Understanding exponential growth

You may have been hearing that the Covid-19 cases are growing exponentially. That means that the number of new cases is proportional to the number of existing cases. So for instance, if you get 100 new cases when you have 1,000 existing cases, then when there are 10,000 cases, there will be 1,000 new cases that day. The numbers quickly blow up and we are overwhelmed.

The point of social distancing is to reduce the transmission rate and break that exponential growth. We will need to continue the shutdown until we have slowed down the transmission to the point that we can track all new cases. But how can we know if it is working?  Let’s look at a graph showing the number of cases per day in New York and California.  Everything is growing so fast that it is hard to see anything else. We can see that New York grew faster than California and that is about it. One way to help is to use a logarithmic scale. Now we can see that New York grew very fast at the beginning of the outbreak and that early burst of cases meant that its total number would end up much higher. But it is still hard to see if we are actually changing the rate with our social distancing. A trick to see whether we have stopped exponential growth is to plot the daily number of cases against the total number of cases, with both scales being logarithmic.  If we are in exponential growth, we are almost plotting a number against itself and the slope of the curve will always be one. We need to use a running average (average of 7 days of data) because daily reporting is uneven.

Here you can see that during March, the slope of the curve in both New York and California is 1, because even though they were growing at different rates, they were still growing exponentially. We can also that the rate in both states is bending over as social distancing is stopping the exponential growth. California is turning over at a lower total number of cases because it started social distancing at a lower total number of cases – 1089 in California on the day social distancing began statewide compared to 10356 in New York. Here are the same curves for 12 states and Los Angeles County (at more than 10 million residents, Los Angeles County is bigger than half of the states). I grouped them by states where the drop from exponential is clear and the second graph are states where the data is just starting to show the effect.  The number for each state is the number of cases it had on the day they started social distancing.

All the data come from the New York Times database of cases: https://github.com/nytimes/covid-19-data