This video is pretty good, and I thought that I’d add some equations to the explanation:

So, in terms of the mathematics, what is going on?

The graph they came up with is “new confirmed cases” on the y-axis (log scale) and total number of cases on the x-axis. Let’s see what this looks like for exponential growth.

Here, letting the total number of cases at time be denoted by , the number of new cases is , the first derivative.

In the case of exponential growth, where is positive.

which is what is being plotted on the y-axis. So with the change of variable we are letting and our new function is , which, of course, is a straight line through the origin. That is, of course, IF the growth is exponential.

To get a feel for what this looks like, suppose we had polynomial growth; say . Then In the case of linear growth we’d have (constant) and for, say, , or a “concave down” function.

Now for the logistic situation in which the number of cases grows exponentially at first and then starts to level out to some steady state value, call it , the relationship between the number of cases and the new number of cases looks like so our which is a quadratic which opens down.

Yes, this gets studied in differential equations class when we study autonomous differential equations.

Now for some graphs:

Here, I tweaked the logistic model to have the same derivative as the exponential model near .

Here: we have linear growth vs the

Here: cubic growth vs.