The challenges of modeling and forecasting the spread of COVID-19

Abstract

The coronavirus disease 2019 (COVID-19) pandemic has placed epidemic modeling at the forefront of worldwide public policy making. Nonetheless, modeling and forecasting the spread of COVID-19 remains a challenge. Here, we detail three regional-scale models for forecasting and assessing the course of the pandemic. This work demonstrates the utility of parsimonious models for early-time data and provides an accessible framework for generating policy-relevant insights into its course. We show how these models can be connected to each other and to time series data for a particular region. Capable of measuring and forecasting the impacts of social distancing, these models highlight the dangers of relaxing nonpharmaceutical public health interventions in the absence of a vaccine or antiviral therapies.

Keywords: COVID-19; branching process; compartmental models; pandemic.

Fig. 1.

(Upper) Exponential model applied to new infection and death data for Italy, Germany, France, Spain, the United Kingdom, and the United States, normalized by the total country population (54). Insets show the same data on a logarithmic scale. Both the normalized infection $i$ and death $d$ data were thresholded to comparable initial conditions for each country; fits are to the first 15 to 20 d of the epidemic after exceeding the threshold. The fitted doubling time is shown for both infections ($T_{d,i}$) and death ($T_{d,d}$) data. Data from Japan and South Korea are shown for comparison. (Lower) Dynamic reproduction number (mean and 95% CI) of COVID-19 for China, Italy, and the United States estimated from reported deaths (17) using a nonparametric branching process (18). Current estimates are as of 1 April 2020 of the reproduction number in New York (NY), California (CA), and Indiana (IN; confirmed cases used instead of mortality for Indiana). Reproduction numbers of COVID-19 vary in different studies and regions of the world (in addition to over time) but have generally been found to be between 1.5 and 6 (19) prior to social distancing.

Fig. 2.

Solution of the dimensionless SIR model (5) with ${\mathcal{R}}_0=2$. Upper Left shows the graphs of $s$ (blue), $i$ (orange), and $r$ (gray) on the vertical axis vs. $\tau$ on the horizontal axis, for different $ϵ$. The corresponding values of $ϵ$ from left to right are $10^{-4}$, $10^{-6}$, $10^{-8}$, $10^{-10}$, respectively. Upper Center shows the time until peak infections vs. $\log \left(ϵ\right)$ for the values shown in Upper Left. This asymptotic tail to the left makes it challenging to fit data to SIR in the early stages. Upper Right is a phase diagram for fraction of infected vs. fraction of susceptible with the direction of increasing $\tau$ indicated by arrows, for three different values of ${\mathcal{R}}_0$. Lower displays a typical set of SIR solution curves over the course of an epidemic, with important quantities labeled.

Fig. 3.

Scenarios for the impact of short-term social distancing: fraction of population vs. date. (Left) California SIR model based on mortality data with parameters from Table 1 (${\mathcal{R}}_0=2.7$, $\gamma =.12$, $I_0=.1$) under two scenarios: ${\mathcal{R}}_0$ constant in time (light blue) and ${\mathcal{R}}_0$ cut in half from 27 March (1 wk from the start of the California shutdown) to 5 May but then returned to its original value, to represent a short-term distancing strategy (dark blue). (Right) New York SIR model with parameters from Table 1 (${\mathcal{R}}_0=4.1$, $\gamma =.1$, $I_0=05$) under the same two scenarios but with short-term distancing occurring over the dates of 30 March (1 wk from the start of the New York shutdown) to 5 May. In both states, the distancing measures suppress the curve and push the peak infected date into the future, but the total number of cases is only slightly reduced.