Replicating and projecting the path of COVID-19 with a model-implied reproduction number

Abstract

We demonstrate a methodology for replicating and projecting the path of COVID-19 using a simple epidemiology model. We fit the model to daily data on the number of infected cases in China, Italy, the United States, and Brazil. These four countries can be viewed as representing different stages, from later to earlier, of a COVID-19 epidemic cycle. We solve for a model-implied effective reproduction number $R_t$ each day so that the model closely replicates the daily number of currently infected cases in each country. For out-of-sample projections, we fit a behavioral function to the in-sample data that allows for the endogenous response of $R_t$ to movements in the lagged number of infected cases. We show that declines in measures of population mobility tend to precede declines in the model-implied reproduction numbers for each country. This pattern suggests that mandatory and voluntary stay-at-home behavior and social distancing during the early stages of the epidemic worked to reduce the effective reproduction number and mitigate the spread of COVID-19.

Keywords: COVID-19; Coronavirus; Epidemics; Reproduction number; SEIR Model.

Fig. 1

China reproduction number. Notes: The peak number of infections for China occurred on February 17 ($t=26$). After this date, the model-implied ${\mathcal{R}}_t$ tracks mostly below 1.0 aside from some brief daily fluctuations. The spike in the model-implied ${\mathcal{R}}_t$ around $t=140$ reflects an outbreak of new cases in the capital city of Beijing.

Fig. 2

Italy reproduction number. Notes: The peak number of infections for Italy occurred on April 19 ($t=54$). After this date, the model-implied ${\mathcal{R}}_t$ tracks below 1.0.

Fig. 3

United States reproduction number. Notes: The model-implied ${\mathcal{R}}_t$ for the United States dropped below 1.0 from May 30 ($t=95$) through June 3 ($t=99$), reflecting a short-lived decline in the number of infected cases. But from June 4 onward, the model-implied ${\mathcal{R}}_t$ for the United States has remained above 1.0, reflecting an upward trend in the number of infected cases.

Fig. 4

Brazil reproduction number. Notes: The model-implied ${\mathcal{R}}_t$ for Brazil exhibits some sharp downward and upward jumps during the middle part of April ($t=40$ to $t=50$), which may reflect reporting errors in the number of infected cases. The model-implied ${\mathcal{R}}_t$ averaged over the most-recent 7 days remains above 1.0 at the end of our data sample, reflecting an upward trend in the number of infected cases.

Fig. 5

Out-of-sample projections: China and Italy. Notes: The top panels show the out-of-sample projections for China (specifically Hubei Province). The peak number of infections occurred on February 17 ($t=26$). At the end of our data sample, the epidemic cycle is nearly complete with only a small number of infected cases. The bottom panels show the out-of-sample projections for Italy. The peak number of infections occurred on April 19 ($t=54$). The projected number of closed cases for Italy at the end of the epidemic is around 260,000.

Fig. 6

Out-of-sample projections: United States and Brazil. Notes: The top panels show the out-of-sample projections for the United States. The peak number of infections is projected to occur on or about August 8 ($t=165$). The projected number of closed cases at the end of the epidemic is around 8.89 million. The bottom panels show the out-of-sample projections for Brazil. The peak number of infections is projected to occur on or about August 10 ($t=162$). The projected number of closed cases at the end of the epidemic is around 4.45 million.

Fig. 7

Quasi real-time projections. Notes: The figure plots sequences of “quasi real-time” projections for the number of infected cases and the number of closed cases in China and the United States. Each projection uses a different end-of-sample starting point indicated by the month-day label. For each end-of-sample starting point, we recalibrate the values of ${\theta }_T,{\mathcal{R}}_0$, and η according to the procedures described in Section 4. The out-of-sample projections can sometimes shift by large amounts from one week to the next, depending on recent incoming data. Dashed lines mark the highest and lowest out-of-sample projections for the number of closed cases at the end of the epidemic.

Fig. 8

Mobility indices and model-implied reproduction numbers. Notes: Declines in measures of population mobility tend to precede declines in the model-implied ${\mathcal{R}}_t$ for each country. This pattern suggests that mandatory and voluntary stay-at-home behavior and social distancing during the early stages of the epidemic worked to reduce the effective reproduction number and mitigate the spread of COVID-19. For plotting purposes, the Google mobility indices are re-normalized to have baseline value of 100 instead of zero.

Figure A.1

Calibrated value of parameter ${\theta }_T$. Notes: Given the common value of γ = 1/20 for all countries, we solve for the value of θ_T so that the model predicted value of R_T exactly matches the end-of-sample smoothed number of closed cases for each country. The figure plots the quasi-real time evolution of ${\theta }_T$ for each country. For the out-of-sample projections (t > T), we assume that ${\theta }_t$ converges towards 1.0, as governed by equation (5) with κ = 0.07, which is estimated from the quasi-real time evolution of ${\theta }_T$ for China. The dashed lines show the out-of-sample paths of ${\theta }_t$ for each country.