Covid-19 Delta Variant % Share of Cases Grew Logistically

Today, as part of my work, I stumbled upon the surprising fact that in many countries the % share of Covid-19 cases that were Delta variant grew almost perfectly logistically from 0% to 100% during 2021.

The graph below, which started all of this, should come up if one follows this Our World In Data link.

The curves clearly look “logistic-like”, meaning sigmoid shaped (s-shaped) and symmetric. However, not everything that looks sigmoid shaped will be modeled well by a logistic function.

The general logistic function is of the form $$f(t)=\frac{L}{1+e^{-k(t-a)}}$$ and its graph looks as follows (figure created by me with Tikz, I’ll show the code in another post!)

There are fancy ways to check if a set of data is logistic, for example, related to the differential equation of logistic growth (see discussion below). However, I have lately grown fond of checking if something is logistic by either just:

• plotting the data points and with sliders for the $L$, $a$ and $k$ trying to fit a logistic curve through the data.
• setting up a spreadsheet in which I plot the data together with the values given by a model with special cells that I reference of the parameters. Then I can tweak the parameters and the plot gets updated, and by trial and error I see if there is a good graphical fit.

This curve-fitting approach is what I set out to do, and I was completely baffled by what I found!

Logistic fit for delta variant % share in the US

A really good model for the delta variant share in the US, where $t=0$ is February 8, 2021 is given by

• $L=1$ (the Delta variant reached $100\%$ share in 2021 — before Omicron hit).
• $k=0.09$ ($k$ affects how fast the logistic rises, and its value $0.09=9\%$ is related to the initial exponential growth rate in the % share of cases).
• $a=139$ (this is just a horizontal shift in the curve and has no mathematically important meaning — it is just the moment of the year in which the growth rate of the delta share was highest, in $\%$ share per day).

The fit is shown in the following figure:

Concisely: $$ \begin{array}{c} \text{Share of delta variant}\cr \text{cases in the US} \cr \text{$t$ days after Feb 8, 2021} \end{array} = \frac{1}{1+e^{-0.09(t-139)}} $$

You can play around with the logistic fit yourself if the desmos app loads below (use the sliders for $a$ and $k$). If it does not load, just follow this link, or download the data and play for yourself in the spreadsheet I created

Logistic fit for delta variant % share in the UK

The very good fit is not just for data for the US. I tried the same with the data for the UK and got a similarly good fit.

In this case, $k=0.11$, $a=104$, $L=1$. It is interesting that that $k$ is different, because it says that the Delta variant overtook other variants slightly faster in the UK than the US.

Here is the plot:

And here is the desmos plot with sliders that I created ( link, in case it does not load). Try changing $k$ to 0.09 to see de difference in the way Delta overtook other variants between the US and UK.

Not all Sigmoid Shaped data can be fit well by a logistic

The best way to see this is with an example!

Try with the sliders in the desmos applet below to fit a logistic model to the total (cumulative) number of cases in the first Covid-19 wave in Italy in 2020 (here is a link in case it does not load). By trying to fit the curve to the data with the sliders you will soon understand why it is not going to work.

(data from John’s Hopkins University).

Here is a screnshot in case the applet does not load:

Other (fancier) ways to tell if something is logistic

Logistic growth is governed by the differential equation $$ \frac{dy}{dt}=ky\left(1-\frac{y}{L}\right)\tag{1} $$ which can also be written as: $$ \frac{y’}{y}=k(1-y/L).\tag{2} $$ This last equation says that the relative growth $y’/y$ (which can be interpreted as percent change in one unit of time) is decreasing linearly as a function of $y$.

Note:

• $y’/y\approx k$ when $y\approx 0$ (corresponding to $y$ growing exponentially with continuous growth rate $k$ at the beginning of the growth)
• $y’/y\approx 0$ when $y\approx L$ (corresponding to $y$ being approximately constant constant) when $y$ approaches $L$.

The two ways $(1)$ and $(2)$ to write the differential equation indicate two ways to see if some data is growing logistically (one can estimate $y’$ numerically from the data by computing the rate of change between the entries in the table).

Is the plot of $y’/y$ against $y$ a line?
Equation $(2)$ says that for logistic growth the plot of $y’/y$ against $y$ should be a line. If one makes this plot for exactly logistic data, this is what one would see:

Is the plot of $y’$ against $y$ a parabola? Equation $(1)$ says that for logistic growth the plot of $y’$ against $y$ should be a parabola going through the origin and the point $(y,y’)=(L,0)$. If one makes this plot for exactly logistic data, this is what one would see:

But once again I say, I have lately grown fond of checking if something is logistic by just plotting the data points and with sliders for the $L$, $a$ and $k$ trying to fit a logistic curve through the data, as I illustrated above.