key: cord-0686766-z9jk1n7o authors: Liu, Xiuli; Hewings, Geoffrey J.D.; Wang, Shouyang; Qin, Minghui; Xiang, Xin; Zheng, Shan; Li, Xuefeng title: Modeling the situation of COVID-19 and effects of different containment strategies in China with dynamic differential equations and parameters estimation date: 2020-03-13 journal: nan DOI: 10.1101/2020.03.09.20033498 sha: ebfdcdc6495fec3c91cab1eb88b2ad557391722d doc_id: 686766 cord_uid: z9jk1n7o This paper proposed a quarantine-susceptible-exposed-infectious-resistant (QSEIR) model which considers the unprecedented strict quarantine measures in almost the whole of China to resist the epidemic. We estimated model parameters from published information with the statistical method and stochastic simulation, we found the parameters that achieved the best simulation test result. The next stage involved quantitative predictions of future epidemic developments based on different containment strategies with the QSEIR model, focused on the sensitivity of the outcomes to different parameter choices in mainland China. The main results are as follows. If the strict quarantine measures are being retained, the peak value of confirmed cases would be in the range of [52438, 64090] and the peak date would be expected in the range February 7 to February 19, 2020. During March18-30, 2020, the epidemic would be controlled. The end date would be in the period from August 20 to September 1, 2020. With 80% probability, our prediction on the peak date was 4 days ahead of the real date, the prediction error of the peak value is 0.43%, both estimates are much closer to the observed values compared with published studies. The sensitive analysis indicated that the quarantine measures (or with vaccination) are the most effective containment strategy to control the epidemic, followed by measures to increase the cured rate (like finding special medicine). The long-term simulation result and sensitive analysis in mainland China showed that the QSEIR model is stable and can be empirically validated. It is suggested that the QSEIR model can be applied to predict the development trend of the epidemic in other regions or countries in the world. In mainland China, the quarantine measures can't be relaxed before the end of March 2020. China can fully resume production with appropriate anti-epidemic measures beginning in early April 2020. The results of this study also implied that other countries now facing the epidemic outbreaks should act more decisively and take in time quarantine measures though it may have negative short-term public and economic consequences. In late December, 2019, an atypical pneumonia case, caused by a virus called COVID-19, was first reported and confirmed in Wuhan, China. Although the initial cases were considered to be associated with the Huanan Seafood Market, the source of the COVID-19 is still unknown. The confirmed cases increased with exponential speed, from 41 on January 10, 2020 to 5,974 on January 28, 2020 in mainland China, far exceeding those of the SARS epidemic in 2003 (see figure 3 (346), Japan (105) and Singapore (86) ranked as the top 3 (figure 2), while 35 cases were reported in United States of America 1 . The transmissibility of COVID-19or at least its geographical distribution (figure 2)seems to be higher and broader than initially expected (Horton, 2020). Compared to SARS-CoV (9.56% mortality) and MERS-CoV (34.4% mortality), the COVID-19 appears to be less virulent at this point except for the elderly and those with underlying health conditions (table 1). COVID-19 was confirmed as subject to human-to-human transmission and it is very contagious. The basic reproduction number R0 for COVID-19 was estimated by WHO and some research institutes in the range of 1.4-6.6 (table 2). This value is slightly higher than that of the 2003 SARS epidemic, and much higher than that of influenza and Ebola. The incubation days of COVID-19 in Wuhan city is 5-10 days with a mean of 7 days (Fan et al., 2020). On average, the duration from confirmed stage to cure or death is 10 days in nation-wide reporting according to Guan et al. (2020) . A long incubation period and an associated large number of patients with mild symptoms increase the difficulty of prevention and control of the epidemic. The likelihood of travel-related risks of the disease spreading has been noted by Bogoch et al. (2020) and Cao et al. (2020a) wherein they indicated the potentials for further regional and global spread (Leung et al., 2020). As the epidemic broke out on the eve of the Spring Festival, large-scale population movements and gatherings of people aggravated the epidemic. After the outbreak, local governments have adopted a series of unprecedented mitigation policies in place to contain the spread of the epidemic. The major local public emergency started with a category Class I response to health incidents, with positively diagnosed cases either quarantine or put under a form of self-quarantine at home (Gan 1 https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports/ 4 et al., 2020). Suspicious cases were confined in monitored house arrest. Most exits and entries into cities were shut down. Certain categories of contact were banned; for instance, universities and schools remained closed, and many businesses remained closed. People were asked to remain in their homes for as much time as possible (Fahrion et al., 2020) . These interventions have reduced the population's contacts to a certain extent, helped to cut off pathways for the spread of the virus and reduce the rate of disease transmission. However, the long-term management and control has brought considerable inconvenience to the daily lives of people. The failure of factories to start on time and run normally after the Spring Festival also had severe effects on Chinese national and global economies. Ayittey et al. (2020) and CNN Business (2020) estimated it would result in China's GDP declining 4.5% year-on-year in Q1 in 2020; the loss in China would be up to $62 billion in the same quarter. Zhang (2020), Huang (2020), Li and Zhang (2020) and IMF News (2020) considered the growth of China's GDP would be 5.0%-5.6% in 2020, decrease 0.5-1.1 percentage points from 2019. IHS Markit (2020) estimated a reduction of global real GDP of 0.8% in Q1 and 0.5% in Q2 in 2020, and the global real GDP would be reduced by 0.4% in 2020. The longer the duration of the epidemic, the more negative the impacts on China and the rest of the world, with the latter effects largely centered on disruptions in increasingly complicated supply chains. Therefore, it is important to estimate the dynamic evolution mechanism of the epidemic in mainland China, to find when the epidemic will end and how this result depends on different containment strategies. These are issues of great significance with important clinical and policy implications (Joseph et al., 2020). The traditional infectious disease dynamics susceptible-exposed-infectious-resistant (SEIR) model has been very popular in analyzing and predicting the development of an epidemic (see 5 Lipsitch et al., 2003; Pastor-Satorras, 2015) . SEIR models the flows of people between four states: susceptible (S), exposed (E), infected (I), and resistant (R). Each of those variables represents the number of people in those groups. Assume that the average number of exposed cases that are generated by one infected person of COVID-19 is β. The parameter β is similar to the basic reproduction number which can be thought of as the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection. Considering the protective measures were taken, β should be smaller than the basic reproduction number in table 2. An individual in the exposed state (type E) will have the probability δ changes to individuals in the infected state (type I), and an individual in the infected state (type I) will change to the cure state (type R) with a probability of γ or to death state (type F) with a probability of η per unit time. In contrast to the traditional SEIR model, we propose a quarantine-susceptible-exposed-infectiousresistant (QSEIR) model that considers the unprecedented strict quarantine measures in mainland China to resist the epidemic. The parameter, α(t), was designed to represent the ratio of people who was not restricted to a specific area and had chances to contact with COVID-19 virus during special period. The α(t) and β(t) vary according to the strength of the prevention and control measures for the epidemic. To make the model accord with reality, contrast with the standard SEIR model, we added two parameters Δ(t) and θ(t). The Δ(t) is the ratio of people with vaccination at time t. θ(t) is the natural mortality of the population in a region at time t (figure 3). The value of δ(t) is closely related with the virus incubation and infectious periods and γ(t) is dependent on the treatment level and patients' health status. It is assumed that the virus incubation period is 7 days and the duration from confirmed stage to cure or death is 10 days based on nation-wide information dR(t)/dt=γ(t)*I(t-10) (4) dF(t)/dt=η(t)*I(t-10) (5) Equation (6) Razum and Becher, 2003) . After the isolation of Wuhan on January 23, 2020 with the stricter requirements of data statistics and the provision of detection levels, the data are more and more reliable. We estimated model parameters reversely with QSEIR model by equations (8)-(12). β(t), γ(t), η(t) and δ(t) can be calculated (see table 4 ). From equations (1)-(6), we obtain: Note that we found some δ(t) in table 4 was>1, which is obviously incorrect, the reason was mainly because biases in the data during the early stages (Cao et al., 2020b) . We deleted these data and calculated the average, median and variance of the rest value of the four parameters in first step. In step 2, we deleted values>1.5 times of the column average. In step 3, we calculated the average, median and variance of the rest value of the four parameters (see Then, we set the values of these parameters in their ranges randomly, and input them to QSEIR model, we got E(i), I(i), R(i), F(i) at each day i, we used the real data I0, E0, R0 and F0 from All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 13, 2020. Set errck= (mean (abs(E0(i)-E(i))) +mean (abs(I0(i)-I(i))) + mean (abs (R0(i)-R(i))) + mean (abs (F0(i)-F(i))))/4 (13) In equation (13) We set January 23, 2020 as the beginning date of the simulation; the initial values of variables were set as of this date (table 6 ). If we set the simulation period D as 300 days, input the best parameters we found, with the MATLAB program of QSEIR model, we can present the results shown in figure 5 . The results showed that with 80% probability, the peak value of I was 58,264 on February 13, 2020. After June 19, 2020, the value of I would be < 50 and from July 29, 2020, the number would be smaller than 5. By August 26, 2020, I would be smaller than 1, implying that the COVID-19 would essentially end. From March 17, 2020, E would be < 5 and, a week later on March 24, the number of E would be < 1, which means the epidemic would be totally controlled since this day, no new infected people would appear. The cumulative confirmed cases of COVID-19 in mainland China was estimated to be 97,653, and the cumulative number of deaths was All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 13, 2020. Furthermore, the existing studies seldom provided estimates of the duration of the epidemic and effects of different containment strategies in mainland China. At the regional level, Wu et al. (2020b) concluded that in Guangdong province, the epidemic would be totally controlled by mid to late March, 2020. The cumulative confirmed cases in Guangdong was ranked second among All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. How would E, I, R, F change if the value of parameters (α(t), β(t), η(t), δ(t), γ(t)) varied or if the beginning date January 23, 2020 of the simulation changed? We conducted sensitivity analyses of them in terms of their impacts on the I index one by one. Figures 6-7 and tables 7-8 showed that the larger the value of β(t) or δ(t), the higher the peak value of the I index and the earlier the peak time. With the increase of β(t) or δ(t), their sensitive coefficient to I index decreased progressively. The sensitivity coefficient of α(t) to I index was the biggest. When α(t) increased 0.001%, 8,596 more confirmed cases will be observed (figure 10 and table 11). These results indicated that quarantine measures (or with vaccination that is not yet available) are the most effective containment strategy to control the epidemic. Figures 8-9 and tables 9-10 showed that the greater the value of γ(t) or η(t), the smaller the peak value of the I index. The peak date of I was not very sensitive to the change of γ(t). When γ(t) increased 1%, confirmed cases will be decrease between 4,395 and 7,432. When η(t) decreased 1%, 4,138 to 4,640 additional confirmed cases could be expected. The average absolute sensitive coefficient of All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 13, 2020. . https://doi.org/10.1101/2020.03.09.20033498 doi: medRxiv preprint γ(t) and η(t) to I ranked the second and third in those of five parameters (tables 7-11). This showed that to improve the rate of cure, the development of special medicine should be the second most effective measure. If the beginning date of the simulation changed from January 23 to January 30 or February 6 in2020 with the value of variables in table 12, together with the same estimated value of parameters in table 5 and, QSEIR program, we can show the main results that started from January 30 in figure 11 . Compared with the baseline, the peak value of the I index increased 0.9% or 1.5%. The peak date of I or the ended date of COVID-19 would be 3 days or 1 day ahead ( figure 12 and table 12 ). Results mean that the simulating results were not sensitive to the initial start date. The QSEIR model system is stable. Due to the downward pressure on the economy, some enterprises resumed work one after another in compliance with the requirements of epidemic prevention and control. Because newly confirm ed cases are decreasing day by day since February 17,2020, the outbreak was gradually brought u nder control, some people began to relax their vigilance. Some began to travel; some went out wi All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 13, 2020. . The paper proposed a QSEIR model that considers the unprecedented strict quarantine measures which are more fit for the epidemic situation in mainland China. Parameter estimation is the most critical part when using this kind of SEIR model to predict the trend of epidemic (Cao et al., 2020b) . We estimated the model parameters reversely for the QSEIR model from published information with statistical methods and stochastic simulations; from these experiments, we found the parameters that achieved the best simulation test results. The application verified that the method is effective. The paper not only predicted the peak number and peak date of confirmed cases, but also provided estimates of the sensitivity of parameters of QSEIR, the duration of the epidemic and effects of different containment strategies at the same time. The long-term simulation result and sensitive analysis in mainland China showed that the QSEIR model is stable and can be empirically validated. It is suggested that the QSEIR model can be applied to predict the development trend of the epidemic in other regions or countries in the world. In QSEIR model, the parameters are dynamically changing for each day. Parameters estimation is the most important part in the kind of SEIR model (Cao et al., 2020b) . The paper illustrated the method to generate the parameter estimations. Given data limitation, we estimated a constant value to each of them with 20% errors in simulation tests, which was the best result in 50000 times stochastic simulation within their statistical ranges. We applied these values in prediction and obtained better results than existed researches. With the improvement of data quality and more data, variable parameters can be estimated and the forecasting accuracy of the model could be enhanced. The vaccine research and development cycle are relatively long, from researching products to large-scale production and promotion, it takes about 6-18 months. It seems that the COVID-19 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 13, 2020. Liu X L designed the QSEIR model, gave method to estimate parameters, compiled MATLAB program, got results and wrote the draft of the manuscript. Hewings G suggested to make sensitive analysis of parameters and estimate effects of different containment strategies. He edited the manuscript. Wang S Y explained some results and provided policy implications. Qin M H, Xiang X, Zheng S and Li X F collected data, some references and analyzed some data, the four of them made equal contributions to the paper. We declare no competing interests. Acknowledgements 3 https://www.cnbeta.com/articles/science/947877.htm All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 13, 2020. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 7 . The peak value and peak date of I index when β(t) was changed and the sensitive coefficient of β(t) to I All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 13, 2020. . https://doi.org/10.1101/2020.03.09.20033498 doi: medRxiv preprint All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 13, 2020. . https://doi.org/10.1101/2020.03.09.20033498 doi: medRxiv preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 13, 2020. . https://doi.org/10.1101/2020.03.09.20033498 doi: medRxiv preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 13, 2020. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 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