**No, it was just around 0.04-0.27 years, the decrease was temporary, and by now, Covid-19 has led to an extra (albeit tiny) increase of our life expectancy on top of the trend-like increase!**

Dear Prof Klaus,

thank you very much for your wonderful message and your question whether it's possible, as the Czech biostatistician Mr Ladislav Dušek claims in a daily, that Covid has shortened life expectancy of Czechs by one year ("in 2020"). It is an excellent question that should be seriously discussed, instead of irrelevant factoids about the number of "cases" in this or that place which people are still fed by the media, aside from the repulsive needles and deceitful arguments in favor of needles.

**First**, I must say that the Czech phrases "střední délka života" and "naděje dožití" ("average length of life" and the "chance of a long life", literally) are considered exactly

**synonymous**in Czech; both are translated as "life expectancy" (click to get the Wikipedia entry, I don't claim that it's the best source I could link to).

There is really only one quantity that may express "how much life is left for an average person". However, one must realize that the word "average" means that we must specify the group that is being averaged (a simple arithmetic average with equal weights) such as the Czech nation. If we choose a subset of the Czech nation, it will have a different life expectancy than the whole nation. In particular, the "remaining life expectancy" will be shorter for older cohorts (groups of people according to an interval of age); but the expected or average age at death (which is the current age plus the remaining life expectancy) is actually

*longer*for the older cohorts because they have already passed a test. It is obvious for those who are 96 because they are guaranteed to die 96-year-old or older; while the average Czech is not guaranteed that outcome of Fred Singer's yet. We must distinguish the remaining life expectancy and the total life expectancy; they simply differ by the average current age of the group we try to describe.

**Second**, I want to emphasize that life expectancy is not being "integrated over time" in any sense. Instead, it is just a prediction of "how much life is left" that is made at a

**single moment**(and we should specify a single moment, not a period, e.g. "July 21st, 2021"). In other words, the beginning and end of a year (or other arbitrary periods defining the calendar, a social convention) play absolutely no special role in calculations of life expectancy! And it is a prediction which means that it is not "directly calculated" from the available data from the past (the data about the age of people who have already deceased would seem most useful). Instead, it is always a somewhat uncertain

**prediction about the future**– it tries to estimate when the deaths will happen in the future. This "future" character of the life expectancy – it is a quantitative expectation, therefore the name – is the reason why the life expectancy would abruptly jump if some miraculous e.g. cancer drug were found; and it would drop e.g. if the nuclear world war became almost certain.

**Third**, life expectancy is always computed for groups of people who are still alive. People who are already dead have the "remaining life expectancy" equal to zero (as of now) because they are already dead. Thomas Jefferson said that "the Earth belongs in usufruct to the living" (which means that democracy mustn't count the votes of the people who are already dead, otherwise we would be ruled by zombies; and it cannot count hypothetical votes of the future people who aren't born yet or who aren't 18 at this moment even though most environmentalists would love to count the not-yet-born kids as their voters, without asking them and even without/before allowing them to be born in the first place).

It may seem comical that I emphasize that the "life expectancy" is only computed for groups of people who are alive because this point seems tautologically obvious but it has one implication that is easy to overlook: life expectancy may abruptly change when a person (especially many people that are not representative within the society) dies because one changes the set for which the average "remaining life span" is computed. The remaining life expectancy would increase discontinuously (but the average expected age at death would decrease discontinuously) if all old people simultaneously died, for example.

**OK, how did life expectancy change in the presence of Covid-19?**

Let me take a bigger picture. As e.g. the first graph here shows, the Czech life expectancy started to grow, especially for men, after the Velvet Revolution, mainly thanks to the positive forces of capitalism. Let me say that since 1991 or so through pre-Covid early 2020 (and the interval is 30 years old), the life expectancy in Czechia grew 7/6 years from 69/76 to 76/82 for men/women, respectively (the rate is about 0.2 years per year). It doesn't make much sense to discuss these numbers with the sub-year precision because the actual uncertainty is larger; however, the small changes in the life expectancy may be assumed to have a better precision. At any rate, I hope that you will agree that the mastermind of the Czechoslovak transformation towards capitalism may be credited with the extension of the Czech lives by a significant fraction of those 6 or 7 years so far. ;-)

OK, the expected average age-at-death of Czech men/women who were alive at a moment in January 2020 was 76/82 years, respectively. How did the life expectancy start to change when it became clear that Covid-19 would claim a not negligible number of lives? The answer is that the result depended on the estimates how many people were going to die and (perhaps) how old they would have been (I originally wanted to argue that the age distribution didn't affect the initial Covid-related drop of life expectancy but I later changed my mind, a high-precision integral-sum over all "groups of people" of the product of the remaining life expectancy, the risk of death in the group, and percentage of population that belongs to the group is needed, see the last P.S. at the very bottom). Depending on these estimates of the "severity of the coming Covid epidemics", one could have gotten a tiny decrease or a huge drop of the Czech life expectancy.

But now we know the total number of Czech lives claimed by Covid-19. The official current number of deaths with Covid-19 is 30342 and it is clearly a nearly final number. Because of the huge fraction of the immune Czechs (and the natural immunity from the disease surival applies to a larger number of Czechs than that from the vaccination so far), it is almost mathematically impossible to see another significant wave of Covid deaths in Fall 2021. The total number of excess deaths is going to be a bit higher than 30,000. In 2020, the excess deaths (about 17,000 people: 129k instead of 112k) were actually often up to 50% higher than the "deaths with Covid" which indicated either "deaths due to the lockdowns etc." or "deaths where the Covid cause was overlooked for some reason", depending on one's belief. However, the situation got reversed since June 2021 and the most recently reported 3 June weeks, Weeks 21-23, already saw a deficit of deaths: the number of people who died was by 50-100 lower than the weekly average for those weeks, around 2,000.

Great. But now we know that Covid-19 has killed 30,000 Czechs and we also know their distribution within the age-sex pyramid. If we had known this "coming future" in early 2020, we could have calculated the effect of Covid-19 on the Czechs' life expectancy. As I said in one of the early disclaimers, at the very moment when we would

*learn*that 30,000 people were going to die from Covid (let me assume that the number of deaths from Covid is very close to the official deaths with Covid because I actually believe it is the case now), the life expectancy would abruptly decrease. How much?

Well, it's easy to give a rather accurate estimate by a trivial argument. The average Czech is about 40 years old (the median is about 43 years), one-half of the life expectancy, and has 40 years ahead in his or her life, also one-half of the life expectancy. During 40 years, some 40 * 112,000 people will die in Czechia, about 4.5 million (assuming some stagnation of everything unrelated to Covid-19 which is a good enough approximation). But the sudden knowledge of the extra 30,000 deaths from Covid increases the expected number of deaths in the next 40 years by those 30,000, too. Because 30,000 is about 0.27 * 112,000, it corresponds to 0.27 years (about 3.2 months) of dying. Almost all these extra 30,000 deaths have occurred in 2020 or 2021 but it doesn't matter. They are extra deaths affecting the life expectancy of everybody who was alive in January 2020, before Covid became known to us, and that is why the "moment when the average person alive in January 2020 dies" would have come 0.27 years earlier than without Covid, too.

So the life expectancy has really abruptly shortened by 30,000 / (112,000/year) = 0.27 years (or 3.2 months) in February 2020, when we hypothetically learned that Covid-19 was going to claim 30,000 lives but before the people started to die. It is significantly less than the figure (one whole year) by Dr Dušek, I am reasonably certain that he is wrong.

I want to argue that the figure around 0.27 years is actually independent of the question whether Covid-19 kills mainly old people, representative people of typical ages at death, or young people. In fact, there is another way to organize this estimate which yields the same result. Approximately 0.3% of the Czech population died from Covid. That has simply increased the probability of the death in the next E years (where E is the life expectancy for a given person) by 0.3% for the average Czech person. Because the probability of premature death has increased by 0.3%, the life expectancy dropped by 0.3% and 0.3% of 80 years is, once again, about 0.25 years. With the probability 99.7%, the average person survives (and has the same remaining life expectancy as he or she had before Covid); with the probability of 0.3%, he or she dies very soon. So the new remaining life expectancy is the weighted average 0.997*E + 0.003*1 where "1" means about "1 year left". Oops, in this counting, I got just 50% of the previous figure, a shortening of the remaining life expectancy 40 years by 0.12 years i.e. 1.6 months.

I apologize, I had to send the homework exercise quickly and I am not certain whether 0.12 years (0.3% of 80 years) or 0.27 years (0.3% of 40 years) is the more accurate zeroth order estimate of the shortening of our life expectancy! ;-)

**Now, 0.12 or 0.27 years may still look like a lot even though the life expectancy is increasing due to the trend of improvements so that 0.25 years is just a bit over 1 year of improvements! Where is the catch?**

The catch is hiding in one of the disclaimers that I have already mentioned. Once the people die, we actually change the set of the people whose average remaining life is being quantified. Now once the 30,000 Czechs are gone, we must calculate the life expectancy for the survivors only. And the survivors don't expect to die from Covid-19 any longer! Assuming that no other sources of death were added (like those from the unknown effects of the vaccination), the reduction of the life expectancy by those 0.12-0.27 years attributed to Covid-19 (through the calculation above) is completely undone by now!

**In fact, the truth is slightly more extreme. Because it is calculated from the survivors only (by averaging), the current life expectancy in Czechia is already a bit higher than it was shortly before Covid-19 started.**

Why? Well, a fraction of the people whose health wasn't in a good enough shape (and it was mostly very old people) are gone and however sad and distasteful this eugenics-like argument may sound, their departure improves the average health and therefore increases the "remaining life expectancy" of the average survivor. Again, this is nothing else than some kind of natural selection in real life. The failure to understand that the life expectancy drop has been completely reversed is a kind of survivorship bias.

How much longer the life expectancy is now than it was shortly before Covid? Well, we are getting the extension by those 0.2 years from the trend of improved well-being (people are getting healthier, especially from the Velvet Revolution). The temporary drop of 0.12 or 0.27 years due to Covid-19 has been undone because the casualties have been removed from the averaging ensemble. On top of that, we may have gotten a few tenths of a year of an extra increase relatively to the trend. I think it is likely that we went from 76/82 for men/women to 77/83 by now.

It is relatively hard to estimate this increase of the life expectancy because the statistics of the casualties according to the sex and age isn't enough. I think that the proportion of the people who died from Covid-19 in various cohorts wasn't too different from the proportion of the people who die at a given age for

*any*cause (old people dominate deaths with Covid as well as without Covid!), and if this were exactly true, we would get no increase of the life expectancy from the Covid-19 deaths. However, the age and sex don't precisely describe "what kind of people have died". The ill people or people in a bad shape (who had a lower "personal" life expectancy) were heavily overrepresented in the deaths and their departure has made the rest significantly healthier.

This extra post-Covid increase of the life expectancy is a prediction – because the life expectancy is an expectation i.e. prediction. But it may be confirmed or falsified by the experimental data that will come in the future (and they are already starting to arrive). And indeed, what will be confirming the data will be the deficit of deaths (below-the-normal rates of deaths per week etc.) that we have already seen in June. I do think that this "deficit of deaths" (the number of deaths below the normal) will continue for a year or several years (about 1/2 of the remaining life expectancy of the very old people, the typical people who dominate the deaths). Note that if the deficit of deaths continues to be 50-100 per week for a year, it is about 2,500-5,000 "saved deaths" per year. The total "excess deaths" from all Covid-related causes have peaked and now the number is going to retrace a little bit.

So with the observed "deficit of deaths", the number of deaths that will be subtracted due to the "healthier population of survivors" will be about 3,000 or 10% of the deaths from Covid-19. Because the deaths from Covid-19 temporarily lowered the life expectancy by 0.12 or 0.27 years, the increase of the life expectancy due to the "improved fitness" will be about 0.01-0.03 years and I would personally neglect it. It is equivalent to a week or a month of the trend-like improvements of the life expectancy that are caused by the improving well-being.

P.S. 1: I encourage the respected reader to decide whether the correct estimate of the temporary life expectancy reduction was 0.12 or 0.27 years! ;-)

P.S. 2: If you think that even those 0.12 or 0.27 years sound "intuitively too much", I fully understand where you're coming from and I would say that this figure looks so high because its naive articulation neglects that it was a temporarily changed expectation only. Covid has shortened the life expectation by this much only for roughly 1 year when Covid lasted; due to the departure of the victims, the reduction has already been undone. The life expectation averaged over the average Czech person's 80-year-long life was only shortened by about 1/80 of that figure (because the shortening lasted 1/80 of a life) i.e. by 0.0015-0.0034 years i.e. between half a day and one day! One must be careful what these figures exactly encode. Concerning "tiny shortening of lives due to infections", I calculated in November 2020 that the average potential infection only shortened the total length of life of all people by minutes, mocking Tommaso Dorigo's totally incorrect result that an average Covid-19 infection was equivalent to the subtracted 10 years of a life, i.e. nearly to a murder.

P.S. 3: Concerning 0.12 or 0.27, the integration over cohorts has to be done, after all. The remaining life expectancy for a cohort was changed from E(age) to (1-RiskOfCovidDeath)*E(age)+RiskOfCovidDeath*0. You may roughly calculate this expression for every cohort or age-sex group according to the age-sex-Covid-death pyramid numbers, and compute the weighted average of these cohorts. The very old people have a much higher fraction of those who died; but they also tend to have a shorter remaining life expectancy. There is a trade-off. Effectively, we effectively had 30,000 deaths of people who may be imagined to be 70 years old. To approximate the integral, we must just substitute some 0.5% for the RiskOfDeath in this approximate cohort but the remaining life expectancy of the 70-year old is some 13 years, see the Table 1 here. The shortening would yield about 0.5% of 13 years which is 0.065 years, another 50% below the lower figure proposed by the simple arguments above. The 80-year-old actually lead to almost the same product because they have about 7 years left in average (Table 1) but about 1.2% risk of death due to Covid (see the 75-84 figure here) and 0.012*7 = 0.084 years. A better homework: do the damn integral, perhaps as a Riemann sum approximation. I think that you will get below 0.1 years of the temporary life expectancy reduction. An even more precise estimate would sum over the two sexes (but the figures for the "average sex" are probably accurate within 10%). This would still be an imperfect estimate because people's identity is given by other traits than the sex and age and there exists no way to make a perfectly precise calculation that sums over "all groups of people". But generally, if the people's health status were taken into account in some clever way, the reduction of the life expectancy would almost certainly be even lower than 0.065-0.084 years obtained a minute ago because most of the Covid casualties (i.e. subgroups for which the reduction is substantial due to the substantial risk-of-death factor) had a much lower remaining life expectancy than (even) the people in the same age-sex group.

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