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()(200444)(November15,2009)2009.11.13-15.1$$$$,_xi(t)=f(xi(t))+c(t)NXj=1aijHxj(t);i=1;2;:::;N;(0.1)xi(t)2Rntif(¢)c(t)0H2Rn£nA=(aij)N£Nijaij=1aij=0i2f1;2;:::;Ngaii=¡PNj=1;j6=iaijA0=¸1¸¸2¸¢¢¢¸NSISSIS_Ik(t)=¸k[1¡Ik(t)]£(t;k)¡Ik(t);(0.2)k=1;2;:::;dmdm(0.2)2£(t;k)=Xk0p(k0jk)Ik0(t);p(k0jk)kk0SIS_c(t)=®I(t)LXl=11Nl¡1Xj2Clkxil(t)¡xj(t)k1+kxil(t)¡xj(t)k;(0.3)i=1;2;:::;Nk=1;2;:::;dm®0I(t)=Pdmk=1p(k)Ik(t)CllClusterNl¸2Clil2ClClL=1(0.1)[5]Lemma1(0.1)c(t)fLyapunovhmaxH=INT0tTc(t)hmaxj¸2j;(0.1)Theorem1SIS(0.1),(0.2),(0.3)¸¸cxi(t);i=1;2;:::;N¸xi(t);i=1;2;:::;N¸c(0.2)epidemicthreshold.)}3References[1]R.M.AndersonandR.M.May,Infectiousdiseasesofhumans,Oxford:OxfordUniversityPress,1992.[2]A.-L.Barab¶asiandR.Albert,Emergenceofscalinginrandomnetworks,Science,286(1999)509-512.[3]M.Bogu~n¶a,R.P.SatorrasandA.Vespignani,Absenceofepidemicthresholdinscale-freenetworkswithdegreecorrelations,Phys.Rev.Lett.,90(2003)028701.[4]D.H.KimandA.E.Motter,Ensembleaverageabilityinnetworkspectra,Phys.Rev.Lett.,98(2007)248701.[5]X.LiandG.Chen,Synchronizationanddesynchronizationofcomplexdynamicalnetworks:anengineeringviewpoint,IEEETrans.Circ.Syst.-I,50(2003)1381-1390.[6]A.L.LloydandR.M.May,Howvirusspreadamongcomputersandpeople,Science,292(2001)1316-1317.[7]R.M.MayandA.L.Lloyd,Infectiondynamicsonscale-freenetworks,Phys.Rev.E,64(2001)066112.[8]Y.Moreno,J.B.G¶omezandA.F.Pacheco,Epidemicincidenceincorrelatedcomplexnetworks,Phys.Rev.E,68(2003)035103(R).[9]Y.Moreno,R.Pastor-SatorrasandA.Vespignani,Epidemicoutbreaksincomplexheterogeneousnetworks,Eur.Phys.J.B,26(2002)521-529.[10]J.D.Murray,MathematicalBiology,SpringerVerlag,Berlin,1993.[11]R.Pastor-SatorrasandA.Vespignani,Epidemicdynamicsandendemicstatesincomplexnetworks,Phys.Rev.E,63(2001)066117.[12]R.Pastor-SatorrasandA.Vespignani,Epidemicspreadinginscale-freenetworks,Phys.Rev.Lett.,86(2001)3200-3203.[13]R.Pastor-SatorrasandA.Vespignani,Epidemicdynamicsin¯nitesizescale-freenetworks,Phys.Rev.E,65(2002)035108(R).[14]C.J.Tessone,M.CenciniandA.Torcini,Synchronizationofextendedchaoticsystemswithlong-rangeinteractions:ananalogytol¶evy-°ightspreadingofepidemics,Phys.Rev.Lett.,97(2006)224101.[15]G.Yan,Z.Q.Fu,J.RenandW.X.Wang,Collectivesynchronizationinducedbyepidemicdynamicsoncomplexnetworkswithcommunities,Phys.Rev.E,75(2007)016108.4