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通过最新公布的流行病学数据估计了易感者-感染者模型参数, 结合百度迁徙数据和公开新闻报道, 刻画了疫情前期武汉市人口流动特征, 并代入提出的支持人口流动特征的时域差分方程模型进行动力学模拟, 得到一些推论: 1)未受干预时传染率在一般环境下以95%的置信度位于区间[0.2068, 0.2073], 拟合优度达到0.999; 对应地, 基本传染数R0位于区间[2.5510, 2.6555]; 极限环境个案推演的传染率极值为0.2862, 相应的R0极值为3.1465; 2)百度迁徙规模指数与铁路发送旅客人数的Pearson相关系数达到0.9108, 有理由作为人口流动的有效估计; 3)提出的模型可有效推演疫情蔓延至外省乃至全国的日期, 其中41.38%的预测误差 ≤ 1 d, 79.31%的预测误差 ≤ 3 d, 96.55%预测误差 ≤ 5 d, 总体平均误差约为 2.14 d.
In this paper, a simple susceptible-infected (SI) model is build for simulating the early phase of COVID-19 transmission process. By using the data collected from the newest epidemiological investigation, the parameters of SI model is estimated and compared with those from some other studies. The population migration data during Spring festival in China are collected from Baidu.com and also extracted from different news sources, the migration characteristic of Wuhan city in the early phase of the epidemic situation is captured, and substituted into a simple difference equation model which is modified from the SI model for supporting migrations. Then several simulations are performed for the spatiotemporal transmission process of COVID-19 in China. Some conclusions are drawn from simulations and experiments below. 1) With 95% confidence, the infection rate of COVID-19 is estimated to be in a range of 0.2068–0.2073 in general situation, and the corresponding basic reproduction number R0 is estimated to be in a range of 2.5510–2.6555. A case study shows that under an extreme condition, the infection rate and R0 are estimated to be 0.2862 and 3.1465, respectively. 2) The Pearson correlation coefficient between Baidu migration index and the number of travelers sent by railway is 0.9108, which indicates a strong linear correlation between them, thus it can be deduced that Baidu migration index is an efficient tool for estimating the migration situation. 3) The epidemic arrival times for different provinces in China are estimated via simulations, specifically, no more than 1 day within an estimation error of 41.38%; no more than 3 days within an error of 79.31%, and no more than 5 days with an error of 95.55%. An average estimation error is 2.14 days. -
Keywords:
- COVID-19 /
- epidemic model /
- traffic flow /
- migration
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图 1 不同的
$\beta $ 取值下实际累积发病人数与预测发病人数的对比 (a) 2019年; (b) 2020年Fig. 1. Comparing the actual cumulative number of cases and its estimations according to different
$\beta $ : (a) Year 2019; (b) year 2020.
图 2 武汉“封城”前夕迁徙指数与2019年的对比 (a)迁入规模指数; (b) 迁出规模指数
Fig. 2. Comparing the migration index of Wuhan before the spring festival with the same period of 2019: (a) Inner migration index; (b) outer migration index.
图 3 各省区首达病例时间预测误差
Fig. 3. The estimation errors of the first arrival times for each provinces.
表 1 2019年累积发病人数
Table 1. The cumulative number of confirmed cases in 2019.
日期 人数 日期 人数 日期 人数 日期 人数 日期 人数 12/09 1 12/17 12 12/21 29 12/25 47 12/29 78 12/11 2 12/18 14 12/22 37 12/26 49 12/30 90 12/12 5 12/19 16 12/23 40 12/27 59 12/31 102 12/15 8 12/20 25 12/24 45 12/28 68 
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表 2 2020年各时间段累积发病人数
Table 2. The cumulative number of confirmed cases in each time slots in 2020.
截至时点 估计发病人数 实际发病人数 上报CCDC人数 软件抓取法 蒙特卡罗法 2019/12/31 102 49 102 0 2020/01/10 738 459 738—781 41 2020/01/20 6162 5800 6143—6187 291 2020/01/31 32661 29654 32633—32677 11821 2020/02/11 44692 69163 44672 44730
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表 3 传染率
$\beta $ 可能的取值Table 3. Possible values of the infection rate.
截止日期 $\beta $ 置信区间 R2 截止日期 $\beta $ 置信区间 R2 2019/12/31 0.2213 [0.2152, 0.2274] 0.868 2020/01/11 0.2066 [0.2056, 0.2274] 0.990 2020/01/01 0.2171 [0.2116, 0.2225] 0.878 2020/01/12 0.2063 [0.2056, 0.2071] 0.993 2020/01/02 0.2168 [0.2127, 0.2209] 0.923 20200/1/13 0.2060 [0.2054, 0.2067] 0.995 2020/01/03 0.2159 [0.2127, 0.2191] 0.949 2020/01/14 0.2059 [0.2054, 0.2064] 0.997 2020/01/04 0.2155 [0.2130, 0.2179] 0.967 2020/01/15 0.2056 [0.2052, 0.2060] 0.998 2020/01/05 0.2138 [0.2118, 0.2159] 0.973 2020/10/16 0.2058 [0.2054, 0.2061] 0.998 2020/01/06 0.2127 [0.2109, 0.2144] 0.980 2020/01/17 0.2060 [0.2057, 0.2063] 0.999 2020/01/07 0.2109 [0.2093, 0.2126] 0.980 2020/01/18 0.2064 [0.2061, 0.2066] 0.999 2020/01/08 0.2091 [0.2075, 0.2107] 0.979 2020/01/19 0.2065 [0.2063, 0.2067] 0.999 2020/01/09 0.2080 [0.2067, 0.2094] 0.984 2020/01/20 0.2066 [0.2064, 0.2068] 0.999 2020/01/10 0.2067 [0.2054, 0.2080] 0.985 2020/01/21 0.2070 [0.2068, 0.2073] 0.999
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表 4 重要时间节点的累积发病人数
Table 4. Cumulative confirmed cases in key time nodes.
截至时点 $\beta $ 0.2213 0.2159 0.2080 0.2066 2019/12/31 130 116 97 95 2020/01/10 1190 1001 777 753 2020/01/20 10870 8666 6220 5964
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表 5 武汉市三大火车站发送旅客人数与迁出指数
Table 5. The Baidu inner migration index and the number of the travelers sent from Wuhan`s major railway stations.
日期 迁徙指数 人数/万 日期 迁徙指数 人数/万 日期 迁徙指数 人数/万 2020/01/10 6.6232 27 2020/01/22 11.8403 29.96 2019/01/29 7.0282 27.2 2020/01/11 7.5612 29.8 2019/01/21 4.5718 21.6 2019/01/30 7.1072 27.7 2020/01/12 6.2165 27 2019/01/22 4.6892 21.4 2019/01/31 7.4800 28.1 2020/01/13 5.7620 24.8 2019/01/23 4.8062 23 2019/02/01 8.7140 29.8 2020/01/15 5.9087 26.5 2019/01/24 4.8605 21.7 2019/02/02 9.6043 31.5 2020/01/16 6.0028 27.7 2019/01/26 7.0436 27 2019/02/03 9.2247 29.1 2020/01/19 7.4060 30 2019/01/28 6.7706 26.8
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表 6 省级区域的首例到达时间
Table 6. The arrival times of each provinces.
省份 $\beta $ 实际日期 省份 $\beta $ 实际日期 0.2213 0.2159 0.2070 0.2213 0.2159 0.2070 安徽 01/06 01/06 01/07 01/07 辽宁 01/11 01/12 01/13 01/09 北京 01/07 01/07 01/08 01/08* 内蒙古 01/13 01/14 01/16 01/16 福建 01/09 01/09 01/10 01/06 宁夏 01/17 01/18 01/20 01/17 甘肃 01/10 01/11 01/12 01/04 青海 01/19 01/21 01/23 01/21 广东 01/05 01/05 01/06 01/04 山东 01/08 01/08 01/09 01/08 广西 01/08 01/09 01/09 01/13 山西 01/09 01/10 01/10 01/14 贵州 01/08 01/08 01/09 01/06 陕西 01/09 01/09 01/10 01/12 海南 01/10 01/11 01/12 01/13 上海 01/08 01/08 01/09 01/10 河北 01/08 01/09 01/09 01/13 四川 01/08 01/08 01/09 01/07 河南 01/04 01/04 01/05 01/03 天津 01/14 01/15 01/16 01/11 黑龙江 01/12 01/13 01/14 01/12 西藏 > 01/23 > 01/23 > 01/23 01/30 湖南 01/05 01/05 01/06 01/05 新疆 01/12 01/12 01/14 01/17 吉林 01/15 01/15 01/17 01/14 云南 01/09 01/11 01/10 01/07 江苏 01/06 01/07 01/07 01/10 浙江 01/07 01/07 01/08 01/04 江西 01/06 01/07 01/07 01/07 重庆 01/08 01/08 01/09 01/06
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[1] Li Q, Guan X H, Wu P, Wang X Y, Zhou L, Tong Y Q, Ren R Q, Leung S, Lau E, Wong J, Xing X S, Xiang N J, Wu Y, Li C, Chen Q, Li D, Liu T, Zhao J, Li M, Tu W X, Chen C D, Jin L M, Yang R, Wang Q, Zhou S H, Wang R, Liu H, Luo Y B, Liu Y, Shao G, Li H, Tao Z F, Yang Y, Deng Z Q, Liu B X, Ma Z T, Zhang Y P, Shi G Q, Lam T, Wu J, Gao G, Cowling B, Yang B, Leung G, Feng Z J 2020 N. Engl. J. Med. 382 1199
Google Scholar
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Google Scholar
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Google Scholar
Cao S L, Feng P H, Shi P P 2020 J. Zhejiang Univ. 49 1
Google Scholar
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[9] Yang Z F, Zeng Z Q, Wang K, Wong S S, Liang W H, Zanin M, Liu P, Cao X D, Gao Z Q, Mai Z T, Liang J Y, Liu X Q, Li S Y, Li Y M, Ye F, Guan W J, Yang Y F, Li F, Luo S M, Xie Y Q, Liu B, Wang Z L, Zhang S B, Wang Y N, Zhong N S, He J X 2020 J. Thorac. Dis. 12 2077
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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