A Time-dependent SIR model for COVID-19 with Undetectable Infected Persons

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 $3\%$. 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 $R_0$.