Exponential Models
Exponential Models
Fitting the Data
We often use models to explain and predict. In the case of a pandemic, prediction takes on special importance as it guides policy. The data showing the number of cases of COVID-19 in the United States over the past month reveal a curve that increases at an increases rate.
A straightforward, and a bit naive, method for predicting is to extrapolate from past data, to fit the curve. If we fit the curve for the number of cases of COVID-19, we see that it closely approximates an exponential curve, where the number of cases, N, equals a constant, C, times a growth rate, G, raised to a power equal to the number of days.
An exponential curve increases by a constant percentage. In this case, the number increases by 1.32. On March 6, there were 331 cases. If we multiply that number by 1.32, we arrive at the number 437. The actual number of cases on March 7 was 444.
An exponential function has a doubling period, the number of days for each doubling of the number of cases. For G=1.32, the doubling period equals 2.5. According to the model, every 2.5 days the number of cases doubles. Continuing with this logic, every five days, the number increases four-fold, and every ten days it increases sixteen-fold.
We can check this against the data. On March 10, there were 1000 cases. On March 26, 10 days later, the model predicts 16,000 cases. The actual number was 18,723, a little higher than predicted.
Weaknesses
This approach suffers from a fundamental weaknesses: it lacks logical foundations. Even though the number of cases approximates an exponential early, we have no logic that explains why. When we construct a model that generates a pattern, the SIR Model, the model will produce an S-shaped pattern.
Expoential Growth, Delay, and Measuring Fatality Rates
Given the delay between catching the virus and a fatality and the early exponential growth in the number of cases, running average fatality rates which divide the number of fatalities by the number of cases, can drastically understate the true fatality rate as see in this video:
Current Cases by Country (as of April 1st, 2020):
Most countries are experiencing near-logarithmic rates of growth, save for South Korea and China.
Current Cases by State (as of March 21st, 2020):
Contributors: Michael Lin, Jonathan Hochberg