The key insight that enables the http://app.jackprior.org forecasting tool is the observation that each state or country has a characteristic rate at which it is flattening the curve.
In the early stage of the pandemic, the virus would spread unconstrained by any distancing so quickly that the number of people infected in a country would increase by a large constant percentage each day, sometimes by as much as 50%.
So if 100 people were infected on day one, you would have 150 on day 2, and then 275 on day three. This is referred to as exponential growth.
![\[ \frac{dN}{dt} = \mu N \]](https://covid19.jackprior.org/wp-content/ql-cache/quicklatex.com-2415c1d67800149789edb68161df70d8_l3.png "Rendered by QuickLaTeX.com")
Once social distancing is implemented in a region, the growth rate u begins to decline at an exponential rate. So the exponential constant is declining at an exponential rate.
![\[ \frac{d\mu}{dt}= -k \]](https://covid19.jackprior.org/wp-content/ql-cache/quicklatex.com-9e4aca1ddf5391ae102bc360408e4d05_l3.png "Rendered by QuickLaTeX.com")
![\[ \mu(t) = \mu o \exp(-k t) \]](https://covid19.jackprior.org/wp-content/ql-cache/quicklatex.com-aadd7465ee26edd26bc4903e251d811e_l3.png "Rendered by QuickLaTeX.com")
We can find k by plotting the percentage change each day on a log scale and fitting a line to that data.
In the figure above you can see that the US is dropping its daily case growth rate by 6.1% per day.
Because each region hasn’t implemented its distancing at the same time or has had a step-change in its distancing at a certain point in time, it is necessary to choose this manually. There is a trade-off to be made between getting a good estimate of k by fitting a lot of data and getting the best estimate of the current value by choosing just the most recent points. The problem is that if you choose too few points, outliers and weekend/weekday reporting effects can swamp the result.
You can make that choice using the “Social Distance Model Window” slider and see how it impacts the fit of the growth slowing curve and the forecasts of cases, hospitalizations, and deaths.
When daily counts can be erratic (worse than Poisson), there is a strong weekly rhythm, and underlying processes can’t change on a dime, it often helps analysis to provide a 7 day moving average or total.
7-day moving average added to daily plots. Thanks for the suggestion, Bob!