具有媒体报道的SVIR传染病模型的生存性分析.pdf

2023,43A(5):15951606http:/kxN?SVIRD/?.?)51o2=3fJ(14O?4 350116;24$4p?-:?4 350116;3g4?OX.dx G1 1XH):T?kLogisticOxN?u)?SVIR?.?.?5,ky?.?)?35,gL?E?o,&;?5.L:?Rs0 1,;mY3.?Re0 1,;361m?.?,L?y?(.c:D/?.;xN?;55;-.MR(2010)Ka:60H10;92B05a:O175.13zI:A?:1003-3998(2023)05-1595-121?;3,/y,T/?;%?&?“;D,-?“LDx,?1“L3xN?$?.,xN?U?D/?D,d,1?b?n?.?ug(I)=I1+aIN?a/?1CzP?A,a?11.?a/?,g(I)Cu1a1,3,21,25,26,28,30,34,35,38.z30k?u)?D/?.,L:?Jp?a/?$.u6,13,14,17,19,24,27,36.?b?,?aCz3mSlLogisticO2,8,15,20,29,?Xm?”?2g?a+,|?u)a/?P?A,vF:2022-04-22;?F:2022-10-31E-mail:78:Ig,7-IS(/)?68(61911530398)!4Ee8(2021L3018)!4g,7(2021J01621)!=I(WM160014,=I?)!=I7(NA160317,-p?7)n”?(EP/K503174/1)Supported by the National Natural Science Foundation of China(61911530398),the Special Projectsof the Central Government Guiding Local Science,Technology Development(2021L3018),the Natu-ral Science Foundation of Fujian Province of China(2021J01621),the Royal Society,UK(WM160014,Royal Society Wolfson Research Merit Award),the Royal Society and the Newton Fund,UK(NA160317,Royal Society-Newton Advanced Fellowship),the EPSRC,the Engineering and PhysicalSciences Research Council(EP/K503174/1)1596n?Vol.43 AdSVIR?.dS(t)=?S?1 SK?+V S SI?dt+1SdB1(t),dV(t)=?S V?1Ib+I?V I1+aI V?dt+2V dB2(t),dI(t)=?SI+?1Ib+I?V I1+aI(+)I?dt+3IdB3(t),dR(t)=(I R)dt+4RdB4(t).(1.1),S(t),V(t),I(t),R(t)OL;L”?;La?;a,b?;B1(t),B2(t),B3(t),B4(t)p?IOK$;1,2,3,4LxD(?r;(,F,Ftt0,P)?Vm,fFtt0v.duE?CzKa!?a/?,?.(1.1)CdS(t)=?S?1 SK?+V S SI?dt+1SdB1(t),dV(t)=?S V?1Ib+I?V I1+aI V?dt+2V dB2(t),dI(t)=?SI+?1Ib+I?V I1+aI(+)I?dt+3IdB3(t).(1.2)e,y?.(1.2)?)?35,H-3?,9;?.2)?355?.(1.2)?)?355?.5.?t 0,?.(1.2)?)PX(t)=(S(t),V(t),I(t)T.dB(t)=(dB1(t),dB2(t),dB3(t)T.?Rnnm,X(t)Rn?g?L,KdX(t)=F(X(t)dt+nXl=1gl(X(t)dBl(t),-X(0)=X0 Rn,gl=(gl1,gl2,gln)T*?A(X)=(aij(X)nn,aij(X)=nPl=1gil(X)gjl(X).?fL=nXl=1Fi(X)Xi+12nXl=1aij(X)2XiXj.z21,34,35,38,N?n2.1,y!.n 2.1u?(S(0),V(0),R(0)T R3+,?.(1.2)3?),T)V133R3+.z40,41?(,?n2.1,yL.No.5o?:kxN?SVIRD/?.?)51597n 2.1?.(1.2)?)X(t)ke5limt1tXi(t)=0,limt1tlnXi(t)6 0(i=1,2,3)a.s.e max21,22,23 1,21 2(),max21,22,23 (1+E)(Rs0 1)0a.s.y?EXeC2V1=a+(S+V+I)c1lnV c2lnI,V2=2(S+V)3KlnS,c1c2?,V1AIt o,?dV1=LV1dt+aS1+dB1(t)+?aV+c1?2dB2(t)+?aI+c2?3dB3(t),LV1 E aI c1SV+c1p2+c2p3+?1Ib+I?c1I1+aI c2V1+aI?c1SVc2?1?V1+aI(1+aI)+1+c1p2+c2p3+E+c1(2 1)I 33pc1c2(1)S+c1p2+c2p3+E+c1(2 1)I+1,(2.3)E=maxSR+naS+?1 SK?c2So=Ka c2(+)24a(+).(2.4)V2AIt o,?dV2=LV2dt+?2S3K1?1dB1(t)+2V3K2dB2(t),LV22S3K?1 SK?+SK 1+I+2123rSK+I?1 212?.(2.5)V3=V1+33pc1c2(1)KV2,kV3AIt o,?(2.3)(2.5),?LV3 1+E+c1p2+c2p3+3I3pc1c2(1)K 33pc1c2(1)K?1 212?+c1(2 1)I.(2.6)1598n?Vol.43 A(2.2),-c1=p1p22p3,c2=p1p2p23,dLV3,2t,(2.7)?1tV3(t)V3(0)0.y.3-?35z38,n3.3,z5,55.1,?n3.1.n 3.1eRs0 1,K?.(1.2)3-,kH5.yk.8D=nX R3+,6 S 61,26 V 612,6 I 61o,(3.1)p?,veAM+6 minnm(m+2)(+),(m+3)2K,m+3m+2o,(3.2)F+1 6 minn,4Km+3,m(+)2m+2,m2m+4o.(3.3)?.(1.2)*?A=diag21S2,22V2,23I2=(aij)33,?.(1.2)?*?=minXD21S2,22V2,23I2 0,No.5o?:kxN?SVIRD/?.?)51599?X D,=(1,2,3)T R3+,knXi,j=1aijij=(1,2,3)A(1,2,3)T=(1S)221+(2V)222+(3I)223 kk2,Xz5,55.1(i)v.?ELyapunovW=M(V3+V4)+V5+V6,V4=A+(V+I),V5=1m+2(S+I+V)m+2,V6=lnV,m 0?vm 0?vM+B+222+2 1a6 2.(3.5)w,W(X)Y,3?W(X),d?EK?C2Q=M(V3+V4)+V5+V6 W(X).(3.6)gV4,V5,V6A It o,?LV4=A+hS (+)V+SI (+)IiAS+ASI+AI,(3.7)LV5=(S+V+I)m+1hS?1 SK?V (+)Ii+m+12(S+V+I)m(21S2+22V2+23I2)S(S+V+I)m+1KSm+3 Vm+2(+)Im+2+m+12(S+V+I)m+2(21 22 23),(3.8)LV6=SV+?1Ib+I?I1+aI+p26 SV+2 1a+p2,(3.9)B=maxXR3+n2KSm+3(1 m)Vm+2(+)(1 m)Im+2+S(S+V+I)m+1+AMS+m+12(S+V+I)m+2(21 22 23)o,(2.7)(3.7)(3.9),?LQ MASI+M 2KSm+3 mVm+2 m(+)Im+2+B SV+2 1a+p2.(3.10)1600n?Vol.43 AR3+D8fm,y LQ 38mSvLQ 1,R3+D=D1 D2D3 D4 D5 D6.D1=nX R3+,0 S o,D2=nX R3+,0 I ,I ,0 V 1o,D5=nX R3+,I 1o,D6=nX R3+,V 12o.(3.11)/1eX D1,KSI 6 I 6 m+1+Im+2m+2,(3.2),(3.5),(3.10),?LQ 2+AM(m+1)(+)(m+2)+hAM(+)(m+2)m(+)iIm+26 1./2eX D2,KSI 6 S 6 m+2+Sm+3m+3,(3.2),(3.5),(3.10),?LQ 2+AM(m+2)(+)(m+3)+?AM(+)(m+3)2K?Sm+36 2+1=1./3eX D3,KL(3.3),(3.10)?LQ SV+F +F 6 1,F=supXR3+?AMSI+4KSm+3m2(+)Im+2+B+p2+2 1a?./4e X D4,KL(3.3),(3.10)?LQ 4KSm+3+F 4Km+3+F 6 1./5eX D5,KL(3.3),(3.10)?LQ m2(+)Im+2+F (+)m2m+2+F 6 1./6eX D6,KL(3.3),(3.10)?LQ mVm+2+F m2m+4+F 6 1.z5,55.1(ii)v;d?.(1.2)3H?-.y.4;?5n 4.1evRe0=K(+2 1)4?+0.523?1,max21,22,23 l0?t,?2t,?1?/1tS(t)S(0)?(.4limsupthSitK4+limsupthV it.(4.3)1 D2(t)=(1S,2V,3I),?.(1.2)nk2t?hV itDS?1 SK?Et+2(t)K4+2(t),(4.4)2(t)=1tZt0D2(s)dB(s)1tS(t)S(0)1tV(t)V(0)1tI(t)I(0),dn2.1 limsupt2(t)=0,(4.4)?(.4?limsupthV itK4.(4.5)y3lnI(t)AIt o,?dlnI(t)=LlnI(t)dt+3dB3(t),(4.6)LlnI(t)=S+?1Ib+I?V I1+aI p3.(4.6)0?t,2t,?1tlnI(t)lnI(0)hSit+(2 1)hV it p3+3B3(t)t,(4.7)rn23?limt3B3(t)t=0,(4.7)?4,Kklimsupt1tlnI(t)limsupthSit+(2 1)limsupthV it p3K4?1+?+(2 1)K4 p3=p3(Re0 1)1,0.15=0.5(21 22 23)=0.00125,0.7=2()21=0.0025.X1,a!?a/3mS;X2,?1Cz,a?O,a/?,?a!?a/E,?;l345?,?OO,a?k?5.L:3D/D?,xN?Zk|u;.0246810 x 10400.511.522.533.544.5MinutesDensitiesSVI 1a!?!a/?505000100001500000.511.522.533.54MinutesDensities for Sbeta1=0.1beta1=0.8505000100001500000.10.20.30.40.50.60.70.80.91MinutesDensities for Vbeta1=0.1beta1=0.8505000100001500000.511.522.5MinutesDensities for Ibeta1=0.1beta1=0.85 2?1O,a!a/!?CzNo.5o?:kxN?SVIRD/?.?)5160305000100001500000.511.5MinutesDensities for Szeta=0.3zeta=0.805000100001500000.10.20.30.40.50.60.70.80.91MinutesDensities for Vzeta=0.3zeta=0.8 3?O,a?Cz05000100001500000.511.5MinutesDensities for Stheta=0.2theta=0.905000100001500000.10.20.30.40.50.60.70.80.91MinutesDensities for Vtheta=0.2theta=0.9 4?O,a!?Cz 5.2;?53df?,?.(1.2)S(0)=1,V(0)=0.8,I(0)=0.8,9?=0.65,K=1.25,=0.9,1=0.1,=0.65,=0.35,=0.3,=0.2,=0.8,=0.5,1=0.05,2=0.05,3=0.05,a=0.85,b=0.3.?vn4.1?Re0 0.9628 0.5(21 22 23)=0.0012.05000100001500000.10.20.30.40.50.60.70.80.91MinutesDensities for Izeta=0.3zeta=0.805000100001500000.10.20.30.40.50.60.70.80.91MinutesDensities for Itheta=0.3theta=0.8 5?O,;?51604n?Vol.43 A00.511.52x 10400.10.20.30.40.50.60.70.80.91MinutesDensities for Itau=0.45tau=0.6505000100001500000.10.20.30.40.50.60.70.80.91MinutesDensities for Isigma3=0.05sigma3=0.3 6?3O,;?500.511.52x 10400.10.20.30.40.50.60.70.80.91MinutesDensities for Idelta=0.35delta=0.5505000100001500000.10.20.30.40.50.60.70.80.91MinutesDensities for Imu=0.5mu=0.8 7?OO,;?505000100001500000.10.20.30.40.50.60.70.80.91MinutesDensities for Ibeta1=0.1beta1=0.8 8?1O,;?5L:?,3O?(57),a/u?m;?1O,xN?;?k?K(8).No.5o?:kxN?SVIRD/?.?)51605 z1 Adimy M,Chekroun A,Kuniya T.Traveling waves of a differential-difference diffusive Kermack-McKendrickepidemic model with age-structured protection phase.J Math Anal Appl,2022,505(1):1254642 Cao Z W,Feng W,Wen X D.Dynamical behavior of a stochastic SEI epidemic model with saturationincidence and logistic growth.Physica A,2019,523:8949073 Capasso V,Serio G.A generalization of the KermackMcKendrick deterministic epidemic model.MathBiosci,1978,42:43614 Caraballo T,El Fatini M,Pettersson R,et al.A stochastic SIRI epidemic model with relapse and mediacoverage.Discrete Cont Dyn Sys B,2018,23(8):348335015 Chen L H,Wei F Y.Persistence and distribution of a stochastic 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of a singlespecies population model with fluctuations and migrationsbetween patches.Appl Math Model,2020,81:11312738 Wei F Y,Xue R.Stability and extinction of SEIR epidemic models with generalized nonlinear incidence.Math Comput Simul,2020,170:11539 Zhang Y,Fan K G,Guo S J,et al.Ergodic stationary distribution of a stochastic SIRS epidemic modelincorporating media coverage and saturated incidence rate.Physica A,2019,514:67168540 Zhao Y N,Jiang D Q.Dynamics of stochastically perturbed SIS epidemic model with vaccination.AbstrAppl Anal,2013,Article number 51743941 Zhao Y N,Jiang D Q.The threshold of a stochastic SIS epidemic model with vaccination.Appl MathComput,2014,243:71872742 Zhao Y,Zhang Y P,Yuan S L.The effect of media coverage on threshold dynamics for a stochastic SISepidemic model.Physica A,2018,512:248260Survival Analysis of an SVIR Epidemic Model with Media Coverage1Li Dan2Wei Fengying3Mao Xuerong(1School of Mathematics and Statistics,Fuzhou University,Fuzhou 350116;2Key Laboratory of Operations Research and Control of Universities in Fujian,Fuzhou University,Fuzhou 350116;3Department of Mathematics and Statistics,University of Strathclyde,Glasgow G1 1XH,UK)Abstract:We consider the long-term properties of a stochastic SVIR epidemic model with mediacoverage and the logistic growth in this paper.We firstly derive the fitness of a unique global positivesolution.Then we construct appropriate Lyapunov functions and obtain the existence of ergodicstationary distribution when Rs0 1 is valid,and also derive sufficient conditions for persistence in themean.Moreover,the exponential extinction to the density of the infected is figured out when Re0 1holds.Key words:Epidemic model;Vaccination;Media coverage;Persistence and extinction;Stationarydistribution.MR(2010)Subject Classification:60H10;92B05