diff --git a/Model/Estimation of R0_Yijun/README.md b/Model/Estimation of R0_Yijun/README.md deleted file mode 100644 index bc15186..0000000 --- a/Model/Estimation of R0_Yijun/README.md +++ /dev/null @@ -1,116 +0,0 @@ -# Estimation of R0 - -#### Author: Yijun Wang -#### Date: 2020-02-09 -#### Last Update: 2020-02-12 - -### Usage -- Download my Jupyter notebook file: [Estimation of R0.ipynb](https://github.com/yijunwang0805/YijunWang/blob/master/Estimation%20of%20R0_Yijun/Estimation%20of%20R0.ipynb) - -### Conclusion -- R0 exhibits a sharp upward trend from 2020-01-24 to 2020-01-29, rising from 2.24 to 2.51. Since the Wuhan city runs out of medical supply to confirm nCoV cases, the rise of R0 could be the consequence of outflow of Wuhan residents before lockdown, who seek for medical assistance outside of the Wuhan city and eventually confirmed by the official -- Then, R0 value fluctuates at 2.50 from 2020-01-30 to 2020-02-03, reaching its peak at 2.51 on 2020-02-03. Time lag between onset of symptons and case confirm is estimated to be 7 to 10 days. The halt of rising trend could be inferred as the consequence of official and social appeal of 'staying at home'. In addition, the fluctuation could be attributed by the inner family infection due to in-house quarantine, where family member infects one and another. -- Thereafter, R0 is trending downward, dropping to 2.39 on 2020-02-12. This can be viewed as the positive effect of social and non-pharmeceutical intervention (stay at home and wear facial mask) -- Sensitivity analysis shows that the value of R0 is highly influenced by the value of Tg. As more cases being collected, the measure of Tg will become more accurate - -![R0](https://user-images.githubusercontent.com/56286591/74337661-aafb6480-4ddb-11ea-8a49-9c433ddc81aa.png) - -### Sensitivity Analysis -Parameter values -| p | rho | Tg | Y(t) | R0 | -| --- | --- | --- | --- | --- | -| 0.695 | 0.4 | 7.5 | 59826 | 2.44 | -| 0.695 | 0.65 | 7.5 | 59826 | 2.42 | -| 0.695 | 0.9 | 7.5 | 59826 | 2.24 | -| 0.695 | 0.4 | 10 | 59826 | 3.05 | -| 0.695 | 0.65 | 10 | 59826 | 3.02 | -| 0.695 | 0.9 | 10 | 59826 | 2.71 | -| 0.8 | 0.4 | 7.5 | 62102 | 2.44 | -| 0.8 | 0.65 | 7.5 | 62102 | 2.42 | -| 0.8 | 0.9 | 7.5 | 62102 | 2.25 | -| 0.8 | 0.4 | 10 | 62102 | 3.06 | -| 0.8 | 0.65 | 10 | 62102 | 3.03 | -| 0.8 | 0.9 | 10 | 62102 | 2.72 | - -### Parameters -- TL is the generation period, which is assumed to be 7.5, taken from reference 1 and 2 -- p is the ratio of susceptible (49) turning into confirmed case (59), which is taken to be 0.695, numbers from [People's Daily Weibo](https://m.weibo.cn/u/2803301701) -- Median incubation period is assumed to be 3 days, according to Guan's [Clinical characteristics of 2019 novel coronavirus infection in China](https://www.medrxiv.org/content/10.1101/2020.02.06.20020974v1) (2020) -- rho is the ratio of incubation period over generation period, which is assumed to be 0.4 in the baseline scenario -- R0 estimation formula refers to reference 3. Mathematical proof is shown in the model section below - -### Background -- R0 is known as basic reproduction number. It can be understood as the expected number of cases infected by one case in a population where all individuals are susceptible to infection. If R0 < 1, an epidemic will not start. If R0 > 1, an epidemic will be able to start spreading out. If R0 = 1, an epidemic will become an endemic. -- The basic reproduction number R0 has important implication. The greater the R0, the harder to control an epidemic. R0 for SARS and Ebola virus is 0.49 and 1.51, respectively (Gerardo 2004; Althaus, 2014). -- Tracing R0 for nCoV throughout time will provide a glance of the change in R0, giving clues of the effect of social and non-pharmaceutical prevention. - -### Fact -- On 2019-12-01, the first case of nCoV exhibits symptons, according to [Huang's Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China](https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(20)30183-5/fulltext#seccestitle10). This date is used as the first day of the epidemic. -- On 2020-1-23, Wuhan city lockdowns [(ifeng news)](http://news.ifeng.com/c/7tpL47zV2Vy). Before Wuhan lockdown, 5 million people leave the city [(Tencent News)](https://new.qq.com/sv1/qd/aoyou.html?cmsid=20200127A0EFXJ00) -- There is a time lag between onset of symptons and case confirmed, particularly due to staff and medical supply shortage - -### Assumption -- This model assume that infected individuals were not infectious during the incubation period (Zhou, 2020). - -### Model -A typical **SEIR** (susceptible, exposed, infectious, removed) model can be described as a system of differential equations - - - -where - -S(t) is the number of susceptible at time t - -E(t) is the number of exposed at time t - -I(t) is the number of infectious at time t - -R(t) is the number of removed, which includes the number of recovered and dead at time t - -N(t) is the population at time t - -N(t) = S(t) + E(t) + I(t) + R(t) - -At the beginning, the ratio of infectious to population is almost negligible. - -Therefore, - - - -R0 can be expressed as - - - -Given that - - - -where - -TL is the time of latent period - -TI is the time of infectious period - -TG is the time of generation period, which is also known as the serial interval - -R0 can be derived as - - - -This will be the formula estimating R0 in the code. Here is the **mathematical proof** for the equation above - - - -### Disclaimer -- Data uses API from [BlankerL](https://github.com/BlankerL/DXY-COVID-19-Crawler), which is an infection data realtime crawler. The data source is [Ding Xiang Yuan](https://3g.dxy.cn/newh5/view/pneumonia). - -### Reference -1. Li, Q., Guan, X., et al. (2020, January 29). [Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus–Infected Pneumonia](https://www.nejm.org/doi/full/10.1056/NEJMoa2001316#article_references). The New England Journal of Medicine. - -2. Wu, Joseph T., et al. (2020, January 28). [Nowcasting and Forecasting the potential domestic and International Spread of the 2019-nCoV Outbreak Originating in Wuhan, China: a Modeling Study](https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(20)30260-9/fulltext). Lancet. - -3. Zhou Tao, Liu, Y., et al. (2020, January 29). [Preliminary Prediction of the Basic Repreduction Number of the Novel Coronavirus 2019-nCoV](http://kns.cnki.net/kcms/detail/51.1656.r.20200204.1640.002.html) - -4. Althaus, Christian L. (2014, June). Estimating the Reproduction Number of Ebola Virus (EBOV) During the 2014 Outbreak in West Africa. PLoS Currents. PMC 4169395. PMID 25642364. - -5. Gerardo Chowell, Carlos, P., et al. (2004, July). Model Parameters and Outbreak Control for SARS.