Fig 1.
Schematic description of the Hawkes process Eq (1).
The occurrence rate λ(t) is increased according to past events occurred at times t = tk (k = 1, 2, …) with the transmission delays t − tk distributed with ϕ(t − tk). R is the reproduction number that represents the average number of events induced by a single event.
Fig 2.
The distribution of transmission delays.
A bar histogram represents the distribution of transmission delays on a daily bases ϕd, which was converted from the log-normal distribution with the mean 4.7 days and SD 2.9 days (a magenta line).
Fig 3.
Synthetic daily cases generated by simulating the Hawkes-type count process and the estimated reproduction number.
(A) Rapid increase followed by a slow decrease. (B) Increase followed by a rapid decrease, and then an increase. (C) Slow increase followed by a decrease, and then another large increase. In the upper panel plotting the number of daily cases (purple line), the rate estimated by the state-space method is also plotted (blue line). In the lower panel, the reproduction number
estimated with the state-space method is plotted in reference to the true reproduction number Ri (purple line). The blue solid line and the shaded area represent the median and 95% range of the posterior distribution, respectively. The reproduction numbers estimated by Wallinga and Teunis (WT: orange line) and by Cori et al. (EpiEstim: green line) are also plotted for reference.
Table 1.
Convergence of the posterior estimate of .
Fig 4.
Variations by day of the week in the number of reported infections {βi} computed for several countries.
Fig 5.
Number of daily new cases and the reproduction number estimated using the state-space method.
Italy, Japan, Saudi Arabia, and the USA.
Fig 6.
Number of daily new cases and the reproduction number estimated using the state-space method.
France, Australia, Iran, and Brazil.
Fig 7.
The difference in the reproduction number at the initial phase.
(A) The reproduction numbers estimated with the proposed state-space method for 10 days until the day before the lockdown measures of each country. Days are counted from 12 days before the confinement measures. (B) Initial variation in the numbers of daily new cases; ni divided by n6. The period is shifted by 5 days, by taking account of the typical transmission delay.
Fig 8.
The minimum reproduction number achieved in each country.
(A) The reproduction number for 10 days whose average takes minimum in each country. (B) Variation in the numbers of daily new cases; ni divided by n6. The period is shifted by 5 days, by taking account of the typical transmission delay.
Fig 9.
Predicting the number of new cases in the future.
The forecasting method was applied to the data of Japanese daily cases, assuming that we were on June 30, 2020. We ran the Hawkes process 100 times to obtain the expected daily cases, by assuming that the reproduction number remains constant R = 1.4, which was obtained using the previous data (orange line). We also examined the cases in which the reproduction number is decreased to R = 0.7 due to confinement measures (green line) or increased to R = 1.8 by liberalization (red line).