I was recently looking at mortality statistics for England and Wales in 2019 and 2020. There’s been a lot of talk about excess mortality due to Covid, but there hasn’t been much discussion of mortality reduction. Given that Covid rarely caused serious symptoms — and hardly ever death — in young people, and given the high proportion of deaths due to accidents — particularly vehicular accidents — you might expect the lockdown to have reduced their overall mortality. And this is indeed what we see.

Age

0

1-14

15-44

45-64

65-74

75-84

85+

Mortality 2019

395

9

65

415

1474

4160

14090

Mortality 2020

375

8

69

470

1651

4744

16120

Relative % excess 2020

-5.2

-12.8

4.6

13.4

12.1

14.0

14.4

Force of mortality in 2019 and 2020, in deaths/100,000

Above age 45 we see a fairly consistent 13% increase in mortality from 2019 to 2020. But there was a 13% decrease in mortality among children under 15. Even in the newborn group there was a 5% decrease. In total there were 116 fewer deaths recorded in 2020 in the 1-14 age group compared with 2019, and 200 fewer deaths in the first year of life (of which about 70 may be attributed to a 3% decrease in the number of births).

Multiple European countries have now suspended use of the Oxford/AstraZeneca vaccine, because of scattered reports of rare clotting disorders following vaccination. In all the talk of “precautionary” approaches the urgency of the situation seems to be suddenly ignored. Every vaccine triggers serious side effects in some small number of individuals, occasionally fatal, and we recognise that in special systems for compensating the victims. It seems worth considering, when looking at the possibility of several-in-a-million complications, how many lives may be lost because of delayed vaccinations.

I start with the case fatality rate (CFR) from this metaanalysis, and multiply them by the current overall weekly case rate, which is 1.78 cases/thousand population in the EU (according to data from the ECDC). This ignores the differences between countries, and differences between age groups in infection rate, and certainly underestimates the infection rate for obvious reasons of selective testing.

Age group

0-34

35-44

45-54

55-64

65-74

75-84

85+

CFR (per thousand)

0.04

0.68

2.3

7.5

25

85

283

Expected fatalities per week per million population

0.07

1.2

4.1

13

45

151

504

Number of days delay to match VFR

1200

70

20

6.4

1.8

0.6

0.2

Let’s assume now that all of the blood clotting problems that have occurred in the EEA — 30 in total, according to this report — among the 5 million receiving the AZ vaccine were actually caused by the vaccine, and suppose (incorrectly) that all of those people had died.* That would produce a vaccine fatality rate (VFR) of 6 per million. We can double that to account for possible additional unreported cases, or other kinds of complications that have not yet been recognised. We can then calculate how many days of delay would cause as many extra deaths as the vaccine itself might cause.

The result is fairly clear: even the most extreme concerns raised about the AZ vaccine could not justify even a one-week delay in vaccination, at least among the population 55 years old and over. (I am also ignoring here the compounding effect of onward transmission prevented by vaccination, which makes the delay even more costly.) As is so often the case, “abundance of caution” turns out to be fundamentally reckless.

* I’m using only European data here, to account for the contention that there may be a specific problem with European production of the vaccine. The UK has used much more of the AZ vaccine, with even fewer problems.

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

17/04/2020

4617

861

16/04/2020

4603

761

15/04/2020

5252

778

14/04/2020

4342

717

13/04/2020

5288

737

12/04/2020

8719

917

11/04/2020

5195

980

10/04/2020

4344

881

09/04/2020

5491

938

08/04/2020

3634

786

07/04/2020

3802

439

06/04/2020

5903

621

05/04/2020

3735

708

04/04/2020

4450

684

03/04/2020

4244

389

02/04/2020

4324

743

01/04/2020

3009

381

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.

Deaths

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.

The US:

Canada:

Italy shows an odd alternating pattern:

Cases

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.

Estimated fatality in deaths per thousand cases (symptomatic and asymptomatic)

These numbers looked somewhat familiar to me, having just lectured a course on life tables and survival analysis. Recent one-year mortality rates in the UK are in the table below:

0-9

10-19

20-29

30-39

40-49

50-59

60-69

70-79

80-89

.12

.17

.43

.80

1.8

4.2

10

28

85

One-year mortality probabilities in the UK, in deaths per thousand population. Neonatal mortality has been excluded from the 0-9 class, and the over-80 class has been cut off at 89.

Depending on how you look at it, the Covid-19 mortality is shifted by a decade, or about double the usual one-year mortality probability for an average UK resident (corresponding to the fact that mortality rates double about every 9 years). If you accept the estimates that around half of the population in most of the world will eventually be infected, and if these mortality rates remain unchanged, this means that effectively everyone will get a double dose of mortality risk this year. Somewhat lower (as may be seen in the plots below) for the younger folk, whereas the over-50s get more like a triple dose.