.(4.34)yw,Ui(t,)Cunif0,t0Cunif0,t0Cunif,t0Cunif0,t0,lim+U3(t,)3ut ,+)lim+Ui(t,)(i=1,2,4)3ut 0,+).-eUi(t,x)=ui(t,x)i(x+ct),i=1,2,3,4,eU10(0,x)=u10(x)1(x),i=1,2,4,eU30(s,x)=u30(s,x)3(x+cs),s ,0,x R.?lim+Ui(t,)=lim+eUi(t,)3ut 0,),KeUi(t,x)=Ui(t,),veU1t(t,)+(+wk4+hk3)eU1(t,)+hk1eU3(t ,)+wk1eU4(t,)=Q1(t ,),eU2t(t,)+(+wk4+hk3)eU2(t,)+wk2eU4(t,)+hk2eU3(t ,)eU1(t,)=Q2(t ,),eU3t(t,)+(+)eU3(t,)(wk4+hk3)eU1(t,)(wk4+hk3)eU2(t,)(hk1+hk2)eU3(t ,)(wk1+wk2)eU4(t,)=Q1(t ,)+Q2(t ,),eU4t(t,)+eU4(t,)=eU3(t,),eU3(s,)=eU30(s,),s ,0,eUi(0,)=eUi0(0,),i=1,2,4.?O?5,kA?5X:u(t,)v(t,)w(t,)z(t,)0=Au(t,)v(t,)w(t,)z(t,)+Bu(t ,)v(t ,)w(t ,)z(t ,),A=wk4 hk300wk1 wk4 hk30wk2wk4+hk3wk4+hk3(+)wk1+wk200,1 2?ak?D/?.?1)-5437B=00hk1000hk2000hk1+hk200000.:(eU10(),eU20(),eU30(s,),eU40()T,s ,0.?|:u(t,)v(t,)w(t,)z(t,)0=Pu(t,)v(t,)w(t,)z(t,),P=A+B./Routh-Hurwitzn,?=?|I P|=0,?P?kA?kK,!.15,?en:n53?02,?0.(4.35)d,30?1,supx0,+)|eUi(t,x)|CM(0)e2t,t 0,).(4.33)n5,3?bC?0?1M2()bCM2(0)e2t,t 0,(4.36)0 min1,2.e?0:=bC,?MV().d(4.36),?ut 0,M()pbCM(0)et.?M(0)0,7kMV()pbCM(0)etpbCM(0)=b.(4.37)y.z1 CODECO C 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delayJ.Tainwanese Journal of Mathematics,2017,21(6):1381-1411.19 ZHANG G B.Global stability of non-monotone traveling wave solutions for a nonlocal dispersalequation with time delayJ.Journal of Mathematical Analysis and Applications,2019,475(1):605-627.20?,=.*?5?.?1)J.A,2023,36(2):327-342.Stability of Traveling Waves of a Cholera Model withVaccination and Time DelayLIAO Shu,HUANG Chenchen(School of Mathematics and Statistics,Chongqing Technology and Business University,chongq