One way to study the spread of covid19 and whether isolation and other measures are succeeding in limiting its spread, is to look at the increase or decrease in the number of daily deaths from the virus.
The daily deaths from the virus are – in the view of many – a more reliable metric than detected cases, since the latter depends simply on who has and has not been tested, and that is limited and variable. However deaths are a metric that lags 3-4 weeks behind the spread of infection.
Mathematically there are 3 ways you can look at the daily death rate from covid19 in different countries. You can look at the number, the rate of change or slope, or you can look at the acceleration – does the increase or decrease get faster or slower? Does the plotted curve curve upward or downward?
Mathematically this can be expressed in a second order polynomial – the simplest way of expressing a line that is straight or curved. The equation is:
y = a x^2 + b x + c
How does y change with x, or how do daily death numbers change per day? a is the square or quadratic term that gives the curvature of the slope, if it’s positive the curve is upward and vice versa. b is the linear slope component and c just a number.
Here’s a graph of the quadratic term a of daily mortality, for several countries for the last two weeks. Each daily number of deaths was a 3-day average, and the polynomial was fitted to the 7 days (all 3-averaged) up to that date:
Here’s the source of the daily mortality numbers as a 3-day rolling average:
Remember, these curves are not daily number of deaths, nor are they the slope or linear change with time of daily mortality. They are the “derivative” or the curvature, and indication of if the rate of change is accelerating or decelerating.
Start with the easy one to see, the blue curve for Italy with big amplitude waves. Death rate is – mercifully – decreasing in that country and that is reflected in the bigger area under the oscillating curve that is in the negative region than in the positive. It seems in Italy there was initially fast growth – in Lombardy and the north, which then levelled off, but then further acceleration – the next excursion above the line – reflected new spread in the south of Italy.
That waviness of the curve is present in all countries in the graph. This may reflect the multiple local outbreaks that comprise the covid19 situation in every country. Different outbreaks may be behind the separate waves of acceleration and deceleration of the death rate.
Clustered in the middle and close to zero, and hard to see, are several north European countries – Germany, Belgium and The Netherlands. They seem to follow a similar wave. Curiously their waves match those of Iran, even though Iran is much further developed in its epidemic and deaths are steadily declining.
Spain has a wave that is almost opposite to that of Italy, although like Italy, the good news is that the most recent quadratic is negative and the rate of daily deaths is curving down. The countries with ongoing and accelerating spread appear to be the UK, USA and France.
Note that the quadratic numbers are normalised to the population of each country.