As we are all watching the daily reports of Covid-19 cases and deaths, one conspicuous fact is that there are enormous fluctuations from day to day. It’s clear that we shouldn’t be too focused on the counts from an individual day to understand the trend. But why do they fluctuate so much?
Here are the numbers for the UK since the start of April (data from the European Centre for Disease Prevention and Control):
|Date||No. cases||No. deaths|
We have 381 deaths one day, then 743 deaths the next day, then 389, then 684. A natural explanation is that it has to do with delays in reporting weekends and/or holidays. But the 381 and 389 were on Wednesday and Friday, and the next peak came on a Saturday and Sunday. I don’t have an explanation, but I wanted to investigate whether this is a real issue. Are the day-to-day fluctuations more than you would expect purely by chance?
It’s not immediately obvious how many we should expect, since there is clearly a trend. I decided to estimate it in two different ways:
- Choose a range s of days — most naturally s=3 — and estimate the predicted cumulative number of events C[x] on day x as the geometric average of the cumulative number of events to day x+s, and the cumulative number to day x-s. This also gives us a natural estimate of the growth rate r over the period, r=(C[x+s]/C[x-s])^(1/2s) -1, and we calculate the Predicted number of events on day x as rC[x]/(1+r). We call this the geometric average method.
- As above, but we estimate r by the maximum likelihood estimator, assuming that the number events on day x is Poisson distributed with parameter rC[x-1], where C[x-1] is the observed cumulative number on day x-1. We call this the cumulative MLE method.
- An approach that addresses more directly the fluctuations in daily events computes the maximum likelihood estimators for a two-parameter model with parameters λ and r, where the number of events on day x+k is Poisson(λr^k), taking as observations the numbers on days x-s,x-s+1,…,x-1,x+1,…,x+s. We call this the single-day MLE method.
Here are the results for the UK. The discrepancy between observed and predicted has been standardised to standard deviation 1, so we should not expect to see results outside of the range (-2.5,+2.5) by chance.
There’s no obvious pattern. The geometric-average increase-rate is always larger, hence the discrepancies are smaller. Not surprisingly, the growth rates estimated from single-day counts are more variable, but the discrepancies are, surprisingly, less extreme. With any method the discrepancies are extreme in both directions, and don’t seem to have an obvious pattern with respect to the weekends. We see extreme positive deviations on weekends, and extreme negative deviations on weekdays. (I have shaded Sunday and Monday as weekends, supposing that most of the deaths in hospitals on Saturday and Sunday would appear in national totals the following day. According to some estimates, the most common delay is 2 days, suggesting that Monday and Tuesday counts should be expected to be low.)
Here is Germany:
These show, perhaps, a more consistent weekly pattern, with negative deviations on Sundays and Mondays.
Italy shows an odd alternating pattern:
The dynamics of new cases are expected to be different. Here is the equivalent plot for new cases in the UK
The fluctuations seem to be increasing, with huge excess number of cases reported on Easter, for some reason — 8719 new cases, where the largest reported on any other day was 5903.
The US also shows this intensification of fluctuations
Germany, Italy, and Canada don’t seem to show this pattern. The plots for Germany and Canada each is dominated by a single very extreme day, possibly corresponding to a shift in counting criteria.