1SummaryBackgroundThe novel coronavirus (SARS-CoV-2) is currently causing concern in the medical, epidemiological and mathematical communities as the virus is rapidly spreading around the world. Internationally, there are more than 1 200 000 cases detected and confirmed in the world on April 6. The asymptomatic and mild symptomatic cases are just going to be really crucial for us to understand what is driving this epidemic to transmit rapidly. Combining a mathematical model of severe (SARS-CoV-transmission with data from China, South Korea, Italy, France, Germany and United Kingdom, we provide the epidemic predictions of the number of reported and unreported cases for the SARS-CoV-2 epidemics and evaluate the effectiveness of control measures for each country.MethodsWe combined a mathematical model with data on cumulative confirmed cases from China, South Korea, Italy, France, Germany and United Kingdom to provide the epidemic predictions and evaluate the effectiveness of control measures. We divide infectious individuals into asymptomatic and symptomatic infectious individuals. The symptomatic infectious phase is also divided into reported (severe symptoms) and unreported (mild symptoms) cases. In fact, there exists a period for the cumulative number of reported cases to grow (approximately) exponentially in the early phase of virus transmission which is around the implementation of the national prevention and control measures. We firstly combine the date of the implementation of the measures with the daily and cumulative data of the reported confirmed cases to find the most consistent period for the cumulative number of reported cases to grow − approximately exponentially with the formula χ1 exp(χ2t) χ3, thus we can determine the parameters χ1, χ2, χ3 in this formula and then determine the parameters and initial conditions for our model by using this formula and the plausible biological parameters for SARS-CoV-2 based on current evidence.We then provide the epidemic predictions, evaluate the effectiveness of control measures by simulations of our model.FindingsBased on the simulations using multiple groups of parameters (d1, d2, N), here [d1, d2] is the consistent period for the cumulative number of reported cases to grow approximately exponentially with the formula χ1 exp(χ2t) χ3 and N is the date at which public intervention measures became effective, we found that the ranges of the turning point, the final size of reported and unreported cases are respectively Feb.6 − 7, 67 000 − 69 000 and 45 000 − 46 000 for China, Feb.29−Mar.1, 9 000 − 9 400and 2 250 − 2 350 for South Korea, Mar.24 − 26, 156 000 − 177 000, and 234 000 − 265 000 for Italy, Mar.30−Apr.9, 104 000 − 212 000, and 177 000 − 318 000 for France, Mar.30−Apr.20, 141 000 − 912 000, and 197 000 − 1 369 000 for Germany, Apr.1−May12, 140 000 − 473 000, and 210 000 − 709 000 for UnitedKingdom. Our prediction relies on the cumulative data of the reported confirmed cases. As more data become available, the ranges become smaller and smaller, that means the prediction becomes better and better. It is evident that our estimates and simulations have shown good correspondence with the distribution of the cumulative data available of the reported confirmed cases for each country and in particularly, the curves plotted by using different parameter groups (d1, d2, N) for reported and unreported cases tend to be consistent in China and South Korea (see (e) in Figures 2-3). For Italy, France, Germany and United Kingdom, the prediction can be updated to higher accuracy with on-going day by day reported case data (see Figures 4-7).InterpretationWe used the plausible biological parameters f, ν, η for SARS-CoV-2 based on current evidence which might be refined as more comprehensive data become available. Our prediction also relies on the cumulative data of the reported confirmed cases. Using multiple groups of parameters (d1, d2, N), we have attempted to make the best possible prediction using the available data. We found that with more cumulative data available, the curves plotted by using different parameter groups (d1, d2, N) for reported and unreported cases will be closer and closer, and finally tend to be consistent. This shows that when we have no enough cumulative data available, we need to use all possible parameter groups to predict the range of turning point, the final size of reported and unreported cases. When we have enough cumulative data, for example, when we get the data after the turning point, we only need to use any one of these parameter groups to get a prediction with high accuracy.FundingNSFC (Grant No. 11871007), NSFC and CNRS (Grant No. 11811530272) and the Fundamental Research Funds for the Central Universities.
Estimation of the final size of the second phase of the coronavirus COVID 19 epidemic by the logistic model
