In this paper, we conduct mathematical and numerical analyses  for COVID-19 . To predict the trend of COVID-19 , we propose a time-dependent SIR model that tracks  the transmission rate and the recovering rate at time t.  Using the data provided by China authority , we show our one-day prediction errors are almost less than 3 %DIFDELCMD < \\%%% %DIF < . The turning point and the total number of confirmed cases in China are predicted under our model. To analyze the impact of the asymptomatic infections on the spread of disease, we extend our SIR model by considering two types of infected persons: detectable infected persons and undetectable infected persons. Whether there is an outbreak is characterized by the spectral radius of a 2 \times 2 matrix that is closely related to the basic reproduction number R_0. We plot the phase transition diagram of an outbreak and show that there are several countries on the verge of COVID-19 outbreaks on Mar. 2, 2020. To illustrate the effectiveness of social distancing, we analyze the independent cascade model for disease propagation in a random network specified by a degree distribution. We show two approaches of social distancing that can lead to a reduction of R_0.\end{abstract} 
In this paper, we conduct mathematical and numerical analyses to address the following crucial questions for COVID-19 : (Q1) Is it possible to contain COVID-19 ? (Q2) If COVID-19 can be contained, when will be the peak and the end of the epidemic? (Q3) How do the asymptomatic infections affect the spread of disease? (Q4) If COVID-19 cannot be contained, what is the ratio of the population that needs to be infected to achieve herd immunity? (Q5) How effective are the social distancing approaches? For (Q1) and (Q2), we propose a time-dependent susceptible-infected-recovered (SIR) model that tracks 2 time series: (i) the transmission rate at time t and (ii) the recovering rate at time t. Such an approach is more adaptive than traditional static SIR models and more robust than direct estimation methods. Using the data provided by the Chinese government , we show that the one-day prediction errors for the numbers of confirmed cases are almost in 3 %DIFDELCMD < \\%%% %DIF < . The turning point and the total number of confirmed cases in China are predicted under our model. To analyze the impact of the asymptomatic infections on the spread of disease, we extend our SIR model by considering two types of infected persons: detectable infected persons and undetectable infected persons. Whether there is an outbreak is characterized by the spectral radius of a 2 \times 2 matrix that is closely related to the basic reproduction number R_0. We plot the phase transition diagram of an outbreak and show that there are several countries on the verge of COVID-19 outbreaks on Mar. 2, 2020. To illustrate the effectiveness of social distancing, we analyze the independent cascade model for disease propagation in a random network specified by a degree distribution. We show two approaches of social distancing that can lead to a reduction of R_0.\end{abstract} \%, and the total number of confirmed cases is precisely predicted. Also, the turning point, defined as the day that the transmission rate is less than the recovering rate can be accurately predicted. After that day, the basic reproduction number R_0 is less than 1. For (Q3), we extend our SIR model by considering 2 types of infected persons: detectable and undetectable infected persons. Whether there is an outbreak in such a model is characterized by the spectral radius of a 2 by 2 matrix that is closely related to R_0. For (Q4), we show that herd immunity can be achieved after at least 1-1/R_0 fraction of individuals being infected. For (Q5), we analyze the independent cascade (IC) model for disease propagation in a configuration random graph. By relating the propagation probabilities in the IC model to the transmission rates and recovering rates in the SIR model, we show 2 approaches of social distancing that can lead to a reduction of R_0.
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