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Chinese Science Bulletin, Volume 65 , Issue 22 : 2334-2341(2020) https://doi.org/10.1360/TB-2020-0413

Progress on the basic reproduction number of SARS-CoV-2

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  • ReceivedApr 14, 2020
  • AcceptedJun 3, 2020
  • PublishedJun 4, 2020

Abstract

Coronavirus disease 2019 (COVID-19) is an emerging infectious diseases caused by a new coronavirus named SARS-CoV-2. In the epidemiology of infectious diseases, the basic reproduction number (R0) is a central quantitative parameter that can be used to measure the transmission potential of infectious diseases. R0 represents the average number of new infections generated by an infectious person in a totally naïve population. If R0>1, the number infected is likely to increase. If R0<1, transmission is likely to die out. In this study, we present a review on the estimation of basic reproduction number of SARS-CoV-2, in order to provide a scientific basis for better understanding the dynamic characteristics of SARS-CoV-2 transmission. R0 is usually estimated with various types of complex mathematical models. It can be calculated and estimated by differential equations in the propagation dynamics model (such as SIR model and SEIR model), or estimated using maximum likelihood method, stochastic model, etc. Modeled R0 values are dependent on model structures and assumptions. With the rapid spread of the global COVID-19 epidemic, there is an increasing evidence on R0 of SARS-CoV-2. However, the results of R0 estimation varied a lot in different studies, due to different model assumptions, parameters settings, and data used in the models. The estimated median R0 is about 3.15 (95%CI: 2.26–6.20) in studies published by peer-reviewed journals, 3.01 (95%CI: 1.99–5.44) in preprinted platform without peer-reviewed and online reports, and 2.55 (95%CI: 1.61–3.55) in studies published in Chinese language, and 3.10 (95%CI: 2.09–6.05) in studies published in English language. With the implementation of comprehensive intervention measures, R0 showed a downward trend. In the basic propagation dynamics model, R0 might be estimated with biases, due to the limitations of its basic assumptions (such as fixed population). Regardless of the methods used, characteristics of the virus should be considered fully during estimation. To better estimate R0, some researchers improved the basic propagation dynamics model and considered the assumption of unfixed population, population mobility, infectivity of the virus during incubation period, and quarantine measures, to make it more in line with the characteristics of SARS-CoV-2. When adopting other methods such as exponential growth model and stochastic model to estimate R0, the estimated results is relatively stable because that it is not restricted by assumptions of the propagation dynamics model. However, the estimation are more susceptible to the data used in the model (such as distribution of the data) because that the inherent characteristics of the spread of infectious diseases are not considered. Thus, it is particularly critical to use appropriate data distribution and parameters during estimation.


Funded by

国家重点研发计划(2020YFC0846300)

国家自然科学基金(71934002)


References

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  • Table 1   Estimation of R0 in China and other countries

    文献

    地区

    研究日期

    方法

    参数估计

    R0 (95%CI)

    Chen等人[19]

    中国

    2020.12.7~2020.1.1

    建立蝙蝠-宿主-病毒储蓄地-人的简化RP模型

    根据已有研究和报告确定一部分参数的取值, 使用Berkeley Madonna方法拟合曲线, 依据公式计算再生数

    2.3~3.58

    Liu等人[12]a)

    中国

    ~2020.1.20

    建立Flow-SEHIR模型, Flow为城市间人员流动, H为核酸检测假阴性的患者

    使用2020年1月20日之前的确诊及人员流动数据模拟疾病的自然传播

    3.56(2.31~4.81)

    Li等人[20]

    中国

    ~2020.1.22

    基于考虑动物源性传播的更新方程对与华南海鲜市场无关的病例建立模型

    根据暴露史和出现症状日期拟合对数正态分布估计潜伏期、根据出现症状的日期、第一次去医院的时间、医院确诊时间拟合Weibull分布来估计相应时间间隔的分布, 使用更新方程构建模型并估计再生数

    2.2(1.4~3.9)

    Shen等人[11]a)

    中国

    2019.12.12~2020.1.22

    建立SEIJR模型, I为有症状感染者, J为接受治疗的隔离者

    使用非线性最小二乘获得参数的点估计, 使用马尔可夫链蒙特卡洛方法与Metropolis-Hastings算法获取参数的分布

    4.71(4.50~4.92)

    Tang等人[21]

    中国

    2020.1.10~22

    建立改良SEIR模型, 包括住院者H, 无症状感染者A, 被隔离的易感者Sq, 被隔离的暴露者Eq与被隔离的感染者Iq

    使用马尔可夫链蒙特卡洛方法与Metropolis-Hasting采样拟合仓室模型并拟合模型关键参数, 基于估计参数使用公式计算基本再生数

    6.47

    (5.71~7.23)

    WHO[4]a)

    中国

    ~2020.1.23

    /

    /

    1.4~2.5

    Li等人[13]

    中国

    2020.1.10~23

    拟合加入旅行数据的, 含有分层人群(记录的和未记录的感染者)的SEI模型

    通过随机采样进行模型试验和参数估计, 并根据延迟报告调整预测数与报告数之间的关系, 使用IF-EAFK方法评估模型有效性

    2.38(2.04~2.77)

    Zhao等人[17]

    中国

    ~2020.1.24

    基于报告率调整病例总数,并拟合指数增长模型

    使用均值为7.5, 标准差为3.4的Gamma分布估计序列间隔, 并根据非线性最小二乘拟合指数模型

    2.13~3.33

    Zhao等人[25]

    中国

    ~2020.1.24

    基于指数泊松过程拟合流行曲线

    使用SARS与MERS的平均代际间隔估计SARS-CoV2的代际间隔, 并使用泊松先验的对数似然法估计指数增长率, 并代入公式计算再生数

    2.56(2.49~2.63)

    Jung等人[26]

    中国

    2019.12.8与2020.1.24

    拟合指数增长模型

    通过马尔可夫链蒙特卡洛方法(MCMC)拟合指数模型, 根据拟合参数计算再生数

    2.10~3.19

    Wu等人[10]

    中国

    2019.12.31~2020.1.25

    构建加入陆地及航空旅行数据与动物源性传播的SEIR模型

    使用马尔可夫链蒙特卡洛方法(MCMC)与吉布斯采样估计基本再生数

    2.68(2.47~2.86)

    Majumder等人[18]a)

    中国

    2019.12.8, 1.26

    病例延迟与指数调整模型

    使用SARS与MERS的中间代际间隔6~10 d建立IDEA模型

    2.2~3.1

    周涛等人[27]

    中国

    ~2020.1.26

    构建SEIR模型

    结合已有研究、报告数据, 带入公式计算

    2.8~3.9

    Riou等人[9]a)

    中国

    2020.1.28

    对早期人传人轨迹进行随机模拟

    基于负二项分布生成二代病例, 基于Gamma分布生成代际间隔, 考虑指示病例数和动物传人的时间, 2019年11月27日对病例周围进行均匀采样, 将模拟结果加和, 与先前预测的感染人数进行比较

    2.2(1.4~3.8)

    Sanche等人[28]

    中国

    2020.1.15~30

    假定暴露/感染的武汉个体数符合指数增长, 建立“第一次到达”数学模型

    假定暴露/感染的武汉个体数符合指数增长, 建立第一个病例到达各省的时间的概率分布函数, 通过使用国内旅行数据计算观察到的到达时间来估计指数增长率, 再使用已有研究的代际间隔与潜伏期, 通过公式计算基本再生数

    5.7

    (3.8~8.9)

    宋倩倩等人[8]

    中国

    2020.1.10~31

    拟合指数增长模型、使用最大似然法、构建SEIR模型

    使用Weibull、Gamma、对数正态模型拟合从报告中获取的潜伏期和代际间隔的概率分布并拟合模型, 根据模型参数计算R0

    3.74(3.63~3.87,指数)

    3.16(2.90~3.43,似然)

    3.91(3.47~4.11, SEIR)

    Tang等人[29]

    中国

    ~2020.2.3

    构建基于代际间隔与每日出现症状数的更新方程

    通过确诊者的流行病学数据拟合症状出现至确诊的时间分布, 使用武汉1月22日之前确诊病例数与该分布估计武汉在1月4日前出现症状的病例数; 使用流行病学数据拟合代际间隔的Gamma分布并代入更新方程估计R0

    3.27(2.98, 3.58)

    耿辉等人[6]

    中国

    2020.1.21~2.8

    构建SEIR模型

    根据报告数据及模型假设确定参数, 代入公式计算

    2.38~2.72

    Lai等人[30]

    中国

    ~2020.2.13

    拟合指数增长模型

    根据指数增长率计算

    2.6(2.1~5.1)

    Rocklov等人[31]

    钻石公主号

    2020.1.21~2.20

    构建SEIR模型

    根据预测的累计病例与观测到的累计病例进行矫正

    14.8

    Fang等人[32]

    中国

    ~2020.2.29

    建立SEIR模型

    根据确诊病例数拟合三次函数, 根据拟合的三次函数的预测病例数来模拟SEIR模型并获得参数估计, 根据公式计算再生数

    2.35~3.21

    Choi等人[14]

    韩国

    ~2020.3.5

    建立SEIHR模型, H指住院者

    根据预测病例数与实际病例数调整模型参数

    4.27(湖北)

    3.47~3.54(韩国)

    Tang等人[21] a)

    意大利伦巴第大区

    2020.1.14~3.8

    假设每日新增病例符合泊松分布, 根据似然函数估计再生数

    利用流行病学数据估计代际间隔等分布, 假设每日新增病例符合泊松分布, 使用马尔可夫链蒙特卡洛方法Metropolis-Hastings采样获取再生数的后验分布

    3.1(2.9~3.2)

    预印本. 非标注的文献均已在线发表; /表示未提及

  • Table 2   Comparison of the estimation of R0 in different provinces

    文献

    研究地区

    研究日期

    方法

    参数估计

    R0 (95%CI)

    Liu等人[33] a)

    广东省

    ~2020.1.23

    拟合指数分布、使用最大似然法

    使用SARS的生成时间通过泊松回归拟合指数增长模型, 使用指数增长率计算再生数、假设二代病例数服从泊松分布, 基于最大似然计算再生数

    2.90(2.32~3.63, 指数)

    2.92(2.28~3.67, 似然)

    Zhao等人[7]

    湖北省

    2020.1.10~24

    基于报告率调整病例总数, 并拟合指数模型

    分别根据MERS的均值为7.7、标准差为3.4的Gamma分布, SARS的均值为8.4、标准差为3.8的Gamma分布和平均之后的均值为8.0、标准差为3.6的Gamma分布来估计代际间隔, 并根据非线性最小二乘拟合指数模型

    2.24~5.71

    Sanche等人[34] a)

    湖北省

    ~2020.1.30

    指数增长模型(模型一);

    基于人群分层的SEIR模型(模型二)

    模型一假设武汉本地的病例指数增长, 使用旅行数据和感染者在各省市首次出现的时间拟合指数增长模型; 模型二假设武汉病例指数增长, 使用随机的基于主体的模型描述感染者的轨迹, 拟合SEIR模型. 参数假设一为传染期等于潜伏期; 参数假设二为潜伏期比传染期短2 d

    6.2(3.3~11.3, 指数1)

    4.7(2.8~7.6, 指数2)

    6.6(4~10.5, SEIR1)

    4.9(3.3~7.2, SEIR2)

    武文韬等人[35]

    广东省

    2020.1.4~2.8

    构建SIR模型

    最小化t时刻真实值和预测值之间的方差

    1.65~2.48

    白尧等人[15]

    山西省

    2019.12.31~2020.2.13

    构建SEIAR模型, 其中A指无症状感染者

    通过文献、实际疫情确定参数范围, 使用Berkeley Madonna软件构建模型

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