A scaling approach to estimate the age-dependent COVID-19 infection fatality ratio from incomplete data

Abstract

SARS-CoV-2 has disrupted the life of billions of people around the world since the first outbreak was officially declared in China at the beginning of 2020. Yet, important questions such as how deadly it is or its degree of spread within different countries remain unanswered. In this work, we exploit the 'universal' increase of the mortality rate with age observed in different countries since the beginning of their respective outbreaks, combined with the results of the antibody prevalence tests in the population of Spain, to unveil both unknowns. We test these results with an analogous antibody rate survey in the canton of Geneva, Switzerland, showing a good agreement. We also argue that the official number of deaths over 70 years old might be importantly underestimated in most of the countries, and we use the comparison between the official records with the number of deaths mentioning COVID-19 in the death certificates to quantify by how much. Using this information, we estimate the infection fatality ratio (IFR) for the different age segments and the fraction of the population infected in different countries assuming a uniform exposure to the virus in all age segments. We also give estimations for the non-uniform IFR using the sero-epidemiological results of Spain, showing a very similar increase of the fatality ratio with age. Only for Spain, we estimate the probability (if infected) of being identified as a case, being hospitalized or admitted in the intensive care units as function of age. In general, we observe a nearly exponential increase of the fatality ratio with age, which anticipates large differences in total IFR in countries with different demographic distributions, with numbers that range from 1.82% in Italy, to 0.62% in China or even 0.14% in middle Africa.

Fig 1. Normalized number of deaths occurred in French hospitals as a function of age.

A We show the evolution with time of the cumulative number of deaths normalized by the number density of individuals in age group α (i.e. ${\widehat{D}}_{\alpha }\left(t\right)$ in Eq (2)). In B, we show ${\widehat{D}}_{\alpha }\left(t\right)/{\widehat{D}}_{50-59}\left(t\right)$ as function of the age group, for all the times in A (the darker the color, the more recent the measurement, and we give some dates in the legend). This quotient is essentially time-independent as discussed in Eq (7), and it lets us estimate the quotient between the UIFR (the IFR under the assumption of uniform attack rate, see Eq (6)) of the two age groups, that is, ${\widehat{f}}_{\alpha }/{\widehat{f}}_{50-59}$.

Fig 2. Under-counting of deaths per age groups.

We show the fraction of under-counted deaths, per age groups, observed when comparing the number of deaths certificates where COVID-19 was mentioned either confirmed or suspected, and the official deaths attributed to COVID-19, relatively to this second number, see Eq (1) for the definition, for England and Wales in A, and for the Community of Madrid B. The horizontal lines mark the mean rate of ‘under-counting’ below 80 years old.

Fig 3. Normalized number of deaths in different countries as a function of age.

A We show the normalized number of deaths per age group (defined in Eq (2)) for a selection of countries affected by the COVID-19 epidemic at very different scales. In B, we show the same data (excluding the Netherlands) but where each country has been multiplied by a constant $\mathcal{D}\left(\mathcal{C}\right)$ so that it collapses with the Spanish curve in the age region in between 30 and 70 years old. The values of each country’s constants are given in S2 Table. In black, we show the country average for each age segment (errors calculated with the boostrap method up to a 95% of confidence), and in C the fit of this average to a pure exponential function, see Eq (8).

Fig 4. Probabilities assuming a uniform attack rate.

A We use the measurements of the number of infections in Spain to estimate the UIFR using Eq (3) in both regions. We fix the constant ${\widehat{f}}_0$ in S2 Table using the estimation of the UIFR in Spain for the age group 50-59 to infer the values of the country average UIFR (from the collapse of Fig 3B). We show this first estimation in red, and in green, we show the UIFR after correcting the under-counting of deaths over 70 years old. We compare these results with the estimation by Verity et al. [25] and the CFR (i.e. the probability of dying for confirmed COVID-19 cases, not the IFR) by age in South Korea. In B, we use the IFR estimations from A̠ and Table 1, to predict the seroprevalence of anti-SARS-CoV-2 antibodies in the population of Geneva, Switzerland, from the official distribution of deaths per age of a total of 277 deceases. The predicted fraction of infections is given in dots (in green, if we used the bare estimation of Eq (9), in violet, if we include the corrections linked to under-counting). In horizontal lines (and the 95% of confidence interval in gray shadow), we show the actual values measured from the antibody survey of Ref. [47] in patients of different age-groups.

Fig 5. Other probabilities as function of age assuming uniform attack rate.

In A we show the probability of being classified as official case, ${\widehat{f}}_C$, being hospitalized, ${\widehat{f}}_H$, admitted in intensive care, ${\widehat{f}}_S$, and dying, ${\widehat{f}}_D$, in Spain in Spring 2020, as function of the age using age segments of 10 years. B, we show the same data but were the kid’s information has been grouped by smaller age-segments, evidencing the severity of the cases in patients under 2 years old. A is generated using the data by the Spanish Health Ministry up to the 22nd of May and B with the data published by the RENAVE, see Materials and S1 Table.

Fig 6. Uniform versus non uniform IFR.

A We show the relative risk of infection for an age segment r_α (see Eq (4) and definition below) taken from the sero-epidemiological study of the Spanish population [46]. While the youngest segments of the population seem to be less hit by the virus, the distribution of the infections is rather similar to that of a uniform attack rate, indicated by the dashed line r_α = 1 here. The 95% confidence interval for r_α is indicated by the red shadow. B We show the estimated uniform and nonuniform IFR for Spain and compare it with the CFR as a function of age. The error for the non-uniform IFR is shown by a red shadow.