关于SARS病毒传播的数学模型

第22卷第2期毕节师范高等专科学校学报JOURNALOFBIJIETEACHERSCOLLEGE2004年6月:(1975)),,,:关于SARS病毒传播的数学模型t(毕节师专数学系,贵州毕节551700):根据SARS病毒传播的特性,建立了SARS病毒传播的常微分方程模型利用数学软件Matlab求解了此建立的SARS病毒传播的常微分方程模型,画出了有关的图形利用相轨线性质,讨论了SARS病毒传播的常微分方程模型解的有关性质,描述了SARS病毒传播的整个过程,预测了SARS病毒传播的高峰期,对SARS病毒传播的预防控制提供了有关的数学模型理论:数学模型;相轨线;特征指数;阈值:O141.4:A:1672-0296(2004)02-0046-07OnMathematicalModeloftheSpreadofSARSVirusLiWei(DepartmentofMathematics,BijieTeachersCollege,Bijie,551700,Guizhou)Abstract:AccordingtothespreadcharacteristicsofSARSvirus,thispaperestablishestheordinarydifferentialequationmodelsofthespreadofSARSvirus,whosesolutionsareobtainedandgraphesconcernedaredrawnbymathe-maticalsoftware,Matlab,takingadvantageofthequalitiesofphasetrajectory,discussesthecharacteristicsconcernedthesolution,describesthewholeprocessofthespreadofSARSvirus,forecaststhepeakperiodofthespreadandprovidestherelevantmathematicalmodeltheoryaboutthepreventionandcontrolofthespreadofSARSvirus.Keywords:MathematicalModel;PhaseTrajectory;CharacteristicExponent;ThresholdValue1,,[1][2][3]SARS,,,,,SARS(SevereAcuteRespiratorySyndrome),,:,21SARS,,,SARS,,SARS,,#46#SARS,/0,/0,/0,SARS,SARS/0,,,,/0,,SARS[4][5]SARSSARS2,,,,,SARShttp:PPmcm.edu.cn[6]:N0,K(K),LL,:N(t)=N0(1+K)tN(t)=N0(1+K)t,,,,,,,K,,,3,:i(t):tSARS;i(0)=i0:SARS;N:;K:;r(t):t();:(1)i(t)t;(2)k0;tt+$ti(t+$t)-i(t)=k0i(t)$t(1)[7]:di(t)d(t)=K0i(t)i(0)=i0(2),(2)i(t)=i0ek0t(3)#47#:(3):SARSSARS,,SARS,,ty],i(t)y]SARS,,,,:k0,,:(1)s(t)t(),i(t)t()(2)t(),k0=ks(t);(3)N,,s(t)+i(t)=N;:di(t)dt=ks(t)i(t)s(t)+i(t)=Ni(0)=i0(4)(4)i(t)=N1+Ni0-1e-kNt(5)i(t)~tdidt~i23#48#:i(t)=12N,didt(5)didt=kN2Ni0-1e-kNt1+Ni0-1e-kNt2(6)d2i(t)dt2=0,tm=lKNlnNi0-1,didtdidtmaxty],i(t)yN,tmK,KN,tm,,SARS,tm,K,,,,SARS,K,SARStm,SARS:ty],i(t)yN,,,,,,:(1),,,3:,i(t)t;,s(t)t;,,,r(t)t(2)N,;(3):k=k0s(t)()l=dr(t)d(t)Pi(t)(l)(7):dr(t)dt=li(t)di(t)dtKs(t)i(t)-dr(t)dtdsdt=-di(t)dt-dr(t)dt(8)i(0)=i0,s(0)=s0,r(0)=r0=N-i0-s0,di(t)dt=Ks(t)i(t)-li(t)ds(t)dt=-Ks(t)i(t)drdt=li(t)i(0)=i0,s(0)=s0,r(0)=N-s0-i0(9)#49#(9),:,s(t)i(t),,(9),k=0.02,Q=0.5,i(0)=0.01,s(0)=0.80,MATLAB[8]:functiony=ill(t,x)a=1;b=0.5;y=[a*x(1)*x(2)-b*x(2),-a*x(1)*x(2)];ts=0:50;x0=[0.01,0.80][t,x]=ode45(cillc,ts,x0;[t,x]plot(x(:1),t,x(:2)),grid,pauseplot(x(:2),x(:,1)),grid.:t123456789i(t)0.29660.18020.15150.10460.11740.11230.11010.07080.1098s(t)0.98000.95250.90190.81690.69270.53480.39950.28390.1493t1015202530354045i(t)0.14370.07270.05070.06030.03430.05000.03210.0434s(t)0.11450.05430.04340.04080.03990.03980.03940.039014i~s,[9]:s(t)i(t),i(t)s(t)i(t)s(t)s~i,D={(s,i)|sE0,iE0,s+iF1}(9)dt:dids=-1+lk#1sQ=lk(Q)dids=Qs-1#50#:i(s)=Qlnss0-s+s0+i0(10)(10),55:(1),s,;(2)s](ty]);(3)s0Q,i(t),s=Q,i(t)im=s0+i0-Q(1+lns0Q)(11),i(t),s(t)s]5p1(s0,i0)(4)s0FQ,,i(t),s(t)s]5P2(s0,i0),Q,,,,,,,œ,SARS,,,,SARS1[10],60,,D=kQ,,k,Q,s0s](10),is=si+i0+1D(l+lnDs0),i0,D=lns0-lns]s0-s],,ky0,Qy0,D=lns0-lns]s0-s]4,[10],430,,4301440,,k,4251440,7,n3100,55,,N(t)=1440(l+0.13913)5=27621897#51#73100,,,,,!,,,,,1-4,,,4.5,0.5,41.7,,,,,,,,,5(1)SARS()(2)(3),,::[1]刘来福,曾文艺.数学模型与数学建模[M].北京:北京师范大学出版社,1997.4.[2]李乔祥,论数学建模竞赛对提高学生综合素质的作用[J].高等理科教育,2004,(1):60-63.[3]林李.赛程编排的数学模型[J].广西商业高等专科学校学报,2003,(4):88-92.[4]张利军,程代展,洪奕光.北京市SARS疫情统计分析[J].数学的实践与认识,2003,(10):102-109.[5]王嘉丽等.SARS病毒的研究进展[J].第三军医大学学报,2003,(9):828-830.[6]SARS疫情分析及对北京疫情走势的预测[Z].http:PP[7]杨启帆方道元.数学建模[M].杭州:浙江大学出版社,1999.69.[8]龚剑,朱亮.MATLAB5.X入门与提高[M].北京:清华大学出版社,2000.45.[9]姜启源.数学模型[M].北京:高等教育出版社,1992.115.[10]北京市疫情的数据http:PP=66070[Z].2003-9.()#52#