Research progress on the basic and effective reproductive number in the epidemiology of infectious diseases
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摘要: 当一种新的传染病出现时,研究者们往往会用一些流行病学指标来衡量它的传播能力,其中一类重要的指标就是基本再生数(R0)和有效再生数(Re)。然而,R0和Re的定义、计算方法和结果解释等在许多场景中被错误地理解,甚至误用。本文围绕R0和Re的定义、计算方法、两者关系、影响因素、流行病学意义、应用注意事项以及常见传染病的基本再生数等进行综述,为卫生决策部门利用R0和Re制定针对性疫情防控措施提供参考。Abstract: With the occurrence of an emerging infectious disease, some epidemiological indicators are used to measure the transmission of the disease. Basic reproduction number (R0) and effective reproduction number (Re) are two crucial indicators among them. However, the definition, calculation, and interpretation of R0 and Re are misunderstood or even misused in many cases. This review introduces the definition, calculation, influence factors, epidemiology significance, notes for application, and the R0 of some common infectious diseases, aiming to provide scientific guidance for health decision-making departments to prevent and control the epidemic of infectious disease.
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表 1 常见传染病的基本再生数
Table 1. Basic reproductive numbers of common infectious diseases
疾病 传播途径 时间(年) 地区 R0值 百日咳 空气、飞沫、接触 1908-1917 美国 12.20 1944-1979 英格兰、威尔士 14.30~17.10 a 麻疹 空气 1912 美国 12.5 1944-1979 英格兰、威尔士 13.70~18.00 a 1996 卢森堡 6.2~7.7 a 天花 空气、飞沫 1967 尼日利亚 3.50~6.00 a SARS 空气、飞沫、粪口 2003 新加坡[20] 2.20~3.60 a 2003 中国香港[21] 2.70 (2.20~3.70) b MERS c 空气、飞沫、接触 2013 英国、法国等[22] 0.60~0.69 a 2013 沙特阿拉伯[23] 2.00~6.70 a EBHF d 体液 2014 几内亚[24] 1.51(1.50~1.52) b 2014 塞拉利昂[24] 2.53 (2.41~2.67) b H1N1流感 空气、飞沫 1918 全球[25] 1.80 (1.47~2.27) b 2009 全球[25] 1.46 (1.30~1.70) b 2009 中国[26] 1.68 (1.45~1.92) b 2009 印度[27] 1.03~1.75 a ZIKVD e 虫媒、体液、母婴 2007 雅普岛等[28] 2.10 (1.80~2.50) b 2015 巴西[29-30] 1.80~5.80 a AIDS 性、血液、母婴 2009 法国 3.65 (3.64~3.66) b 2009 西德 4.08 (4.02~4.14) b 2009 英国 3.67 (3.66~3.69) b COVID-19 飞沫、接触、粪口 2020 中国[31] 1.40~3.58 a 2020 韩国[32] 2.10 (1.84~2.42) b 2020 钻石公主号游轮[33] 5.70 (4.23~7.79) b 注:a R0值的范围,即Min~Max;b R0 (95% CI)值;c中东呼吸综合征(Middle East respiratory syndrome, MERS);d埃博拉出血热(ebola hemorrhagic ferer, EBHF);e寨卡病毒病(Zika virus diseases, ZIKVD)。 表 2 差分方程和Python软件实现代码
Table 2. Differential equation and implementation code in Python
类别 描述 差分方程 dS/dt=-βSI/N dE/dt=βSI/N-σE dI/dt=σE-γI dR/dt=γI 实现代码 import scipy.integrate as spi import numpy as np import matplotlib.pyplot as plt N=11212000 beta=0.6 gamma=0.1 Te=4.0 I_0=1 E_0=0 R_0=0 S_0=N-I_0-E_0-R_0 T=160 INI=(S_0, E_0, I_0, R_0) def funcSEIR(inivalue, _): Y=np.zeros(4) X=inivalue Y[0]=-(beta*X[0]*X[2])/N Y[1]=(beta*X[0]*X[2])/N-X[1]/Te Y[2]=X[1]/Te-gamma*X[2] Y[3]=gamma*X[2] return Y T_range=np.arange(0, T+1) RES=spi.odeint(funcSEIR, INI, T_range) plt.plot(RES[: , 0], color =’darkblue’, label=’Susceptible’, marker=’.’) plt.plot(RES[: , 1], color=’orange’, label=’Exposed’, marker=’.’) plt.plot(RES[: , 2], color =’red’, label=’Infection’, marker=’.’) plt.plot(RES[: , 3], color=’green’, label =’Recovery’, marker=’.’) plt.title(’SEIR Model’) plt.legend() plt.xlabel(’Day’) plt.ylabel(’Number’) plt.show() -
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