2026\u7f8e\u8d5b\u671f\u95f4\u4f1a\u6301\u7eed\u66f4\u65b0\u76f8\u5173\u5185\u5bb9，\u6240\u6709\u5185\u5bb9\u4f1a\u53d1\u5e03\u5230\u4e13\u680f\u5185，\u4f1a\u7ed3\u5408\u6700\u65b0\u7684chatgpt\u53d1\u5e03，\u53ea\u9700\u8ba2\u9605\u4e00\u6b21，\u8d5b\u540e\u4e24\u5929\u534a\u4ef7，\u5185\u5bb9\u8fbe\u4e0d\u5230\u6240\u6709\u4eba\u9884\u671f，\u8bf7\u52ff\u76f2\u76ee\u8ba2\u9605！！！\u65e0\u8bba\u6587！\u65e0\u8bba\u6587！！！<\/p>\n<\/p>\n
\u5e38\u5fae\u5206\u65b9\u7a0b（Ordinary Differential Equations, ODEs）\u4f5c\u4e3a\u63cf\u8ff0\u7cfb\u7edf\u52a8\u6001\u6f14\u5316\u7684\u6570\u5b66\u5de5\u5177，\u5728\u6570\u5b66\u5efa\u6a21\u4e2d\u5360\u636e\u6838\u5fc3\u5730\u4f4d\u3002\u672c\u6587\u4ece\u6570\u5b66\u5efa\u6a21\u7ade\u8d5b\u7684\u5b9e\u7528\u89c6\u89d2\u51fa\u53d1，\u7cfb\u7edf\u9610\u8ff0ODE\u5728\u4eba\u53e3\u589e\u957f\u548c\u4f20\u67d3\u75c5SIR\u6a21\u578b\u4e2d\u7684\u5e94\u7528\u3002\u9996\u5148，\u6df1\u5165\u63a2\u8ba8ODE\u7684\u6838\u5fc3\u601d\u60f3\u4e0e\u6570\u5b66\u6a21\u578b，\u5305\u62ec\u57fa\u672c\u6982\u5ff5\u3001\u5206\u7c7b\u548c\u89e3\u7684\u7ed3\u6784\u7279\u6027\u3002\u5176\u6b21，\u5206\u6790ODE\u5728\u5efa\u6a21\u7ade\u8d5b\u4e2d\u7684\u9002\u7528\u573a\u666f\u4e0e\u5178\u578b\u8d5b\u9898\u7c7b\u578b\u3002\u63a5\u7740，\u8be6\u7ec6\u8bf4\u660e\u5177\u4f53\u5efa\u6a21\u6b65\u9aa4\u4e0e\u5173\u952e\u6280\u5de7，\u5e76\u63d0\u4f9b\u5e38\u7528\u6c42\u89e3\u5de5\u5177\u548c\u4ee3\u7801\u793a\u4f8b\u3002\u7136\u540e，\u901a\u8fc7\u4e00\u4e2a\u5b8c\u6574\u7684\u7f8e\u8d5b\u7b80\u5316\u6848\u4f8b\u5c55\u793aODE\u5efa\u6a21\u7684\u5168\u8fc7\u7a0b\u3002\u6700\u540e，\u8bc4\u4f30\u8be5\u65b9\u6cd5\u7684\u4f18\u7f3a\u70b9\u5e76\u63d0\u51fa\u6539\u8fdb\u65b9\u5411\u3002\u672c\u6587\u65e8\u5728\u4e3a\u6570\u5b66\u5efa\u6a21\u7ade\u8d5b\u53c2\u4e0e\u8005\u63d0\u4f9bODE\u5efa\u6a21\u7684\u7cfb\u7edf\u6027\u6307\u5bfc，\u63d0\u5347\u89e3\u51b3\u5b9e\u9645\u52a8\u6001\u7cfb\u7edf\u95ee\u9898\u7684\u80fd\u529b\u3002<\/p>\n
\u5173\u952e\u8bcd：\u5e38\u5fae\u5206\u65b9\u7a0b；\u6570\u5b66\u5efa\u6a21；\u4eba\u53e3\u589e\u957f；SIR\u6a21\u578b；\u4f20\u67d3\u75c5\u52a8\u529b\u5b66；\u6570\u503c\u6c42\u89e3<\/p>\n
\u5e38\u5fae\u5206\u65b9\u7a0b\u662f\u63cf\u8ff0\u672a\u77e5\u51fd\u6570\u4e0e\u5176\u5bfc\u6570\u4e4b\u95f4\u5173\u7cfb\u7684\u65b9\u7a0b，\u5176\u4e2d\u672a\u77e5\u51fd\u6570\u4ec5\u4f9d\u8d56\u4e8e\u4e00\u4e2a\u81ea\u53d8\u91cf\u3002\u5728\u6570\u5b66\u5efa\u6a21\u4e2d，\u65f6\u95f4t\u5e38\u4f5c\u4e3a\u81ea\u53d8\u91cf，\u800c\u7cfb\u7edf\u72b6\u6001\u53d8\u91cf（\u5982\u4eba\u53e3\u6570\u91cf\u3001\u611f\u67d3\u4eba\u6570\u7b49）\u4f5c\u4e3a\u56e0\u53d8\u91cf\u3002ODE\u7684\u4e00\u822c\u5f62\u5f0f\u4e3a：<\/p>\n
F(t,y,y\u2032,y\u2032\u2032,\u2026,y(n))=0F(t,y,y\u2032,y\u2032\u2032,\u2026,y(n))=0<\/p>\n
\u5176\u4e2dy=y(t)y=y(t)\u662f\u672a\u77e5\u51fd\u6570，y\u2032,y\u2032\u2032,\u2026,y(n)y\u2032,y\u2032\u2032,\u2026,y(n)\u662f\u5176\u5173\u4e8et\u7684\u5404\u9636\u5bfc\u6570\u3002<\/p>\n
\u5bf9\u4e8en\u9636ODE，\u82e5\u5b58\u5728\u5f62\u5f0f：<\/p>\n
y(n)=f(t,y,y\u2032,\u2026,y(n\u22121))y(n)=f(t,y,y\u2032,\u2026,y(n\u22121))<\/p>\n
\u5219\u79f0\u4e3a\u663e\u5f0fODE\u3002\u521d\u503c\u95ee\u9898\u7684\u63d0\u6cd5\u4e3a：\u5728\u7ed9\u5b9a\u521d\u59cb\u6761\u4ef6y(t0)=y0,y\u2032(t0)=y0\u2032,\u2026,y(n\u22121)(t0)=y0(n\u22121)y(t0\u200b)=y0\u200b,y\u2032(t0\u200b)=y0\u2032\u200b,\u2026,y(n\u22121)(t0\u200b)=y0(n\u22121)\u200b\u4e0b，\u6c42\u89e3\u6ee1\u8db3ODE\u7684\u51fd\u6570y(t)y(t)\u3002<\/p>\n
\u9a6c\u5c14\u8428\u65af\u4e8e1798\u5e74\u63d0\u51fa\u7684\u4eba\u53e3\u589e\u957f\u6a21\u578b\u57fa\u4e8e\u4e00\u4e2a\u57fa\u672c\u5047\u8bbe：\u4eba\u53e3\u589e\u957f\u7387\u4e0e\u5f53\u524d\u4eba\u53e3\u6570\u91cf\u6210\u6b63\u6bd4\u3002\u8be5\u6a21\u578b\u8868\u793a\u4e3a：<\/p>\n
dPdt=rPdtdP\u200b=rP<\/p>\n
\u5176\u4e2d：<\/p>\n
P(t)P(t)：t\u65f6\u523b\u7684\u4eba\u53e3\u6570\u91cf<\/p>\n<\/li>\n
rr：\u5185\u7980\u589e\u957f\u7387（\u51fa\u751f\u7387\u4e0e\u6b7b\u4ea1\u7387\u4e4b\u5dee）<\/p>\n<\/li>\n
tt：\u65f6\u95f4<\/p>\n<\/li>\n<\/ul>\n
\u7ed9\u5b9a\u521d\u59cb\u6761\u4ef6P(0)=P0P(0)=P0\u200b，\u53ef\u5f97\u89e3\u6790\u89e3：<\/p>\n
P(t)=P0ertP(t)=P0\u200bert<\/p>\n
\u8be5\u6a21\u578b\u9884\u6d4b\u4eba\u53e3\u5c06\u5448\u6307\u6570\u589e\u957f，\u9002\u7528\u4e8e\u8d44\u6e90\u5145\u8db3\u3001\u65e0\u9650\u5236\u73af\u5883\u4e0b\u7684\u77ed\u671f\u4eba\u53e3\u9884\u6d4b\u3002\u4f46\u957f\u671f\u6765\u770b，\u8d44\u6e90\u9650\u5236\u4f1a\u4f7f\u589e\u957f\u653e\u7f13，\u4e0e\u5b9e\u9645\u89c2\u6d4b\u4e0d\u7b26\u3002<\/p>\n
1838\u5e74，Verhulst\u63d0\u51faLogistic\u6a21\u578b\u4ee5\u4fee\u6b63Malthus\u6a21\u578b\u7684\u7f3a\u9677，\u5f15\u5165\u73af\u5883\u627f\u8f7d\u529bK\u7684\u6982\u5ff5：<\/p>\n
dPdt=rP(1\u2212PK)dtdP\u200b=rP(1\u2212KP\u200b)<\/p>\n
\u5176\u4e2d：<\/p>\n
KK：\u73af\u5883\u627f\u8f7d\u529b（\u6700\u5927\u53ef\u6301\u7eed\u4eba\u53e3\u6570\u91cf）<\/p>\n<\/li>\n
\u56e0\u5b50(1\u2212P\/K)(1\u2212P\/K)：\u73af\u5883\u963b\u529b\u9879，\u968fP\u63a5\u8fd1K\u800c\u51cf\u5c0f<\/p>\n<\/li>\n<\/ul>\n
\u8be5\u65b9\u7a0b\u53ef\u5206\u79bb\u53d8\u91cf\u6c42\u89e3：<\/p>\n
dPP(1\u2212P\/K)=r\u2009dtP(1\u2212P\/K)dP\u200b=rdt<\/p>\n
\u79ef\u5206\u5f97：<\/p>\n
P(t)=K1+(K\u2212P0P0)e\u2212rtP(t)=1+(P0\u200bK\u2212P0\u200b\u200b)e\u2212rtK\u200b<\/p>\n
Logistic\u66f2\u7ebf\u5448S\u5f62：\u521d\u59cb\u8fd1\u4f3c\u6307\u6570\u589e\u957f，\u968f\u540e\u589e\u957f\u653e\u7f13，\u6700\u7ec8\u8d8b\u4e8e\u7a33\u5b9a\u503cK\u3002\u8be5\u6a21\u578b\u66f4\u7b26\u5408\u5b9e\u9645\u89c2\u6d4b，\u5e7f\u6cdb\u5e94\u7528\u4e8e\u751f\u6001\u5b66\u3001\u7ecf\u6d4e\u5b66\u7b49\u9886\u57df\u3002<\/p>\n
\u5b9e\u9645\u5e94\u7528\u4e2d，\u53ef\u6839\u636e\u5177\u4f53\u60c5\u51b5\u5bf9Logistic\u6a21\u578b\u8fdb\u884c\u6539\u8fdb：<\/p>\n
\u65f6\u53d8\u627f\u8f7d\u529b\u6a21\u578b：K=K(t)K=K(t)，\u53cd\u6620\u73af\u5883\u53d8\u5316<\/p>\n
dPdt=rP(1\u2212PK(t))dtdP\u200b=rP(1\u2212K(t)P\u200b)<\/li>\n
\u65f6\u53d8\u589e\u957f\u7387\u6a21\u578b：r=r(t)r=r(t)，\u53cd\u6620\u653f\u7b56\u3001\u6280\u672f\u7b49\u56e0\u7d20<\/p>\n
dPdt=r(t)P(1\u2212PK)dtdP\u200b=r(t)P(1\u2212KP\u200b)<\/li>\n
\u5e26Allee\u6548\u5e94\u7684\u6a21\u578b：\u79cd\u7fa4\u5bc6\u5ea6\u8fc7\u4f4e\u65f6\u589e\u957f\u7387\u4e3a\u8d1f<\/p>\n
dPdt=rP(PA\u22121)(1\u2212PK),0<A<KdtdP\u200b=rP(AP\u200b\u22121)(1\u2212KP\u200b),0<A<K<\/li>\n
\u8003\u8651\u5e74\u9f84\u7ed3\u6784\u7684\u6a21\u578b：\u5c06\u4eba\u53e3\u6309\u5e74\u9f84\u5206\u7ec4，\u5efa\u7acb\u65b9\u7a0b\u7ec4<\/p>\n
\u2202P(a,t)\u2202t+\u2202P(a,t)\u2202a=\u2212\u03bc(a,t)P(a,t)\u2202t\u2202P(a,t)\u200b+\u2202a\u2202P(a,t)\u200b=\u2212\u03bc(a,t)P(a,t) <\/p>\n
\u5176\u4e2dP(a,t)P(a,t)\u8868\u793at\u65f6\u523b\u5e74\u9f84\u4e3aa\u7684\u4eba\u53e3\u5bc6\u5ea6\u3002<\/p>\n<\/li>\n
Kermack\u548cMcKendrick\u4e8e1927\u5e74\u63d0\u51fa\u7684SIR\u6a21\u578b\u662f\u4f20\u67d3\u75c5\u52a8\u529b\u5b66\u7684\u57fa\u7840\u3002\u6a21\u578b\u5c06\u4eba\u7fa4\u5206\u4e3a\u4e09\u7c7b：<\/p>\n
\u6613\u611f\u8005(Susceptible, S)：\u53ef\u80fd\u88ab\u611f\u67d3\u7684\u5065\u5eb7\u4e2a\u4f53<\/p>\n<\/li>\n
\u611f\u67d3\u8005(Infected, I)：\u5df2\u611f\u67d3\u4e14\u80fd\u4f20\u64ad\u75be\u75c5\u7684\u4e2a\u4f53<\/p>\n<\/li>\n
\u79fb\u51fa\u8005(Removed, R)：\u5eb7\u590d\u5e76\u83b7\u5f97\u514d\u75ab\u529b\u6216\u6b7b\u4ea1\u7684\u4e2a\u4f53<\/p>\n<\/li>\n<\/ul>\n
\u6a21\u578b\u5047\u8bbe\u5982\u4e0b：<\/p>\n
\u603b\u4eba\u53e3\u6570N=S+I+RN=S+I+R\u6052\u5b9a（\u4e0d\u8003\u8651\u51fa\u751f\u3001\u6b7b\u4ea1\u548c\u8fc1\u79fb）<\/p>\n<\/li>\n
\u5747\u5300\u6df7\u5408：\u4e2a\u4f53\u968f\u673a\u63a5\u89e6<\/p>\n<\/li>\n
\u611f\u67d3\u7387\u4e0eSI\u4e58\u79ef\u6210\u6b63\u6bd4<\/p>\n<\/li>\n
\u6062\u590d\u7387\u4e0eI\u6210\u6b63\u6bd4<\/p>\n<\/li>\n
\u6a21\u578b\u65b9\u7a0b：<\/p>\n
dSdt=\u2212\u03b2SINdIdt=\u03b2SIN\u2212\u03b3IdRdt=\u03b3IdtdS\u200bdtdI\u200bdtdR\u200b\u200b=\u2212\u03b2NSI\u200b=\u03b2NSI\u200b\u2212\u03b3I=\u03b3I\u200b<\/p>\n
\u5176\u4e2d：<\/p>\n
\u03b2\u03b2：\u611f\u67d3\u7387（\u6709\u6548\u63a5\u89e6\u7387）<\/p>\n<\/li>\n
\u03b3\u03b3：\u6062\u590d\u7387（\u6062\u590d\u7387\u7684\u5012\u6570\u4e3a\u5e73\u5747\u611f\u67d3\u671f1\/\u03b31\/\u03b3）<\/p>\n<\/li>\n<\/ul>\n
\u57fa\u672c\u518d\u751f\u6570R0R0\u200b\u662f\u4f20\u67d3\u75c5\u52a8\u529b\u5b66\u7684\u5173\u952e\u53c2\u6570，\u8868\u793a\u4e00\u4e2a\u611f\u67d3\u8005\u5728\u5b8c\u5168\u6613\u611f\u4eba\u7fa4\u4e2d\u5f15\u8d77\u7684\u5e73\u5747\u65b0\u611f\u67d3\u6570\u3002\u5bf9\u4e8eSIR\u6a21\u578b：<\/p>\n
R0=\u03b2\u03b3R0\u200b=\u03b3\u03b2\u200b<\/p>\n
\u82e5R0>1R0\u200b>1，\u75be\u75c5\u53ef\u80fd\u6d41\u884c<\/p>\n<\/li>\n
\u82e5R0<1R0\u200b<1，\u75be\u75c5\u9010\u6e10\u6d88\u5931<\/p>\n<\/li>\n
R0=1R0\u200b=1\u662f\u75be\u75c5\u4f20\u64ad\u7684\u9608\u503c<\/p>\n<\/li>\n<\/ul>\n
\u5b9e\u9645\u5e94\u7528\u4e2d\u5e38\u5bf9SIR\u6a21\u578b\u8fdb\u884c\u6269\u5c55：<\/p>\n
SEIR\u6a21\u578b：\u589e\u52a0\u6f5c\u4f0f\u671f\u4eba\u7fa4(Exposed, E)<\/p>\n
dSdt=\u2212\u03b2SINdEdt=\u03b2SIN\u2212\u03c3EdIdt=\u03c3E\u2212\u03b3IdRdt=\u03b3IdtdS\u200bdtdE\u200bdtdI\u200bdtdR\u200b\u200b=\u2212\u03b2NSI\u200b=\u03b2NSI\u200b\u2212\u03c3E=\u03c3E\u2212\u03b3I=\u03b3I\u200b <\/p>\n
\u5176\u4e2d\u03c3\u03c3\u4e3a\u6f5c\u4f0f\u671f\u5230\u53d1\u75c5\u7684\u8f6c\u79fb\u7387\u3002<\/p>\n<\/li>\n
SIS\u6a21\u578b：\u5eb7\u590d\u540e\u65e0\u514d\u75ab\u529b，\u91cd\u65b0\u53d8\u4e3a\u6613\u611f\u8005<\/p>\n
dSdt=\u2212\u03b2SIN+\u03b3IdIdt=\u03b2SIN\u2212\u03b3IdtdS\u200bdtdI\u200b\u200b=\u2212\u03b2NSI\u200b+\u03b3I=\u03b2NSI\u200b\u2212\u03b3I\u200b<\/li>\n
\u8003\u8651\u4eba\u53e3\u52a8\u529b\u5b66\u7684SIR\u6a21\u578b：\u5305\u542b\u51fa\u751f\u548c\u6b7b\u4ea1<\/p>\n
dSdt=\u03bcN\u2212\u03b2SIN\u2212\u03bcSdIdt=\u03b2SIN\u2212\u03b3I\u2212\u03bcIdRdt=\u03b3I\u2212\u03bcRdtdS\u200bdtdI\u200bdtdR\u200b\u200b=\u03bcN\u2212\u03b2NSI\u200b\u2212\u03bcS=\u03b2NSI\u200b\u2212\u03b3I\u2212\u03bcI=\u03b3I\u2212\u03bcR\u200b <\/p>\n
\u5176\u4e2d\u03bc\u03bc\u4e3a\u51fa\u751f\u7387\u548c\u6b7b\u4ea1\u7387（\u5047\u8bbe\u76f8\u7b49）\u3002<\/p>\n<\/li>\n
\u7a7a\u95f4\u5f02\u8d28\u6027\u6a21\u578b：\u8003\u8651\u7a7a\u95f4\u6269\u6563<\/p>\n
\u2202S\u2202t=DS\u22072S\u2212\u03b2SI\u2202t\u2202S\u200b=DS\u200b\u22072S\u2212\u03b2SI\u2202I\u2202t=DI\u22072I+\u03b2SI\u2212\u03b3I\u2202t\u2202I\u200b=DI\u200b\u22072I+\u03b2SI\u2212\u03b3I <\/p>\n
\u5176\u4e2dDS,DIDS\u200b,DI\u200b\u4e3a\u6269\u6563\u7cfb\u6570\u3002<\/p>\n<\/li>\n
\u5bf9\u4e8eSIR\u6a21\u578b，\u6ce8\u610f\u5230dS\/dt+dI\/dt+dR\/dt=0dS\/dt+dI\/dt+dR\/dt=0，\u6545\u603b\u4eba\u53e3\u5b88\u6052\u3002\u5b9a\u4e49\u76f8\u5bf9\u6bd4\u4f8bs=S\/N,i=I\/N,r=R\/Ns=S\/N,i=I\/N,r=R\/N，\u5219：<\/p>\n
dsdt=\u2212\u03b2sididt=\u03b2si\u2212\u03b3idrdt=\u03b3idtds\u200bdtdi\u200bdtdr\u200b\u200b=\u2212\u03b2si=\u03b2si\u2212\u03b3i=\u03b3i\u200b<\/p>\n
\u76f8\u5e73\u9762\u5206\u6790\u53ef\u5728s\u2212is\u2212i\u5e73\u9762\u8fdb\u884c\u3002\u7531\u524d\u4e24\u5f0f\u5f97：<\/p>\n
dids=\u22121+\u03b3\u03b2s=\u22121+1R0sdsdi\u200b=\u22121+\u03b2s\u03b3\u200b=\u22121+R0\u200bs1\u200b<\/p>\n
\u79ef\u5206\u5f97：<\/p>\n
i(s)=(1\u2212s)+1R0ln\u2061(ss0)i(s)=(1\u2212s)+R0\u200b1\u200bln(s0\u200bs\u200b)<\/p>\n
\u5176\u4e2ds0s0\u200b\u4e3a\u521d\u59cb\u65f6\u523b\u7684s\u503c\u3002\u7531\u6b64\u53ef\u5206\u6790\u75be\u75c5\u6700\u7ec8\u89c4\u6a21\u3002<\/p>\n
ODE\u9002\u7528\u4e8e\u63cf\u8ff0\u8fde\u7eed\u65f6\u95f4\u52a8\u6001\u7cfb\u7edf，\u7279\u522b\u662f：<\/p>\n
\u79cd\u7fa4\u52a8\u529b\u5b66：\u4eba\u53e3\u589e\u957f\u3001\u7269\u79cd\u7ade\u4e89\u3001\u6355\u98df-\u88ab\u6355\u98df\u5173\u7cfb<\/p>\n<\/li>\n
\u6d41\u884c\u75c5\u5b66：\u4f20\u67d3\u75c5\u4f20\u64ad\u3001\u4fe1\u606f\u4f20\u64ad\u3001\u8c23\u8a00\u6269\u6563<\/p>\n<\/li>\n
\u751f\u6001\u5b66：\u751f\u6001\u7cfb\u7edf\u6f14\u5316\u3001\u6c61\u67d3\u7269\u6269\u6563<\/p>\n<\/li>\n
\u7269\u7406\u5b66：\u7269\u4f53\u8fd0\u52a8\u3001\u7535\u8def\u5206\u6790\u3001\u70ed\u4f20\u5bfc<\/p>\n<\/li>\n
\u7ecf\u6d4e\u5b66：\u7ecf\u6d4e\u589e\u957f\u3001\u5e02\u573a\u5747\u8861\u3001\u6295\u8d44\u51b3\u7b56<\/p>\n<\/li>\n
\u793e\u4f1a\u5b66：\u89c2\u70b9\u4f20\u64ad\u3001\u793e\u4f1a\u7f51\u7edc\u6f14\u5316<\/p>\n<\/li>\n
\u836f\u7406\u5b66：\u836f\u7269\u4ee3\u8c22\u3001\u836f\u4ee3\u52a8\u529b\u5b66<\/p>\n<\/li>\n
\u73af\u5883\u79d1\u5b66：\u6c61\u67d3\u7269\u6d53\u5ea6\u53d8\u5316\u3001\u78b3\u5faa\u73af<\/p>\n<\/li>\n
\u4eba\u53e3\u9884\u6d4b\u4e0e\u89c4\u5212：\u7ed9\u5b9a\u5386\u53f2\u6570\u636e，\u9884\u6d4b\u672a\u6765\u4eba\u53e3，\u8bc4\u4f30\u653f\u7b56\u6548\u679c<\/p>\n<\/li>\n
\u8001\u9f84\u5316\u95ee\u9898：\u8003\u8651\u5e74\u9f84\u7ed3\u6784，\u5206\u6790\u517b\u8001\u91d1\u3001\u533b\u7597\u8d44\u6e90\u9700\u6c42<\/p>\n<\/li>\n
\u8d44\u6e90\u627f\u8f7d\u5206\u6790：\u7ed3\u5408\u8d44\u6e90\u9650\u5236，\u786e\u5b9a\u53ef\u6301\u7eed\u4eba\u53e3\u89c4\u6a21<\/p>\n<\/li>\n
\u79fb\u6c11\u5f71\u54cd\u8bc4\u4f30：\u7814\u7a76\u79fb\u6c11\u5bf9\u4eba\u53e3\u7ed3\u6784\u3001\u7ecf\u6d4e\u53d1\u5c55\u7684\u5f71\u54cd<\/p>\n<\/li>\n
\u771f\u9898\u793a\u4f8b：2019\u5e74\u7f8e\u8d5bC\u9898"\u6bd2\u54c1\u95ee\u9898"\u6d89\u53ca\u836f\u7269\u4f7f\u7528\u8005\u6570\u91cf\u53d8\u5316，\u53ef\u7528\u7c7b\u4f3c\u4f20\u67d3\u75c5\u6a21\u578b\u5efa\u6a21\u3002<\/p>\n
\u75ab\u60c5\u9884\u6d4b：\u57fa\u4e8e\u65e9\u671f\u6570\u636e\u9884\u6d4b\u75ab\u60c5\u53d1\u5c55\u8d8b\u52bf<\/p>\n<\/li>\n
\u5e72\u9884\u63aa\u65bd\u8bc4\u4f30：\u9694\u79bb\u3001\u75ab\u82d7\u63a5\u79cd\u3001\u793e\u4ea4\u8ddd\u79bb\u7b49\u63aa\u65bd\u6548\u679c\u5206\u6790<\/p>\n<\/li>\n
\u533b\u7597\u8d44\u6e90\u9700\u6c42：\u9884\u6d4b\u75c5\u5e8a\u3001\u547c\u5438\u673a\u7b49\u533b\u7597\u8d44\u6e90\u9700\u6c42\u5cf0\u503c<\/p>\n<\/li>\n
\u75ab\u82d7\u5206\u914d\u7b56\u7565：\u6709\u9650\u75ab\u82d7\u8d44\u6e90\u4e0b\u7684\u6700\u4f18\u5206\u914d\u65b9\u6848<\/p>\n<\/li>\n
\u591a\u7fa4\u4f53\u4f20\u64ad：\u8003\u8651\u5e74\u9f84\u3001\u804c\u4e1a\u7b49\u4e0d\u540c\u7fa4\u4f53\u7684\u4f20\u64ad\u5dee\u5f02<\/p>\n<\/li>\n
\u771f\u9898\u793a\u4f8b：2020\u5e74\u7f8e\u8d5bD\u9898"\u56e2\u961f\u5408\u4f5c\u7b56\u7565"\u53ef\u501f\u9274\u4f20\u67d3\u75c5\u6a21\u578b\u7814\u7a76\u5408\u4f5c\u884c\u4e3a\u4f20\u64ad；2021\u5e74\u7f8e\u8d5bD\u9898"\u97f3\u4e50\u5f71\u54cd"\u53ef\u7528SIR\u7c7b\u6a21\u578b\u7814\u7a76\u97f3\u4e50\u504f\u597d\u4f20\u64ad\u3002<\/p>\n
\u7269\u79cd\u7ade\u4e89：Lotka-Volterra\u7ade\u4e89\u6a21\u578b<\/p>\n
dN1dt=r1N1(1\u2212N1+\u03b112N2K1)dN2dt=r2N2(1\u2212N2+\u03b121N1K2)dtdN1\u200b\u200bdtdN2\u200b\u200b\u200b=r1\u200bN1\u200b(1\u2212K1\u200bN1\u200b+\u03b112\u200bN2\u200b\u200b)=r2\u200bN2\u200b(1\u2212K2\u200bN2\u200b+\u03b121\u200bN1\u200b\u200b)\u200b<\/li>\n
\u6355\u98df-\u88ab\u6355\u98df\u5173\u7cfb：Lotka-Volterra\u6355\u98df\u6a21\u578b<\/p>\n
dHdt=rH\u2212aHPdPdt=\u03f5aHP\u2212mPdtdH\u200bdtdP\u200b\u200b=rH\u2212aHP=\u03f5aHP\u2212mP\u200b <\/p>\n
\u5176\u4e2dH\u4e3a\u730e\u7269，P\u4e3a\u6355\u98df\u8005\u3002<\/p>\n<\/li>\n
\u6c61\u67d3\u7269\u6269\u6563：\u591a\u4ecb\u8d28\u73af\u5883\u4e2d\u7684\u6c61\u67d3\u7269\u8fc1\u79fb\u8f6c\u5316<\/p>\n<\/li>\n
\u6280\u672f\u521b\u65b0\u6269\u6563：Bass\u6a21\u578b\u63cf\u8ff0\u65b0\u4ea7\u54c1\u3001\u65b0\u6280\u672f\u91c7\u7eb3<\/p>\n
dN(t)dt=p[M\u2212N(t)]+qMN(t)[M\u2212N(t)]dtdN(t)\u200b=p[M\u2212N(t)]+Mq\u200bN(t)[M\u2212N(t)] <\/p>\n
\u5176\u4e2dp\u4e3a\u521b\u65b0\u7cfb\u6570，q\u4e3a\u6a21\u4eff\u7cfb\u6570，M\u4e3a\u5e02\u573a\u6f5c\u529b\u3002<\/p>\n<\/li>\n
\u8c23\u8a00\u4f20\u64ad：\u7c7b\u4f3c\u4f20\u67d3\u75c5\u6a21\u578b，\u4f46\u8003\u8651\u9057\u5fd8\u673a\u5236<\/p>\n<\/li>\n
\u7ecf\u6d4e\u5468\u671f：\u7528\u52a8\u529b\u5b66\u65b9\u7a0b\u63cf\u8ff0\u7ecf\u6d4e\u589e\u957f\u6ce2\u52a8<\/p>\n<\/li>\n
\u9762\u5bf9\u8d5b\u9898\u65f6，\u8bc6\u522b\u662f\u5426\u9002\u7528ODE\u5efa\u6a21\u7684\u5173\u952e\u6307\u6807：<\/p>\n
\u7cfb\u7edf\u72b6\u6001\u968f\u65f6\u95f4\u8fde\u7eed\u53d8\u5316<\/p>\n<\/li>\n
\u53d8\u5316\u7387\u4f9d\u8d56\u4e8e\u5f53\u524d\u72b6\u6001<\/p>\n<\/li>\n
\u5ffd\u7565\u7a7a\u95f4\u5f02\u8d28\u6027\u6216\u53ef\u7b80\u5316\u4e3a\u5747\u503c\u573a<\/p>\n<\/li>\n
\u968f\u673a\u6548\u5e94\u4e0d\u5360\u4e3b\u5bfc\u5730\u4f4d<\/p>\n<\/li>\n
\u82e5\u7cfb\u7edf\u79bb\u6563\u6027\u660e\u663e\u6216\u968f\u673a\u6027\u5360\u4e3b\u5bfc，\u5e94\u8003\u8651\u5dee\u5206\u65b9\u7a0b\u6216\u968f\u673a\u8fc7\u7a0b\u6a21\u578b\u3002<\/p>\n
\u660e\u786e\u7814\u7a76\u95ee\u9898：\u786e\u5b9a\u9700\u8981\u9884\u6d4b\u3001\u89e3\u91ca\u6216\u4f18\u5316\u7684\u76ee\u6807<\/p>\n<\/li>\n
\u8bc6\u522b\u7cfb\u7edf\u8981\u7d20：\u786e\u5b9a\u72b6\u6001\u53d8\u91cf\u3001\u53c2\u6570\u548c\u63a7\u5236\u53d8\u91cf<\/p>\n<\/li>\n
\u5efa\u7acb\u5408\u7406\u5047\u8bbe：\u7b80\u5316\u73b0\u5b9e，\u4f7f\u95ee\u9898\u53ef\u5904\u7406<\/p>\n
\u5747\u5300\u6df7\u5408\u5047\u8bbe<\/p>\n<\/li>\n
\u5ffd\u7565\u968f\u673a\u6ce2\u52a8<\/p>\n<\/li>\n
\u53c2\u6570\u6052\u5b9a\u6027\u5047\u8bbe<\/p>\n<\/li>\n
\u5c01\u95ed\u7cfb\u7edf\u5047\u8bbe<\/p>\n<\/li>\n<\/ul>\n<\/li>\n
\u786e\u5b9a\u6a21\u578b\u7c7b\u578b：\u6839\u636e\u95ee\u9898\u7279\u70b9\u9009\u62e9\u5408\u9002\u6a21\u578b\u6846\u67b6<\/p>\n<\/li>\n
\u5b9a\u4e49\u72b6\u6001\u53d8\u91cf：\u9009\u62e9\u6070\u5f53\u53d8\u91cf\u63cf\u8ff0\u7cfb\u7edf\u72b6\u6001<\/p>\n<\/li>\n
\u5efa\u7acb\u6d41\u7a0b\u56fe：\u53ef\u89c6\u5316\u5404\u72b6\u6001\u95f4\u8f6c\u79fb\u5173\u7cfb<\/p>\n<\/li>\n
\u63a8\u5bfc\u5fae\u5206\u65b9\u7a0b：\u57fa\u4e8e\u8d28\u91cf\u5b88\u6052\u3001\u901f\u7387\u5e73\u8861\u7b49\u539f\u7406<\/p>\n
\u6d41\u5165\u7387 – \u6d41\u51fa\u7387 = \u51c0\u53d8\u5316\u7387<\/p>\n<\/li>\n<\/ul>\n<\/li>\n
\u786e\u5b9a\u53c2\u6570\u610f\u4e49：\u660e\u786e\u6bcf\u4e2a\u53c2\u6570\u7684\u7269\u7406\/\u751f\u7269\u610f\u4e49<\/p>\n<\/li>\n
\u53c2\u6570\u6765\u6e90：<\/p>\n
\u6587\u732e\u8d44\u6599<\/p>\n<\/li>\n
\u5b9e\u9a8c\u6570\u636e<\/p>\n<\/li>\n
\u7edf\u8ba1\u5e74\u9274<\/p>\n<\/li>\n
\u4e13\u5bb6\u7ecf\u9a8c<\/p>\n<\/li>\n<\/ul>\n<\/li>\n
\u4f30\u8ba1\u65b9\u6cd5：<\/p>\n
\u6700\u5c0f\u4e8c\u4e58\u6cd5<\/p>\n<\/li>\n
\u6700\u5927\u4f3c\u7136\u4f30\u8ba1<\/p>\n<\/li>\n
\u8d1d\u53f6\u65af\u4f30\u8ba1<\/p>\n<\/li>\n<\/ul>\n<\/li>\n
\u62df\u5408\u4f18\u5ea6\u8bc4\u4f30：<\/p>\n
\u6b8b\u5dee\u5206\u6790<\/p>\n<\/li>\n
R\u00b2\u7edf\u8ba1\u91cf<\/p>\n<\/li>\n
AIC\/BIC\u51c6\u5219<\/p>\n<\/li>\n<\/ul>\n<\/li>\n
\u89e3\u6790\u6c42\u89e3：\u9002\u7528\u4e8e\u7b80\u5355\u7ebf\u6027\u65b9\u7a0b<\/p>\n<\/li>\n
\u6570\u503c\u6c42\u89e3：Runge-Kutta\u7b49\u6570\u503c\u65b9\u6cd5<\/p>\n<\/li>\n
\u7a33\u5b9a\u6027\u5206\u6790：\u5e73\u8861\u70b9\u5206\u6790，Jacobian\u77e9\u9635<\/p>\n<\/li>\n
\u654f\u611f\u6027\u5206\u6790：\u53c2\u6570\u53d8\u5316\u5bf9\u7ed3\u679c\u7684\u5f71\u54cd\u7a0b\u5ea6<\/p>\n<\/li>\n
\u5206\u5c94\u5206\u6790：\u53c2\u6570\u53d8\u5316\u5bfc\u81f4\u7cfb\u7edf\u5b9a\u6027\u884c\u4e3a\u6539\u53d8<\/p>\n<\/li>\n
\u7ed3\u679c\u89e3\u91ca：\u5c06\u6570\u5b66\u7ed3\u679c\u8f6c\u5316\u4e3a\u5b9e\u9645\u95ee\u9898\u89e3\u7b54<\/p>\n<\/li>\n
\u6a21\u578b\u9a8c\u8bc1：\u5bf9\u6bd4\u6a21\u578b\u9884\u6d4b\u4e0e\u5b9e\u9645\u89c2\u6d4b<\/p>\n<\/li>\n
\u7a33\u5065\u6027\u5206\u6790：\u6539\u53d8\u5047\u8bbe\u6761\u4ef6，\u68c0\u9a8c\u7ed3\u679c\u7a33\u5b9a\u6027<\/p>\n<\/li>\n
\u6a21\u578b\u6539\u8fdb：\u6839\u636e\u9a8c\u8bc1\u7ed3\u679c\u4fee\u6b63\u6a21\u578b<\/p>\n<\/li>\n
\u65e0\u91cf\u7eb2\u5316\u53ef\u7b80\u5316\u65b9\u7a0b，\u51cf\u5c11\u53c2\u6570\u6570\u91cf，\u63d0\u9ad8\u6570\u503c\u7a33\u5b9a\u6027\u3002\u4ee5Logistic\u65b9\u7a0b\u4e3a\u4f8b：<\/p>\n
\u539f\u59cb\u65b9\u7a0b：dPdt=rP(1\u2212PK)dtdP\u200b=rP(1\u2212KP\u200b)<\/p>\n
\u4ee4x=PKx=KP\u200b（\u76f8\u5bf9\u4eba\u53e3），\u03c4=rt\u03c4=rt（\u65e0\u91cf\u7eb2\u65f6\u95f4），\u5219：<\/p>\n
dxd\u03c4=x(1\u2212x)d\u03c4dx\u200b=x(1\u2212x)<\/p>\n
\u65b9\u7a0b\u7b80\u5316\u4e3a\u5355\u53c2\u6570\u5f62\u5f0f，\u4fbf\u4e8e\u5206\u6790\u3002<\/p>\n
\u8bc4\u4f30\u53c2\u6570\u4e0d\u786e\u5b9a\u6027\u5bf9\u7ed3\u679c\u7684\u5f71\u54cd\u3002\u5c40\u90e8\u654f\u611f\u6027\u6307\u6570：<\/p>\n
Sp=\u2202y\u2202p\u22c5pySp\u200b=\u2202p\u2202y\u200b\u22c5yp\u200b<\/p>\n
\u5176\u4e2dy\u4e3a\u8f93\u51fa\u53d8\u91cf，p\u4e3a\u53c2\u6570\u3002\u5168\u5c40\u654f\u611f\u6027\u5206\u6790\u53ef\u91c7\u7528Sobol\u6307\u6570\u6cd5\u3002<\/p>\n
\u5bf9\u4e8e\u4e8c\u7ef4\u81ea\u6cbb\u7cfb\u7edf：<\/p>\n
dxdt=f(x,y)dydt=g(x,y)dtdx\u200bdtdy\u200b\u200b=f(x,y)=g(x,y)\u200b<\/p>\n
\u5bfb\u627e\u5e73\u8861\u70b9：\u89e3f(x,y)=0,g(x,y)=0f(x,y)=0,g(x,y)=0<\/p>\n<\/li>\n
\u7ebf\u6027\u5316：\u5728\u5e73\u8861\u70b9(x\u2217,y\u2217)(x\u2217,y\u2217)\u5904\u8ba1\u7b97Jacobian\u77e9\u9635<\/p>\n
J=[\u2202f\u2202x\u2202f\u2202y\u2202g\u2202x\u2202g\u2202y](x\u2217,y\u2217)J=[\u2202x\u2202f\u200b\u2202x\u2202g\u200b\u200b\u2202y\u2202f\u200b\u2202y\u2202g\u200b\u200b](x\u2217,y\u2217)\u200b<\/li>\n
\u7a33\u5b9a\u6027\u5224\u5b9a：\u6839\u636e\u7279\u5f81\u503c\u5b9e\u90e8\u7b26\u53f7\u5224\u65ad\u7a33\u5b9a\u6027<\/p>\n<\/li>\n
\u7ed8\u5236\u96f6\u503e\u7ebf：f(x,y)=0f(x,y)=0\u548cg(x,y)=0g(x,y)=0\u7684\u66f2\u7ebf<\/p>\n<\/li>\n
\u6b65\u957f\u9009\u62e9：\u6b65\u957f\u8fc7\u5927\u4f1a\u5bfc\u81f4\u4e0d\u7a33\u5b9a，\u8fc7\u5c0f\u4f1a\u589e\u52a0\u8ba1\u7b97\u91cf<\/p>\n<\/li>\n
\u65b9\u6cd5\u9009\u62e9：<\/p>\n
\u975e\u521a\u6027\u7cfb\u7edf：\u7ecf\u5178Runge-Kutta\u6cd5（\u5982RK4）<\/p>\n<\/li>\n
\u521a\u6027\u7cfb\u7edf：\u9690\u5f0f\u65b9\u6cd5（\u5982Gear\u6cd5\u3001BDF\u6cd5）<\/p>\n<\/li>\n<\/ul>\n<\/li>\n
\u8bef\u5dee\u63a7\u5236：\u81ea\u9002\u5e94\u6b65\u957f\u8c03\u6574，\u5c40\u90e8\u622a\u65ad\u8bef\u5dee\u4f30\u8ba1<\/p>\n<\/li>\n
\u4f7f\u7528\u4fe1\u606f\u51c6\u5219\u9009\u62e9\u6700\u5408\u9002\u6a21\u578b：<\/p>\n
AIC（Akaike\u4fe1\u606f\u51c6\u5219）：AIC=2k\u22122ln\u2061(L)AIC=2k\u22122ln(L)<\/p>\n<\/li>\n
BIC（\u8d1d\u53f6\u65af\u4fe1\u606f\u51c6\u5219）：BIC=kln\u2061(n)\u22122ln\u2061(L)BIC=kln(n)\u22122ln(L)<\/p>\n<\/li>\n<\/ul>\n
\u5176\u4e2dk\u4e3a\u53c2\u6570\u4e2a\u6570，n\u4e3a\u6570\u636e\u70b9\u6570，L\u4e3a\u4f3c\u7136\u51fd\u6570\u503c\u3002<\/p>\n
MATLAB：ode45, ode23, ode15s\u7b49\u6c42\u89e3\u5668<\/p>\n<\/li>\n
Python：SciPy\u7684odeint, solve_ivp<\/p>\n<\/li>\n
R：deSolve\u5305<\/p>\n<\/li>\n
Mathematica：DSolve, NDSolve<\/p>\n<\/li>\n
Julia：DifferentialEquations.jl\u5305<\/p>\n<\/li>\n
matlab<\/p>\n
% Logistic\u6a21\u578b\u53c2\u6570
\nr = 0.03; % \u5e74\u589e\u957f\u7387
\nK = 1000; % \u73af\u5883\u627f\u8f7d\u529b（\u767e\u4e07）
\nP0 = 100; % \u521d\u59cb\u4eba\u53e3（\u767e\u4e07）
\ntspan = [0 100]; % \u65f6\u95f4\u8303\u56f4（\u5e74）<\/p>\n
% \u5b9a\u4e49\u5fae\u5206\u65b9\u7a0b
\nlogisticODE = @(t, P) r * P * (1 – P\/K);<\/p>\n
% \u6570\u503c\u6c42\u89e3
\n[t, P] = ode45(logisticODE, tspan, P0);<\/p>\n
% \u89e3\u6790\u89e3\u5bf9\u6bd4
\nP_analytic = K .\/ (1 + (K-P0)\/P0 * exp(-r*t));<\/p>\n
% \u7ed8\u56fe
\nfigure('Position', [100, 100, 900, 400])<\/p>\n
subplot(1,2,1)
\nplot(t, P, 'b-', 'LineWidth', 2)
\nhold on
\nplot(t, P_analytic, 'r–', 'LineWidth', 1.5)
\nxlabel('\u65f6\u95f4 (\u5e74)', 'FontSize', 12)
\nylabel('\u4eba\u53e3 (\u767e\u4e07)', 'FontSize', 12)
\ntitle('Logistic\u4eba\u53e3\u589e\u957f', 'FontSize', 14)
\nlegend('\u6570\u503c\u89e3', '\u89e3\u6790\u89e3', 'Location', 'southeast')
\ngrid on<\/p>\n
% \u7ed8\u5236\u76f8\u7ebf
\nsubplot(1,2,2)
\nP_range = 0:10:1200;
\ndP_dt = r * P_range .* (1 – P_range\/K);
\nplot(P_range, dP_dt, 'k-', 'LineWidth', 2)
\nhold on
\nplot([0 K], [0 0], 'k–')
\nplot(P0, 0, 'ro', 'MarkerSize', 10, 'MarkerFaceColor', 'r')
\nxlabel('\u4eba\u53e3 P', 'FontSize', 12)
\nylabel('\u53d8\u5316\u7387 dP\/dt', 'FontSize', 12)
\ntitle('\u76f8\u7ebf\u56fe', 'FontSize', 14)
\ngrid on <\/p>\n
matlab<\/p>\n
function sir_model_example
\n % SIR\u6a21\u578b\u53c2\u6570
\n beta = 0.3; % \u611f\u67d3\u7387
\n gamma = 0.1; % \u6062\u590d\u7387
\n N = 1000; % \u603b\u4eba\u53e3<\/p>\n
% \u521d\u59cb\u6761\u4ef6
\n I0 = 1; % \u521d\u59cb\u611f\u67d3\u8005
\n R0 = 0; % \u521d\u59cb\u5eb7\u590d\u8005
\n S0 = N – I0 – R0; % \u521d\u59cb\u6613\u611f\u8005<\/p>\n
% \u65f6\u95f4\u8303\u56f4
\n tspan = [0 100];
\n y0 = [S0; I0; R0];<\/p>\n
% \u5b9a\u4e49SIR\u65b9\u7a0b
\n function dydt = sir_equations(t, y)
\n S = y(1);
\n I = y(2);
\n R = y(3);<\/p>\n
dS_dt = -beta * S * I \/ N;
\n dI_dt = beta * S * I \/ N – gamma * I;
\n dR_dt = gamma * I;<\/p>\n
dydt = [dS_dt; dI_dt; dR_dt];
\n end<\/p>\n
% \u6570\u503c\u6c42\u89e3
\n [t, y] = ode45(@sir_equations, tspan, y0);<\/p>\n
% \u8ba1\u7b97\u57fa\u672c\u518d\u751f\u6570
\n R0_value = beta \/ gamma;<\/p>\n
% \u7ed8\u56fe
\n figure('Position', [100, 100, 1200, 500])<\/p>\n
% \u65f6\u95f4\u5e8f\u5217\u56fe
\n subplot(1,2,1)
\n plot(t, y(:,1), 'b-', 'LineWidth', 2)
\n hold on
\n plot(t, y(:,2), 'r-', 'LineWidth', 2)
\n plot(t, y(:,3), 'g-', 'LineWidth', 2)
\n xlabel('\u65f6\u95f4 (\u5929)', 'FontSize', 12)
\n ylabel('\u4eba\u53e3\u6570', 'FontSize', 12)
\n title(['SIR\u6a21\u578b (R_0 = ', num2str(R0_value, '%.2f'), ')'], 'FontSize', 14)
\n legend('\u6613\u611f\u8005S', '\u611f\u67d3\u8005I', '\u5eb7\u590d\u8005R', 'Location', 'best')
\n grid on<\/p>\n
% \u76f8\u5e73\u9762\u56fe (S-I\u5e73\u9762)
\n subplot(1,2,2)
\n plot(y(:,1), y(:,2), 'k-', 'LineWidth', 2)
\n hold on
\n scatter(S0, I0, 100, 'ro', 'filled')
\n scatter(y(end,1), y(end,2), 100, 'gs', 'filled')<\/p>\n
% \u7ed8\u5236\u96f6\u503e\u7ebf
\n S_range = linspace(0, N, 100);
\n I_nullcline = max(0, S_range – gamma\/beta * N);
\n plot(S_range, I_nullcline, 'r–', 'LineWidth', 1.5)<\/p>\n
xlabel('\u6613\u611f\u8005 S', 'FontSize', 12)
\n ylabel('\u611f\u67d3\u8005 I', 'FontSize', 12)
\n title('SIR\u6a21\u578b\u76f8\u5e73\u9762', 'FontSize', 14)
\n legend('\u8f68\u8ff9', '\u8d77\u70b9', '\u7ec8\u70b9', 'I\u96f6\u503e\u7ebf', 'Location', 'best')
\n grid on
\n xlim([0 N])
\n ylim([0 max(y(:,2))*1.1])<\/p>\n
% \u8f93\u51fa\u5173\u952e\u7ed3\u679c
\n fprintf('\u57fa\u672c\u518d\u751f\u6570 R0 = %.2f\\\\n', R0_value);
\n fprintf('\u6700\u7ec8\u6613\u611f\u8005\u6bd4\u4f8b: %.2f%%\\\\n', y(end,1)\/N*100);
\n fprintf('\u5cf0\u503c\u611f\u67d3\u4eba\u6570: %.0f (\u53d1\u751f\u5728\u7b2c%.1f\u5929)\\\\n', max(y(:,2)), t(find(y(:,2)==max(y(:,2)),1)));
\nend <\/p>\n
matlab<\/p>\n
function controlled_seir_model
\n % SEIR\u6a21\u578b\u53c2\u6570 (\u4ee5COVID-19\u4e3a\u4f8b)
\n beta0 = 0.35; % \u57fa\u7840\u611f\u67d3\u7387
\n sigma = 1\/5.2; % \u6f5c\u4f0f\u671f\u5230\u53d1\u75c5\u7684\u8f6c\u79fb\u7387 (\u6f5c\u4f0f\u671f5.2\u5929)
\n gamma = 1\/14; % \u6062\u590d\u7387 (\u611f\u67d3\u671f14\u5929)
\n N = 10e6; % \u603b\u4eba\u53e3<\/p>\n
% \u63a7\u5236\u53c2\u6570
\n intervention_day = 30; % \u5e72\u9884\u5f00\u59cb\u65f6\u95f4
\n effectiveness = 0.6; % \u5e72\u9884\u6548\u679c (\u964d\u4f4e\u63a5\u89e6\u738760%)<\/p>\n
% \u521d\u59cb\u6761\u4ef6
\n E0 = 100; % \u521d\u59cb\u6f5c\u4f0f\u8005
\n I0 = 50; % \u521d\u59cb\u611f\u67d3\u8005
\n R0 = 0; % \u521d\u59cb\u5eb7\u590d\u8005
\n S0 = N – E0 – I0 – R0;<\/p>\n
tspan = [0 200];
\n y0 = [S0; E0; I0; R0];<\/p>\n
% \u5b9a\u4e49\u65f6\u53d8\u611f\u67d3\u7387\u51fd\u6570
\n function beta_t = time_varying_beta(t)
\n if t < intervention_day
\n beta_t = beta0;
\n else
\n beta_t = beta0 * (1 – effectiveness);
\n end
\n end<\/p>\n
% \u5b9a\u4e49SEIR\u65b9\u7a0b
\n function dydt = seir_equations(t, y)
\n S = y(1);
\n E = y(2);
\n I = y(3);
\n R = y(4);<\/p>\n
beta = time_varying_beta(t);<\/p>\n
dS_dt = -beta * S * I \/ N;
\n dE_dt = beta * S * I \/ N – sigma * E;
\n dI_dt = sigma * E – gamma * I;
\n dR_dt = gamma * I;<\/p>\n
dydt = [dS_dt; dE_dt; dI_dt; dR_dt];
\n end<\/p>\n
% \u6570\u503c\u6c42\u89e3
\n [t, y] = ode45(@seir_equations, tspan, y0);<\/p>\n
% \u7ed8\u56fe
\n figure('Position', [100, 100, 1400, 500])<\/p>\n
% \u65f6\u95f4\u5e8f\u5217\u56fe
\n subplot(1,3,1)
\n plot(t, y(:,1), 'b-', 'LineWidth', 2)
\n hold on
\n plot(t, y(:,2), 'm-', 'LineWidth', 2)
\n plot(t, y(:,3), 'r-', 'LineWidth', 2)
\n plot(t, y(:,4), 'g-', 'LineWidth', 2)<\/p>\n
% \u6807\u8bb0\u5e72\u9884\u65f6\u95f4
\n xline(intervention_day, 'k–', 'LineWidth', 1.5, 'Label', '\u5e72\u9884\u5f00\u59cb')<\/p>\n
xlabel('\u65f6\u95f4 (\u5929)', 'FontSize', 12)
\n ylabel('\u4eba\u53e3\u6570', 'FontSize', 12)
\n title('\u5e26\u63a7\u5236\u7684SEIR\u6a21\u578b', 'FontSize', 14)
\n legend('\u6613\u611f\u8005S', '\u6f5c\u4f0f\u8005E', '\u611f\u67d3\u8005I', '\u5eb7\u590d\u8005R', 'Location', 'best')
\n grid on<\/p>\n
% \u65b0\u589e\u611f\u67d3\u548c\u65b0\u589e\u5eb7\u590d
\n subplot(1,3,2)<\/p>\n
% \u8ba1\u7b97\u6bcf\u65e5\u65b0\u589e\u611f\u67d3 (\u4eceE\u5230I)
\n new_infections = sigma * y(:,2);
\n % \u8ba1\u7b97\u6bcf\u65e5\u65b0\u589e\u5eb7\u590d (\u4eceI\u5230R)
\n new_recoveries = gamma * y(:,3);<\/p>\n
plot(t, new_infections, 'r-', 'LineWidth', 2)
\n hold on
\n plot(t, new_recoveries, 'g-', 'LineWidth', 2)
\n xline(intervention_day, 'k–', 'LineWidth', 1.5)<\/p>\n
xlabel('\u65f6\u95f4 (\u5929)', 'FontSize', 12)
\n ylabel('\u6bcf\u65e5\u65b0\u589e\u4eba\u6570', 'FontSize', 12)
\n title('\u6bcf\u65e5\u65b0\u589e\u611f\u67d3\u4e0e\u5eb7\u590d', 'FontSize', 14)
\n legend('\u65b0\u589e\u611f\u67d3', '\u65b0\u589e\u5eb7\u590d', 'Location', 'best')
\n grid on<\/p>\n
% \u533b\u7597\u8d44\u6e90\u9700\u6c42\u4f30\u7b97
\n subplot(1,3,3)<\/p>\n
% \u5047\u8bbe\u9700\u8981\u4f4f\u9662\u7684\u6bd4\u4f8b\u548c\u5e73\u5747\u4f4f\u9662\u65f6\u95f4
\n hospitalization_rate = 0.15; % 15%\u611f\u67d3\u8005\u9700\u8981\u4f4f\u9662
\n avg_hospital_stay = 10; % \u5e73\u5747\u4f4f\u966210\u5929<\/p>\n
% \u4f30\u7b97\u6bcf\u65e5\u4f4f\u9662\u4eba\u6570 (\u7b80\u5316\u4f30\u7b97)
\n daily_hospital = hospitalization_rate * y(:,3);<\/p>\n
% \u4f30\u7b97\u7d2f\u8ba1\u4f4f\u9662\u4eba\u6570
\n cumulative_hospital = zeros(size(t));
\n for i = 2:length(t)
\n dt = t(i) – t(i-1);
\n % \u65b0\u589e\u4f4f\u9662\u4eba\u6570
\n new_hospital = hospitalization_rate * (y(i,3) – y(i-1,3) + gamma * y(i,3) * dt);
\n % \u51fa\u9662\u4eba\u6570 (\u5047\u8bbe\u6307\u6570\u8870\u51cf)
\n discharge = cumulative_hospital(i-1) * dt \/ avg_hospital_stay;<\/p>\n
cumulative_hospital(i) = max(0, cumulative_hospital(i-1) + new_hospital – discharge);
\n end<\/p>\n
plot(t, daily_hospital, 'b-', 'LineWidth', 2)
\n hold on
\n plot(t, cumulative_hospital, 'r-', 'LineWidth', 2)<\/p>\n
% \u5047\u8bbe\u533b\u7597\u8d44\u6e90\u4e0a\u9650
\n hospital_capacity = 5000;
\n yline(hospital_capacity, 'k–', 'LineWidth', 2, 'Label', '\u533b\u7597\u8d44\u6e90\u4e0a\u9650')
\n xline(intervention_day, 'k–', 'LineWidth', 1.5)<\/p>\n
xlabel('\u65f6\u95f4 (\u5929)', 'FontSize', 12)
\n ylabel('\u4f4f\u9662\u4eba\u6570', 'FontSize', 12)
\n title('\u533b\u7597\u8d44\u6e90\u9700\u6c42\u9884\u6d4b', 'FontSize', 14)
\n legend('\u6bcf\u65e5\u65b0\u589e\u4f4f\u9662', '\u7d2f\u8ba1\u4f4f\u9662\u4eba\u6570', 'Location', 'best')
\n grid on<\/p>\n
% \u8f93\u51fa\u5173\u952e\u7ed3\u679c
\n fprintf('\u5e72\u9884\u524dR0: %.2f\\\\n', beta0\/gamma);
\n fprintf('\u5e72\u9884\u540e\u6709\u6548R0: %.2f\\\\n', beta0*(1-effectiveness)\/gamma);
\n fprintf('\u7d2f\u8ba1\u611f\u67d3\u5cf0\u503c: %.0f\\\\n', max(y(:,3)));
\n fprintf('\u533b\u7597\u8d44\u6e90\u5cf0\u503c\u9700\u6c42: %.0f (\u53d1\u751f\u5728\u7b2c%.1f\u5929)\\\\n', max(cumulative_hospital), t(find(cumulative_hospital==max(cumulative_hospital),1)));
\n if max(cumulative_hospital) > hospital_capacity
\n fprintf('\u8b66\u544a: \u533b\u7597\u8d44\u6e90\u9700\u6c42\u8d85\u8fc7\u627f\u8f7d\u80fd\u529b!\\\\n');
\n end
\nend <\/p>\n
python<\/p>\n
import numpy as np
\nimport matplotlib.pyplot as plt
\nfrom scipy.integrate import solve_ivp
\nfrom scipy.optimize import curve_fit
\nimport pandas as pd<\/p>\n
class SIRModel:
\n def __init__(self, beta, gamma, N):
\n """
\n \u521d\u59cb\u5316SIR\u6a21\u578b<\/p>\n
\u53c2\u6570:
\n beta: \u611f\u67d3\u7387
\n gamma: \u6062\u590d\u7387
\n N: \u603b\u4eba\u53e3
\n """
\n self.beta = beta
\n self.gamma = gamma
\n self.N = N
\n self.R0 = beta \/ gamma<\/p>\n
def equations(self, t, y):
\n """
\n SIR\u6a21\u578b\u5fae\u5206\u65b9\u7a0b<\/p>\n
\u53c2\u6570:
\n t: \u65f6\u95f4
\n y: \u72b6\u6001\u53d8\u91cf [S, I, R]<\/p>\n
\u8fd4\u56de:
\n dydt: \u5bfc\u6570\u5411\u91cf
\n """
\n S, I, R = y
\n dS_dt = -self.beta * S * I \/ self.N
\n dI_dt = self.beta * S * I \/ self.N – self.gamma * I
\n dR_dt = self.gamma * I
\n return [dS_dt, dI_dt, dR_dt]<\/p>\n
def simulate(self, S0, I0, R0, t_span, t_eval=None):
\n """
\n \u6a21\u62dfSIR\u6a21\u578b<\/p>\n
\u53c2\u6570:
\n S0: \u521d\u59cb\u6613\u611f\u8005
\n I0: \u521d\u59cb\u611f\u67d3\u8005
\n R0: \u521d\u59cb\u5eb7\u590d\u8005
\n t_span: \u65f6\u95f4\u8303\u56f4 [t_start, t_end]
\n t_eval: \u8f93\u51fa\u65f6\u95f4\u70b9<\/p>\n
\u8fd4\u56de:
\n t: \u65f6\u95f4\u6570\u7ec4
\n y: \u72b6\u6001\u77e9\u9635
\n """
\n y0 = [S0, I0, R0]<\/p>\n
sol = solve_ivp(
\n self.equations,
\n t_span,
\n y0,
\n method='RK45',
\n t_eval=t_eval,
\n dense_output=True
\n )<\/p>\n
return sol.t, sol.y<\/p>\n
def fit_to_data(self, time_data, I_data, S0=None, I0=None, R0=None):
\n """
\n \u4f7f\u7528\u5b9e\u9645\u6570\u636e\u62df\u5408SIR\u6a21\u578b\u53c2\u6570<\/p>\n
\u53c2\u6570:
\n time_data: \u65f6\u95f4\u6570\u636e
\n I_data: \u611f\u67d3\u8005\u6570\u636e
\n S0, I0, R0: \u521d\u59cb\u6761\u4ef6 (\u5982\u679c\u4e3aNone\u5219\u4ece\u6570\u636e\u4f30\u8ba1)<\/p>\n
\u8fd4\u56de:
\n popt: \u6700\u4f18\u53c2\u6570 [beta, gamma]
\n """
\n if I0 is None:
\n I0 = I_data[0]
\n if S0 is None:
\n S0 = self.N – I0
\n if R0 is None:
\n R0 = 0<\/p>\n
def model_function(t, beta, gamma):
\n """\u7528\u4e8e\u62df\u5408\u7684\u6a21\u578b\u51fd\u6570"""
\n self.beta = beta
\n self.gamma = gamma
\n _, y = self.simulate(S0, I0, R0, [min(time_data), max(time_data)], t_eval=t)
\n return y[1] # \u8fd4\u56de\u611f\u67d3\u8005I<\/p>\n
# \u521d\u59cb\u53c2\u6570\u731c\u6d4b
\n p0 = [0.3, 0.1]<\/p>\n
# \u53c2\u6570\u8fb9\u754c
\n bounds = ([0, 0], [10, 10])<\/p>\n
# \u62df\u5408\u53c2\u6570
\n popt, pcov = curve_fit(
\n model_function,
\n time_data,
\n I_data,
\n p0=p0,
\n bounds=bounds,
\n maxfev=5000
\n )<\/p>\n
# \u66f4\u65b0\u6a21\u578b\u53c2\u6570
\n self.beta, self.gamma = popt
\n self.R0 = self.beta \/ self.gamma<\/p>\n
return popt, pcov<\/p>\n
def plot_results(self, t, y, title="SIR\u6a21\u578b\u6a21\u62df\u7ed3\u679c"):
\n """
\n \u7ed8\u5236SIR\u6a21\u578b\u7ed3\u679c<\/p>\n
\u53c2\u6570:
\n t: \u65f6\u95f4\u6570\u7ec4
\n y: \u72b6\u6001\u77e9\u9635
\n title: \u56fe\u8868\u6807\u9898
\n """
\n fig, axes = plt.subplots(1, 3, figsize=(15, 4))<\/p>\n
# \u65f6\u95f4\u5e8f\u5217\u56fe
\n axes[0].plot(t, y[0], 'b-', label='\u6613\u611f\u8005S', linewidth=2)
\n axes[0].plot(t, y[1], 'r-', label='\u611f\u67d3\u8005I', linewidth=2)
\n axes[0].plot(t, y[2], 'g-', label='\u5eb7\u590d\u8005R', linewidth=2)
\n axes[0].set_xlabel('\u65f6\u95f4 (\u5929)')
\n axes[0].set_ylabel('\u4eba\u53e3\u6570')
\n axes[0].set_title(f'{title}\\\\nR0 = {self.R0:.2f}')
\n axes[0].legend()
\n axes[0].grid(True, alpha=0.3)<\/p>\n
# \u76f8\u5e73\u9762\u56fe
\n axes[1].plot(y[0], y[1], 'k-', linewidth=2)
\n axes[1].scatter(y[0,0], y[1,0], color='red', s=100, zorder=5, label='\u8d77\u70b9')
\n axes[1].scatter(y[0,-1], y[1,-1], color='green', s=100, zorder=5, label='\u7ec8\u70b9')
\n axes[1].set_xlabel('\u6613\u611f\u8005 S')
\n axes[1].set_ylabel('\u611f\u67d3\u8005 I')
\n axes[1].set_title('\u76f8\u5e73\u9762\u56fe (S-I\u5e73\u9762)')
\n axes[1].legend()
\n axes[1].grid(True, alpha=0.3)<\/p>\n
# \u65b0\u589e\u611f\u67d3\u56fe
\n new_infections = -np.diff(y[0]) # -dS\/dt\u8fd1\u4f3c\u65b0\u589e\u611f\u67d3
\n axes[2].plot(t[1:], new_infections, 'r-', linewidth=2)
\n axes[2].set_xlabel('\u65f6\u95f4 (\u5929)')
\n axes[2].set_ylabel('\u6bcf\u65e5\u65b0\u589e\u611f\u67d3')
\n axes[2].set_title('\u6bcf\u65e5\u65b0\u589e\u611f\u67d3')
\n axes[2].grid(True, alpha=0.3)<\/p>\n
plt.tight_layout()
\n plt.show()<\/p>\n
# \u8f93\u51fa\u5173\u952e\u6307\u6807
\n print(f"\u57fa\u672c\u518d\u751f\u6570 R0 = {self.R0:.2f}")
\n print(f"\u611f\u67d3\u5cf0\u503c = {np.max(y[1]):.0f}")
\n print(f"\u6700\u7ec8\u611f\u67d3\u6bd4\u4f8b = {y[1,-1]\/self.N*100:.1f}%")
\n print(f"\u6700\u7ec8\u6613\u611f\u8005\u6bd4\u4f8b = {y[0,-1]\/self.N*100:.1f}%")<\/p>\n
# \u4f7f\u7528\u793a\u4f8b
\nif __name__ == "__main__":
\n # \u521b\u5efaSIR\u6a21\u578b\u5b9e\u4f8b
\n N = 1000 # \u603b\u4eba\u53e3
\n sir = SIRModel(beta=0.3, gamma=0.1, N=N)<\/p>\n
# \u6a21\u62df\u53c2\u6570
\n S0 = N – 1
\n I0 = 1
\n R0 = 0
\n t_span = [0, 100]
\n t_eval = np.linspace(0, 100, 200)<\/p>\n
# \u8fd0\u884c\u6a21\u62df
\n t, y = sir.simulate(S0, I0, R0, t_span, t_eval)<\/p>\n
# \u7ed8\u5236\u7ed3\u679c
\n sir.plot_results(t, y)<\/p>\n
# \u53c2\u6570\u62df\u5408\u793a\u4f8b（\u4f7f\u7528\u6a21\u62df\u6570\u636e\u52a0\u566a\u58f0）
\n np.random.seed(42)
\n I_data_with_noise = y[1] + np.random.normal(0, 5, len(y[1]))<\/p>\n
# \u62df\u5408\u53c2\u6570
\n popt, pcov = sir.fit_to_data(t, I_data_with_noise, S0=S0, I0=I0, R0=R0)<\/p>\n
print(f"\\\\n\u62df\u5408\u7ed3\u679c:")
\n print(f"beta = {popt[0]:.4f} \u00b1 {np.sqrt(pcov[0,0]):.4f}")
\n print(f"gamma = {popt[1]:.4f} \u00b1 {np.sqrt(pcov[1,1]):.4f}")
\n print(f"R0 = {popt[0]\/popt[1]:.2f}") <\/p>\n
python<\/p>\n
import numpy as np
\nimport matplotlib.pyplot as plt
\nfrom scipy.integrate import odeint
\nimport seaborn as sns<\/p>\n
class AgeStructuredSEIR:
\n def __init__(self, age_groups, contact_matrix, beta, sigma, gamma, N):
\n """
\n \u521d\u59cb\u5316\u5e74\u9f84\u7ed3\u6784SEIR\u6a21\u578b<\/p>\n
\u53c2\u6570:
\n age_groups: \u5e74\u9f84\u7ec4\u5217\u8868
\n contact_matrix: \u5e74\u9f84\u7ec4\u95f4\u63a5\u89e6\u77e9\u9635
\n beta: \u611f\u67d3\u7387\u6807\u91cf\u6216\u5411\u91cf
\n sigma: \u6f5c\u4f0f\u671f\u5230\u53d1\u75c5\u7684\u8f6c\u79fb\u7387
\n gamma: \u6062\u590d\u7387
\n N: \u5404\u5e74\u9f84\u7ec4\u4eba\u53e3\u5411\u91cf
\n """
\n self.age_groups = age_groups
\n self.n_age = len(age_groups)
\n self.contact_matrix = contact_matrix
\n self.beta = beta
\n self.sigma = sigma
\n self.gamma = gamma
\n self.N = N
\n self.total_N = np.sum(N)<\/p>\n
def equations(self, y, t):
\n """
\n \u5e74\u9f84\u7ed3\u6784SEIR\u6a21\u578b\u5fae\u5206\u65b9\u7a0b<\/p>\n
\u53c2\u6570:
\n y: \u72b6\u6001\u53d8\u91cf\u5411\u91cf [S1, E1, I1, R1, S2, E2, I2, R2, …]
\n t: \u65f6\u95f4<\/p>\n
\u8fd4\u56de:
\n dydt: \u5bfc\u6570\u5411\u91cf
\n """
\n # \u91cd\u5851\u72b6\u6001\u53d8\u91cf
\n S = y[0:self.n_age]
\n E = y[self.n_age:2*self.n_age]
\n I = y[2*self.n_age:3*self.n_age]
\n R = y[3*self.n_age:4*self.n_age]<\/p>\n
# \u8ba1\u7b97\u5404\u5e74\u9f84\u7ec4\u7684\u603b\u611f\u67d3\u538b\u529b
\n if np.isscalar(self.beta):
\n beta_matrix = self.beta * self.contact_matrix
\n else:
\n # beta\u4e3a\u5411\u91cf\u65f6，\u8003\u8651\u5e74\u9f84\u7279\u5f02\u6027\u611f\u67d3\u7387
\n beta_matrix = np.diag(self.beta) @ self.contact_matrix<\/p>\n
# \u8ba1\u7b97\u5404\u5e74\u9f84\u7ec4\u7684\u611f\u67d3\u529b
\n force_of_infection = beta_matrix @ (I \/ self.N)<\/p>\n
# \u8ba1\u7b97\u5bfc\u6570
\n dS_dt = -S * force_of_infection
\n dE_dt = S * force_of_infection – self.sigma * E
\n dI_dt = self.sigma * E – self.gamma * I
\n dR_dt = self.gamma * I<\/p>\n
# \u5408\u5e76\u5bfc\u6570
\n dydt = np.concatenate([dS_dt, dE_dt, dI_dt, dR_dt])<\/p>\n
return dydt<\/p>\n
def simulate(self, S0, E0, I0, R0, t):
\n """
\n \u6a21\u62df\u5e74\u9f84\u7ed3\u6784SEIR\u6a21\u578b<\/p>\n
\u53c2\u6570:
\n S0, E0, I0, R0: \u5404\u5e74\u9f84\u7ec4\u521d\u59cb\u6761\u4ef6
\n t: \u65f6\u95f4\u6570\u7ec4<\/p>\n
\u8fd4\u56de:
\n y: \u72b6\u6001\u77e9\u9635
\n """
\n # \u5408\u5e76\u521d\u59cb\u6761\u4ef6
\n y0 = np.concatenate([S0, E0, I0, R0])<\/p>\n
# \u6c42\u89e3ODE
\n y = odeint(self.equations, y0, t)<\/p>\n
return y<\/p>\n
def calculate_R0(self):
\n """\u8ba1\u7b97\u5e74\u9f84\u7ed3\u6784\u6a21\u578b\u7684\u57fa\u672c\u518d\u751f\u6570R0"""
\n # \u4e0b\u4e00\u4ee3\u77e9\u9635\u65b9\u6cd5
\n if np.isscalar(self.beta):
\n beta_matrix = self.beta * self.contact_matrix
\n else:
\n beta_matrix = np.diag(self.beta) @ self.contact_matrix<\/p>\n
# \u6784\u9020\u4e0b\u4e00\u4ee3\u77e9\u9635
\n F = beta_matrix \/ self.gamma
\n V = np.diag(self.N) # \u8003\u8651\u5e74\u9f84\u7ec4\u4eba\u53e3\u5dee\u5f02<\/p>\n
G = F @ np.linalg.inv(V)<\/p>\n
# \u8ba1\u7b97\u8c31\u534a\u5f84\u4f5c\u4e3aR0
\n eigenvalues = np.linalg.eigvals(G)
\n R0 = np.max(np.abs(eigenvalues))<\/p>\n
return R0<\/p>\n
def plot_results(self, t, y, title="\u5e74\u9f84\u7ed3\u6784SEIR\u6a21\u578b"):
\n """
\n \u7ed8\u5236\u5e74\u9f84\u7ed3\u6784SEIR\u6a21\u578b\u7ed3\u679c<\/p>\n
\u53c2\u6570:
\n t: \u65f6\u95f4\u6570\u7ec4
\n y: \u72b6\u6001\u77e9\u9635
\n title: \u56fe\u8868\u6807\u9898
\n """
\n # \u63d0\u53d6\u5404\u72b6\u6001\u53d8\u91cf
\n S = y[:, 0:self.n_age]
\n E = y[:, self.n_age:2*self.n_age]
\n I = y[:, 2*self.n_age:3*self.n_age]
\n R = y[:, 3*self.n_age:4*self.n_age]<\/p>\n
# \u8ba1\u7b97\u603b\u548c
\n S_total = np.sum(S, axis=1)
\n E_total = np.sum(E, axis=1)
\n I_total = np.sum(I, axis=1)
\n R_total = np.sum(R, axis=1)<\/p>\n
fig, axes = plt.subplots(2, 3, figsize=(15, 10))<\/p>\n
# \u603b\u4eba\u53e3\u65f6\u95f4\u5e8f\u5217
\n axes[0, 0].plot(t, S_total, 'b-', label='\u6613\u611f\u8005', linewidth=2)
\n axes[0, 0].plot(t, E_total, 'm-', label='\u6f5c\u4f0f\u8005', linewidth=2)
\n axes[0, 0].plot(t, I_total, 'r-', label='\u611f\u67d3\u8005', linewidth=2)
\n axes[0, 0].plot(t, R_total, 'g-', label='\u5eb7\u590d\u8005', linewidth=2)
\n axes[0, 0].set_xlabel('\u65f6\u95f4 (\u5929)')
\n axes[0, 0].set_ylabel('\u4eba\u53e3\u6570')
\n axes[0, 0].set_title('\u603b\u4eba\u53e3\u52a8\u6001')
\n axes[0, 0].legend()
\n axes[0, 0].grid(True, alpha=0.3)<\/p>\n
# \u5404\u5e74\u9f84\u7ec4\u611f\u67d3\u8005\u65f6\u95f4\u5e8f\u5217
\n for i in range(self.n_age):
\n axes[0, 1].plot(t, I[:, i], label=f'\u5e74\u9f84\u7ec4 {self.age_groups[i]}', linewidth=2)
\n axes[0, 1].set_xlabel('\u65f6\u95f4 (\u5929)')
\n axes[0, 1].set_ylabel('\u611f\u67d3\u8005\u6570')
\n axes[0, 1].set_title('\u5404\u5e74\u9f84\u7ec4\u611f\u67d3\u8005\u52a8\u6001')
\n axes[0, 1].legend()
\n axes[0, 1].grid(True, alpha=0.3)<\/p>\n
# \u5404\u5e74\u9f84\u7ec4\u6700\u7ec8\u611f\u67d3\u6bd4\u4f8b
\n final_infected_ratio = np.sum(I + R, axis=0) \/ self.N
\n axes[0, 2].bar(range(self.n_age), final_infected_ratio * 100)
\n axes[0, 2].set_xlabel('\u5e74\u9f84\u7ec4')
\n axes[0, 2].set_ylabel('\u6700\u7ec8\u611f\u67d3\u6bd4\u4f8b (%)')
\n axes[0, 2].set_title('\u5404\u5e74\u9f84\u7ec4\u6700\u7ec8\u611f\u67d3\u6bd4\u4f8b')
\n axes[0, 2].set_xticks(range(self.n_age))
\n axes[0, 2].set_xticklabels(self.age_groups)
\n axes[0, 2].grid(True, alpha=0.3, axis='y')<\/p>\n
# \u70ed\u56fe：\u63a5\u89e6\u77e9\u9635
\n im = axes[1, 0].imshow(self.contact_matrix, cmap='YlOrRd', aspect='auto')
\n axes[1, 0].set_xlabel('\u63a5\u89e6\u5e74\u9f84\u7ec4')
\n axes[1, 0].set_ylabel('\u88ab\u63a5\u89e6\u5e74\u9f84\u7ec4')
\n axes[1, 0].set_title('\u5e74\u9f84\u63a5\u89e6\u77e9\u9635')
\n plt.colorbar(im, ax=axes[1, 0])<\/p>\n
# \u70ed\u56fe：\u5404\u5e74\u9f84\u7ec4\u611f\u67d3\u8005\u65f6\u95f4\u6f14\u5316
\n im = axes[1, 1].imshow(I.T, aspect='auto', cmap='Reds',
\n extent=[t[0], t[-1], self.n_age-0.5, -0.5])
\n axes[1, 1].set_xlabel('\u65f6\u95f4 (\u5929)')
\n axes[1, 1].set_ylabel('\u5e74\u9f84\u7ec4')
\n axes[1, 1].set_title('\u5404\u5e74\u9f84\u7ec4\u611f\u67d3\u8005\u65f6\u95f4\u6f14\u5316')
\n axes[1, 1].set_yticks(range(self.n_age))
\n axes[1, 1].set_yticklabels(self.age_groups)
\n plt.colorbar(im, ax=axes[1, 1])<\/p>\n
# \u5404\u5e74\u9f84\u7ec4\u5728\u5cf0\u503c\u65f6\u7684\u8d21\u732e
\n peak_time_idx = np.argmax(I_total)
\n peak_age_distribution = I[peak_time_idx, :] \/ I_total[peak_time_idx] * 100
\n axes[1, 2].pie(peak_age_distribution, labels=self.age_groups, autopct='%1.1f%%')
\n axes[1, 2].set_title(f'\u5cf0\u503c\u65f6\u5404\u5e74\u9f84\u7ec4\u611f\u67d3\u8005\u6bd4\u4f8b\\\\n(\u7b2c{t[peak_time_idx]:.0f}\u5929)')<\/p>\n
plt.suptitle(f'{title} – R0 = {self.calculate_R0():.2f}', fontsize=16)
\n plt.tight_layout()
\n plt.show()<\/p>\n
# \u8f93\u51fa\u5173\u952e\u7ed3\u679c
\n print(f"\u57fa\u672c\u518d\u751f\u6570 R0 = {self.calculate_R0():.2f}")
\n print(f"\u603b\u611f\u67d3\u5cf0\u503c = {np.max(I_total):.0f}")
\n print(f"\u6700\u7ec8\u603b\u611f\u67d3\u6bd4\u4f8b = {(I_total[-1] + R_total[-1])\/self.total_N*100:.1f}%")<\/p>\n
# \u5404\u5e74\u9f84\u7ec4\u8be6\u7ec6\u7ed3\u679c
\n print("\\\\n\u5404\u5e74\u9f84\u7ec4\u611f\u67d3\u60c5\u51b5:")
\n for i in range(self.n_age):
\n age_infected_ratio = (I[-1, i] + R[-1, i]) \/ self.N[i] * 100
\n print(f" \u5e74\u9f84\u7ec4 {self.age_groups[i]}: \u6700\u7ec8\u611f\u67d3\u6bd4\u4f8b = {age_infected_ratio:.1f}%")<\/p>\n
# \u4f7f\u7528\u793a\u4f8b
\nif __name__ == "__main__":
\n # \u5b9a\u4e49\u5e74\u9f84\u7ec4
\n age_groups = ['0-9', '10-19', '20-29', '30-39', '40-49', '50-59', '60-69', '70+']
\n n_age = len(age_groups)<\/p>\n
# \u5b9a\u4e49\u4eba\u53e3\u5206\u5e03（\u767e\u4e07）
\n N = np.array([12.0, 11.5, 10.8, 10.2, 9.5, 8.8, 7.2, 5.0])
\n total_N = np.sum(N)<\/p>\n
# \u5b9a\u4e49\u63a5\u89e6\u77e9\u9635（\u7b80\u5316\u793a\u4f8b）
\n # \u5bf9\u89d2\u7ebf\u4e0a\u662f\u540c\u5e74\u9f84\u7ec4\u5185\u63a5\u89e6，\u975e\u5bf9\u89d2\u662f\u4e0d\u540c\u5e74\u9f84\u7ec4\u95f4\u63a5\u89e6
\n np.random.seed(42)
\n contact_matrix = np.zeros((n_age, n_age))<\/p>\n
# \u540c\u5e74\u9f84\u7ec4\u5185\u63a5\u89e6\u8f83\u5f3a
\n np.fill_diagonal(contact_matrix, 8)<\/p>\n
# \u76f8\u90bb\u5e74\u9f84\u7ec4\u95f4\u6709\u4e2d\u7b49\u63a5\u89e6
\n for i in range(n_age-1):
\n contact_matrix[i, i+1] = contact_matrix[i+1, i] = 3<\/p>\n
# \u5bb6\u5ead\u5185\u63a5\u89e6\u6a21\u5f0f（\u513f\u7ae5\u4e0e\u6210\u4eba）
\n contact_matrix[0, 2:5] = [2, 2, 1]
\n contact_matrix[2:5, 0] = [2, 2, 1]<\/p>\n
# \u5de5\u4f5c\u573a\u6240\u63a5\u89e6（\u6210\u4eba\u4e4b\u95f4）
\n contact_matrix[2:6, 2:6] += 4
\n np.fill_diagonal(contact_matrix[2:6, 2:6], 8)<\/p>\n
# \u521b\u5efa\u6a21\u578b\u5b9e\u4f8b
\n beta = 0.25 # \u611f\u67d3\u7387
\n sigma = 1\/5.2 # \u6f5c\u4f0f\u671f5.2\u5929
\n gamma = 1\/14 # \u6062\u590d\u671f14\u5929<\/p>\n
model = AgeStructuredSEIR(age_groups, contact_matrix, beta, sigma, gamma, N)<\/p>\n
# \u8bbe\u7f6e\u521d\u59cb\u6761\u4ef6
\n # \u5047\u8bbe\u75ab\u60c5\u4ece20-29\u5c81\u5e74\u9f84\u7ec4\u5f00\u59cb
\n S0 = N.copy()
\n E0 = np.zeros(n_age)
\n I0 = np.zeros(n_age)
\n R0 = np.zeros(n_age)<\/p>\n
# \u572820-29\u5c81\u5e74\u9f84\u7ec4\u5f15\u5165\u521d\u59cb\u611f\u67d3\u8005
\n initial_infected_age_idx = 2 # 20-29\u5c81
\n initial_infected = 10
\n I0[initial_infected_age_idx] = initial_infected
\n S0[initial_infected_age_idx] -= initial_infected<\/p>\n
# \u65f6\u95f4\u6570\u7ec4
\n t = np.linspace(0, 180, 181) # 180\u5929，\u6bcf\u5929\u4e00\u4e2a\u70b9<\/p>\n
# \u8fd0\u884c\u6a21\u62df
\n y = model.simulate(S0, E0, I0, R0, t)<\/p>\n
# \u7ed8\u5236\u7ed3\u679c
\n model.plot_results(t, y, title="\u5e74\u9f84\u7ed3\u6784SEIR\u6a21\u578b – \u4f20\u67d3\u75c5\u4f20\u64ad\u6a21\u62df") <\/p>\n
\u80cc\u666f：\u67d0\u5927\u5b66\u6821\u56ed\u670915,000\u540d\u5b66\u751f\u548c\u6559\u804c\u5de5\u30022023\u5e74\u79cb\u5b63\u5b66\u671f，\u4e00\u79cd\u65b0\u578b\u547c\u5438\u9053\u4f20\u67d3\u75c5\u5728\u6821\u56ed\u5185\u5f00\u59cb\u4f20\u64ad\u3002\u6821\u533b\u9662\u8d44\u6e90\u6709\u9650，\u53ea\u6709100\u5f20\u9694\u79bb\u75c5\u5e8a\u53ef\u7528\u4e8e\u4f20\u67d3\u75c5\u60a3\u8005\u3002\u5b66\u6821\u7ba1\u7406\u5c42\u9700\u8981\u5236\u5b9a\u6709\u6548\u7684\u5e72\u9884\u7b56\u7565\u6765\u63a7\u5236\u75ab\u60c5\u4f20\u64ad，\u540c\u65f6\u5c3d\u91cf\u51cf\u5c11\u5bf9\u6b63\u5e38\u6559\u5b66\u6d3b\u52a8\u7684\u5e72\u6270\u3002<\/p>\n
\u5177\u4f53\u4efb\u52a1：<\/p>\n
\u5efa\u7acb\u4f20\u67d3\u75c5\u4f20\u64ad\u6a21\u578b，\u9884\u6d4b\u65e0\u5e72\u9884\u60c5\u51b5\u4e0b\u7684\u75ab\u60c5\u53d1\u5c55\u8d8b\u52bf\u3002<\/p>\n<\/li>\n
\u8bc4\u4f30\u4e0d\u540c\u5e72\u9884\u63aa\u65bd（\u53e3\u7f69\u5f3a\u5236\u4ee4\u3001\u7ebf\u4e0a\u6559\u5b66\u3001\u75ab\u82d7\u63a5\u79cd\u7b49）\u7684\u6548\u679c\u3002<\/p>\n<\/li>\n
\u63d0\u51fa\u6700\u4f18\u5e72\u9884\u7b56\u7565，\u5e73\u8861\u75ab\u60c5\u63a7\u5236\u6548\u679c\u4e0e\u7ecf\u6d4e\u3001\u793e\u4f1a\u6210\u672c\u3002<\/p>\n<\/li>\n
\u9884\u6d4b\u533b\u7597\u8d44\u6e90\u9700\u6c42，\u786e\u4fdd\u4e0d\u8d85\u8fc7\u6821\u533b\u9662\u627f\u8f7d\u80fd\u529b\u3002<\/p>\n<\/li>\n
\u8003\u8651\u5230\u6821\u56ed\u73af\u5883\u7684\u7279\u6b8a\u6027，\u6211\u4eec\u5728SEIR\u6a21\u578b\u57fa\u7840\u4e0a\u589e\u52a0\u4ee5\u4e0b\u72b6\u6001：<\/p>\n
\u9694\u79bb\u8005(Q)：\u88ab\u68c0\u6d4b\u5e76\u9694\u79bb\u7684\u611f\u67d3\u8005<\/p>\n<\/li>\n
\u91cd\u75c7\u60a3\u8005(D)：\u9700\u8981\u4f4f\u9662\u6cbb\u7597\u7684\u60a3\u8005<\/p>\n<\/li>\n<\/ul>\n
\u6a21\u578b\u72b6\u6001\u53d8\u91cf：<\/p>\n
S(t)S(t)：\u6613\u611f\u8005<\/p>\n<\/li>\n
E(t)E(t)：\u6f5c\u4f0f\u8005（\u5df2\u611f\u67d3\u4f46\u65e0\u75c7\u72b6\u4e14\u65e0\u4f20\u67d3\u6027）<\/p>\n<\/li>\n
Is(t)Is\u200b(t)：\u6709\u75c7\u72b6\u611f\u67d3\u8005<\/p>\n<\/li>\n
Ia(t)Ia\u200b(t)：\u65e0\u75c7\u72b6\u611f\u67d3\u8005<\/p>\n<\/li>\n
Q(t)Q(t)：\u9694\u79bb\u8005<\/p>\n<\/li>\n
D(t)D(t)：\u91cd\u75c7\u60a3\u8005（\u9700\u4f4f\u9662）<\/p>\n<\/li>\n
R(t)R(t)：\u5eb7\u590d\u8005<\/p>\n<\/li>\n
V(t)V(t)：\u75ab\u82d7\u63a5\u79cd\u8005<\/p>\n<\/li>\n<\/ul>\n
\u603b\u4eba\u53e3：N=S+E+Is+Ia+Q+D+R+VN=S+E+Is\u200b+Ia\u200b+Q+D+R+V<\/p>\n
dSdt=\u2212\u03b2(t)S(Is+\u03b8Ia)N\u2212\u03bd(t)S+\u03c9VdVdt=\u03bd(t)S\u2212\u03c9V\u2212\u03b2V\u03b2(t)V(Is+\u03b8Ia)NdEdt=\u03b2(t)[S+\u03b2VV](Is+\u03b8Ia)N\u2212\u03c3EdIsdt=p\u03c3E\u2212(\u03b3s+\u03b4s+\u03b1s)IsdIadt=(1\u2212p)\u03c3E\u2212(\u03b3a+\u03b4a)IadQdt=\u03b4sIs+\u03b4aIa\u2212\u03b3qQ\u2212\u03b1qQdDdt=\u03b1sIs+\u03b1qQ\u2212\u03b3dD\u2212\u03bcDdRdt=\u03b3sIs+\u03b3aIa+\u03b3qQ+\u03b3dDdtdS\u200bdtdV\u200bdtdE\u200bdtdIs\u200b\u200bdtdIa\u200b\u200bdtdQ\u200bdtdD\u200bdtdR\u200b\u200b=\u2212\u03b2(t)SN(Is\u200b+\u03b8Ia\u200b)\u200b\u2212\u03bd(t)S+\u03c9V=\u03bd(t)S\u2212\u03c9V\u2212\u03b2V\u200b\u03b2(t)VN(Is\u200b+\u03b8Ia\u200b)\u200b=\u03b2(t)[S+\u03b2V\u200bV]N(Is\u200b+\u03b8Ia\u200b)\u200b\u2212\u03c3E=p\u03c3E\u2212(\u03b3s\u200b+\u03b4s\u200b+\u03b1s\u200b)Is\u200b=(1\u2212p)\u03c3E\u2212(\u03b3a\u200b+\u03b4a\u200b)Ia\u200b=\u03b4s\u200bIs\u200b+\u03b4a\u200bIa\u200b\u2212\u03b3q\u200bQ\u2212\u03b1q\u200bQ=\u03b1s\u200bIs\u200b+\u03b1q\u200bQ\u2212\u03b3d\u200bD\u2212\u03bcD=\u03b3s\u200bIs\u200b+\u03b3a\u200bIa\u200b+\u03b3q\u200bQ+\u03b3d\u200bD\u200b<\/p>\n
\u5176\u4e2d：<\/p>\n
\u03b2(t)\u03b2(t)：\u65f6\u53d8\u611f\u67d3\u7387，\u53cd\u6620\u5e72\u9884\u63aa\u65bd\u6548\u679c<\/p>\n<\/li>\n
\u03b8\u03b8：\u65e0\u75c7\u72b6\u611f\u67d3\u8005\u76f8\u5bf9\u4f20\u67d3\u6027（0 < \u03b8 < 1）<\/p>\n<\/li>\n
\u03bd(t)\u03bd(t)：\u75ab\u82d7\u63a5\u79cd\u7387<\/p>\n<\/li>\n
\u03c9\u03c9：\u75ab\u82d7\u4fdd\u62a4\u8870\u51cf\u7387<\/p>\n<\/li>\n
\u03b2V\u03b2V\u200b：\u75ab\u82d7\u63a5\u79cd\u8005\u76f8\u5bf9\u6613\u611f\u6027（0 < \u03b2_V < 1）<\/p>\n<\/li>\n
\u03c3\u03c3：\u6f5c\u4f0f\u671f\u5230\u53d1\u75c5\u7684\u8f6c\u79fb\u7387<\/p>\n<\/li>\n
pp：\u6709\u75c7\u72b6\u611f\u67d3\u6bd4\u4f8b<\/p>\n<\/li>\n
\u03b3s,\u03b3a,\u03b3q,\u03b3d\u03b3s\u200b,\u03b3a\u200b,\u03b3q\u200b,\u03b3d\u200b：\u5404\u7c7b\u4eba\u7fa4\u6062\u590d\u7387<\/p>\n<\/li>\n
\u03b4s,\u03b4a\u03b4s\u200b,\u03b4a\u200b：\u68c0\u6d4b\u9694\u79bb\u7387<\/p>\n<\/li>\n
\u03b1s,\u03b1q\u03b1s\u200b,\u03b1q\u200b：\u53d1\u5c55\u4e3a\u91cd\u75c7\u7684\u6bd4\u7387<\/p>\n<\/li>\n
\u03bc\u03bc：\u91cd\u75c7\u6b7b\u4ea1\u7387<\/p>\n<\/li>\n<\/ul>\n
\u57fa\u4e8e\u6587\u732e\u548c\u7c7b\u4f3c\u75ab\u60c5\u6570\u636e：<\/p>\n
| NN<\/td>\n | \u603b\u4eba\u53e3<\/td>\n | 15,000<\/td>\n | \u95ee\u9898\u7ed9\u5b9a<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b20\u03b20\u200b<\/td>\n | \u521d\u59cb\u611f\u67d3\u7387<\/td>\n | 0.35\/\u5929<\/td>\n | \u57fa\u4e8eR0=3.5\u4f30\u8ba1<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b8\u03b8<\/td>\n | \u65e0\u75c7\u72b6\u8005\u4f20\u67d3\u6027<\/td>\n | 0.5<\/td>\n | \u6587\u732e\u4f30\u8ba1<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03c3\u03c3<\/td>\n | \u6f5c\u4f0f\u671f\u5012\u6570<\/td>\n | 1\/5.2 \u5929\u207b\u00b9<\/td>\n | \u6f5c\u4f0f\u671f5.2\u5929<\/td>\n<\/tr>\n | ||||||||||||||||||||
| pp<\/td>\n | \u6709\u75c7\u72b6\u6bd4\u4f8b<\/td>\n | 0.6<\/td>\n | \u6587\u732e\u4f30\u8ba1<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b3s\u03b3s\u200b<\/td>\n | \u6709\u75c7\u72b6\u8005\u6062\u590d\u7387<\/td>\n | 1\/10 \u5929\u207b\u00b9<\/td>\n | \u611f\u67d3\u671f10\u5929<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b3a\u03b3a\u200b<\/td>\n | \u65e0\u75c7\u72b6\u8005\u6062\u590d\u7387<\/td>\n | 1\/7 \u5929\u207b\u00b9<\/td>\n | \u611f\u67d3\u671f7\u5929<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b4s\u03b4s\u200b<\/td>\n | \u6709\u75c7\u72b6\u8005\u68c0\u6d4b\u7387<\/td>\n | 0.3\/\u5929<\/td>\n | \u5047\u8bbe\u503c<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b4a\u03b4a\u200b<\/td>\n | \u65e0\u75c7\u72b6\u8005\u68c0\u6d4b\u7387<\/td>\n | 0.1\/\u5929<\/td>\n | \u5047\u8bbe\u503c<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b1s\u03b1s\u200b<\/td>\n | \u6709\u75c7\u72b6\u8005\u91cd\u75c7\u7387<\/td>\n | 0.05\/\u5929<\/td>\n | \u5047\u8bbe5%\u53d1\u5c55\u4e3a\u91cd\u75c7<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b1q\u03b1q\u200b<\/td>\n | \u9694\u79bb\u8005\u91cd\u75c7\u7387<\/td>\n | 0.02\/\u5929<\/td>\n | \u9694\u79bb\u540e\u91cd\u75c7\u7387\u964d\u4f4e<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b3q\u03b3q\u200b<\/td>\n | \u9694\u79bb\u8005\u6062\u590d\u7387<\/td>\n | 1\/14 \u5929\u207b\u00b9<\/td>\n | \u9694\u79bb\u671f14\u5929<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b3d\u03b3d\u200b<\/td>\n | \u91cd\u75c7\u8005\u6062\u590d\u7387<\/td>\n | 1\/21 \u5929\u207b\u00b9<\/td>\n | \u4f4f\u9662\u671f21\u5929<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03bc\u03bc<\/td>\n | \u91cd\u75c7\u6b7b\u4ea1\u7387<\/td>\n | 0.01\/\u5929<\/td>\n | \u5047\u8bbe1%\u6b7b\u4ea1\u7387<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03b2V\u03b2V\u200b<\/td>\n | \u75ab\u82d7\u964d\u4f4e\u6613\u611f\u6027<\/td>\n | 0.3<\/td>\n | \u75ab\u82d7\u6709\u6548\u738770%<\/td>\n<\/tr>\n | ||||||||||||||||||||
| \u03c9\u03c9<\/td>\n | \u514d\u75ab\u8870\u51cf\u7387<\/td>\n | 1\/180 \u5929\u207b\u00b9<\/td>\n | \u4fdd\u62a4\u671f\u7ea66\u4e2a\u6708<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u57fa\u672c\u518d\u751f\u6570\u8ba1\u7b97：<\/p>\n R0=\u03b20\u03b3s\u22c5[p+(1\u2212p)\u03b8\u03b3s\u03b3a]\u22483.5R0\u200b=\u03b3s\u200b\u03b20\u200b\u200b\u22c5[p+(1\u2212p)\u03b8\u03b3a\u200b\u03b3s\u200b\u200b]\u22483.5<\/p>\n 5.2.4 \u5e72\u9884\u63aa\u65bd\u5efa\u6a21<\/h5>\n\u5b9a\u4e49\u5e72\u9884\u51fd\u6570\u03b2(t)\u03b2(t)：<\/p>\n \u03b2(t)=\u03b20\u22c5\u220fi=1n[1\u2212\u03b7i\u22c5fi(t)]\u03b2(t)=\u03b20\u200b\u22c5i=1\u220fn\u200b[1\u2212\u03b7i\u200b\u22c5fi\u200b(t)]<\/p>\n \u5176\u4e2d\u03b7i\u03b7i\u200b\u4e3a\u7b2ci\u79cd\u63aa\u65bd\u7684\u6548\u679c，fi(t)fi\u200b(t)\u4e3a\u5b9e\u65bd\u65f6\u95f4\u51fd\u6570\u3002<\/p>\n \u8003\u8651\u4e09\u79cd\u63aa\u65bd：<\/p>\n \u53e3\u7f69\u5f3a\u5236\u4ee4：\u964d\u4f4e\u4f20\u64ad\u738730%，\u03b71=0.3\u03b71\u200b=0.3<\/p>\n<\/li>\n 50%\u8bfe\u7a0b\u7ebf\u4e0a：\u964d\u4f4e\u63a5\u89e6\u738740%，\u03b72=0.4\u03b72\u200b=0.4<\/p>\n<\/li>\n \u75ab\u82d7\u63a5\u79cd：\u63d0\u9ad8\u75ab\u82d7\u63a5\u79cd\u7387\u03bd(t)\u03bd(t)<\/p>\n<\/li>\n 5.3 \u6a21\u578b\u6c42\u89e3\u4e0e\u5206\u6790<\/h4>\n |
| \u6d3b\u8dc3\u75c5\u4f8b\u5cf0\u503c<\/td>\n | 2,850<\/td>\n | 420<\/td>\n | -85.3%<\/td>\n<\/tr>\n |
| \u5cf0\u503c\u65f6\u95f4<\/td>\n | \u7b2c45\u5929<\/td>\n | \u7b2c78\u5929<\/td>\n | +33\u5929\u5ef6\u8fdf<\/td>\n<\/tr>\n |
| \u4f4f\u9662\u9700\u6c42\u5cf0\u503c<\/td>\n | 142<\/td>\n | 18<\/td>\n | -87.3%<\/td>\n<\/tr>\n |
| \u6700\u7ec8\u611f\u67d3\u7387<\/td>\n | 94.2%<\/td>\n | 32.5%<\/td>\n | -61.7%<\/td>\n<\/tr>\n |
| \u603b\u6b7b\u4ea1\u4eba\u6570<\/td>\n | 12.8<\/td>\n | 0.9<\/td>\n | -92.9%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n5.4.2 \u5173\u952e\u53d1\u73b0<\/h5>\n\u65e0\u5e72\u9884\u60c5\u51b5：\u75ab\u60c5\u8fc5\u901f\u7206\u53d1，\u7ea645\u5929\u8fbe\u5230\u5cf0\u503c，\u8fd195%\u4eba\u53e3\u6700\u7ec8\u611f\u67d3，\u4f4f\u9662\u9700\u6c42\u8fdc\u8d85\u533b\u9662\u627f\u8f7d\u80fd\u529b(100\u5e8a)，\u5c06\u5bfc\u81f4\u533b\u7597\u7cfb\u7edf\u5d29\u6e83\u3002<\/p>\n<\/li>\n \u7ec4\u5408\u5e72\u9884\u6548\u679c\u663e\u8457：<\/p>\n
|