Infection Fatality by Age

Abstract

If the overall Infection Fatality Rate (IFR) is .006, what does this mean for the IFR by age? Assuming that the infection rate is the same at all ages, the IFR at an age equals the overall IFR times the ratio of the COVID death rate at that age to the crude COVID death rate. For the US, the IFR’s range from .00004 at age 0 up to .10 at 85+. An infected 75 to 84 year old would have only about one in thirty chance of dying from COVID-19. Because younger people are probably less cautious than older, let’s assume that their infection rate is ten times as high below age 45 as it is above it. This change in assumptions would hardly change the results at all, raising the IFR for those above 45 by a factor of 1.024.

Introduction

There is a great deal of uncertainty about the infection fatality rate (IFR) for COVID-19. The WHO recently convened a meeting of over a thousand epidemiologists from around the world and the prevailing view at that meeting according to the NYT (July 4) was that the IFR was around .006. We can be quite certain, however, that the IFR varies strongly by age and therefore that it will vary from country to country according to their population age distributions. Presumably, other demographic characteristics also matter such as gender, socioeconomic status, urban-rural residence, and so on. But for present purposes, I will ignore such problems, and simply assume that the average IFR for the US is .006. What would this mean for the IFR by age? In particular, since the IFR is close to zero for many younger people, must the IFR for the elderly be frighteningly high? To get a rough idea of the answer to this question I first sketch my estimation strategy and then make some simple estimates for the US. The bottom line is that an 80-year-old who was infected by COVID-19 would have only about 1/30^th chance of dying from it. 

There are serologically based estimates of infection fatality rates by age, for example in Levin et al (2020). These appear to be somewhat higher than the ones reported below, which could arise if the infection rates declined with age at higher ages. There may be some advantage (as well as disadvantages) to the more indirect method I suggest below, just because the serological samples are sometimes smaller, so the number of deaths in the sample will be quite small, particularly when broken down by age. The method I suggest below derives all age detail from the full set of age specific COVID-19 death rates so sample size is not an issue. The method could certainly be improved if it were teamed with serological data on the age incidence of infections, which I have not yet tried to do. 

Assumptions and Notation

Some variables apply to a given period that could be a year, a month, or other. Results should not depend on the period or the number of deaths. 

$i(x)$ = COVID-19 infection rate, which I initially assume is constant at $i$ across age
$c(x)$ = Infection Fatality Rate (IFR) at age $x$; $c$ is the average IFR for the population =.006.
$m(x)$ = estimated COVID mortality rate by age;
$\bar{m}$ is the population-weighted average COVID-19 death rate, $D/N$.
$N$ = total population
$D$ = total COVID-19 deaths

Assuming the infection rate is constant by age, the infection rate for the population $i$ times the proportion infected that dies (IFR) gives the crude COVID-19 death rate, $D/N$: 

$$ i \times c = D/N $$

The age-specific COVID death rate is the infection rate times the IFR at age x.

$$ m(x) = i \times c(x) $$

Solving  for i and substituting in and rearranging we get:

or more simply:

$$ c(x) = c \left[ \frac{m(x)}{D/N} \right] $$

or even more simply: 

$$ c(x) = c \left[ {m(x) \over\bar{m} }\right] $$

Here $\bar{m}$ is the population-weighted average of the standard age schedule of COVID-19 death rates $m(x)$ which is simply the crude COVID death rate, $D/N$. 

This result has a simple interpretation. On (population-weighted) average, the age-specific IFR c(x) must equal the overall IFR, c. At those ages when COVID-19 death rates are lower than the average (the quantity in brackets <1) the IFR must also be lower than average since I have assumed the infection rate is the same at all ages. At the older ages where $m(x)$ is way above average, the IFR must also be way above average. 

Estimates for US

I have calculated the relevant quantities based on the reported number of deaths by age for the US as reported on the INED website on July 29, giving cumulated deaths up June 27, at which point the death count was 130,245. I used US population data to calculate COVID-19 age-specific death rates. I then calculated the ratio of each of these to the crude COVID-19 death rate, 230,245 divided by 330 million. Those ratios can then be multiplied times the consensus value of .006 for the IFR. The age-specific IFR’s estimated in this way are shown in the table below and range from .00004 at age 0 up to .098 at 85+. It remains below 1/1000 until age group 45-54. These calculations suggest that an infected 75 to 84 year old would have only about one in thirty chance of dying from COVID-19.  

Age	Est IFR
0	0.000043
1–4	0.000008
5–14	0.000006
15–24	0.000067
25–34	0.000308
35–44	0.000880
45–54	0.002377
55–64	0.005662
65–74	0.013428
75–84	0.033677
85+	0.098251

The assumption that the infection rate i is the same for all ages is a strong one, and since the infection rate depends on the behavior and circumstances of those at each age it is unlikely to be true. In particular, the young appear to take less care in avoiding crowds, bars, restaurants, political demonstrations, and such, and so are likely to have higher infection rates. These will result in higher m(x) than otherwise, although once infected their IFR may be unaffected. It will also lead to an overestimate of the IFR at younger ages. If the true IFR is then actually lower at younger ages, then the IFR at older ages will have to be higher if the overall population-weighted average must be the given number, .006. This is a strange way to view the matter, to be sure, since the overall IFR must reflect what happens at each age, not the reverse as assumed here. Nonetheless, it will be useful to ask how large this effect might be. 

Suppose the young are more likely to become infected?

Suppose that the infection rate is 10 times as great at each age under 45. It is straightforward to calculate how much higher the IFR’s for ages 45 and above would have to be to keep the average at .006. The answer is that these IFR’s would only have to be higher by a factor of 1.024, a tiny amount. Why is this? If the infection rate is ten times as high for those under 45, then their IFR must be only one-tenth as high, since their age-specific COVID death rate is given. Since then, these IFRs below 45 are lower, the IFRs above 45 must be higher to keep the overall average constant at .6%. But since the IFTs at these younger ages are so very much lower in any case, reducing them by a factor of 10 requires only a very small proportional upward adjustment in the much higher IFRs at older rates. So this calculation is quite robust. An implication is that the behavior and infection rates of the younger part of the population have very little effect on the overall IFR estimate which is largely driven by the older ages.

References

Levin, Andrew T., Kensington B. Cochran, and Seamus P. Walsh (2020) “ASSESSING THE AGE SPECIFICITY OF INFECTION FATALITY RATES FOR COVID-19: META-ANALYSIS & PUBLIC POLICY IMPLICATIONS” NBER Working Paper 27597 http://www.nber.org/papers/w27597