Maximum entropy method for estimating the reproduction number: An investigation for COVID-19 in China and the United States

The key parameter that characterizes the transmissibility of a disease is the reproduction number $R$. If it exceeds 1, the number of incident cases will inevitably grow over time, and a large epidemic is possible. To prevent the expansion of an epidemic, $R$ must be reduced to a level below 1. To estimate the reproduction number, the probability distribution function of the generation interval of an infectious disease is required to be available; however, this distribution is often unknown. In this paper, given the incomplete information for the generation interval, we propose a maximum entropy method to estimate the reproduction number. Based on this method, given the mean value and variance of the generation interval, we first determine its probability distribution function and in turn estimate the real-time values of the reproduction number of COVID-19 in China and the United States. By applying these estimated reproduction numbers into the susceptible-infectious-removed epidemic model, we simulate the evolutionary tracks of the epidemics in China and the United States, both of which are in accordance with that of the real incident cases.

Figure 1

The function shape of Eq. (28) is shown by a blue curve. The sample data of the generation interval of the SARS in Singapore (2003) is shown by a red histogram, and the data resource refers to [9].

Figure 2

The number of real-time (confirmed) infected cases in China for each day. Data resource: see [29].

Figure 3

The evolutionary tendency of reproduction numbers in China from January 9, 2020 to June 10, 2020.

Figure 4

The SIR simulation result by using the reproduction numbers in Fig. 3 is shown by red diamonds, where $N\approx 1.4\times {10}^9$ and $\tau =7.95$. The real-time infected cases in China are shown by black circles.

Figure 5

The evolutionary tendency of reproduction numbers in the United States from February 23, 2020 to June 13, 2020. Data resource: see [30].

Figure 6

The SIR simulation result by using the reproduction numbers in Fig. 5 is shown by red diamonds, where $N\approx 3.3\times {10}^8$ and $\tau =10.50$. The real-time infected cases in the United States are shown by black circles. Data resource: see [30].