Using excess deaths and testing statistics to improve estimates of COVID-19 mortalities

Abstract

Factors such as non-uniform definitions of mortality, uncertainty in disease prevalence, and biased sampling complicate the quantification of fatality during an epidemic. Regardless of the employed fatality measure, the infected population and the number of infection-caused deaths need to be consistently estimated for comparing mortality across regions. We combine historical and current mortality data, a statistical testing model, and an SIR epidemic model, to improve estimation of mortality. We find that the average excess death across the entire US is 13$\%$ higher than the number of reported COVID-19 deaths. In some areas, such as New York City, the number of weekly deaths is about eight times higher than in previous years. Other countries such as Peru, Ecuador, Mexico, and Spain exhibit excess deaths significantly higher than their reported COVID-19 deaths. Conversely, we find negligible or negative excess deaths for part and all of 2020 for Denmark, Germany, and Norway.

FIG. 1:. Examples of seasonal mortality and excess deaths.

The evolution of weekly deaths in (a) New York City (six years) and (b) Germany (five years) derived from data in Refs. [26, 27]. Grey solid lines and shaded regions represent the historical numbers of deaths and corresponding confidence intervals defined in Eq. (1). Blue solid lines indicate weekly deaths, and weekly deaths that lie outside the confidence intervals are indicated by solid red lines. The red shaded regions represent statistically significant mean cumulative excess deaths D_e. The reported weekly confirmed deaths $d_{\text{c}}^{(0)}(i)$ (dashed black curves), reported cumulative confirmed deaths D_c(k) (dashed dark red curves), weekly excess deaths ${\overline{d}}_{\text{e}}(i)$ (solid grey curves), and cumulative excess deaths ${\overline{D}}_{\text{e}}(k)$ (solid red curves) are plotted in units of per 100,000 in (c) and (d) for NYC and Germany, respectively. The excess deaths and the associated 95% confidence intervals given by the error bars are constructed from historical death data in (a-b) and defined in Eqs. (1) and (2). In NYC there is clearly a significant number of excess deaths that can be safely attributed to COVID-19, while to date in Germany, there have been no significant excess deaths. Excess death data from other jurisdictions are shown in the Supplementary Information and typically show excess deaths greater than reported confirmed deaths (with Germany an exception as shown in (d)).

FIG. 2:. Biased and unbiased testing of a population.

A hypothetical scenario of testing a population (total N = 54 individuals) within a jurisdiction (solid black boundary). Filled red circles represent the true number of infected individuals who tested positive and the black-filled red circles indicate individuals who have died from the infection. Open red circles denote uninfected individuals who were tested positive (false positives) while filled red circles with dark gray borders are infected individuals who were tested negative (false negatives). In the jurisdiction of interest 5 have died of the infection while 16 are truly infected. The true fraction f of infected in the entire population is thus f = 16/54 and the true IFR=5/16. However, under testing (green and blue) samples, a false positive is shown to arise. If the apparent positive fraction ${\tilde{f}}_{\text{b}}$ is derived from a biased sample (blue), the estimated apparent IFR can be quite different from the true one. For a less biased (more random) testing sample (green sample), a more accurate estimate of the total number of infected individuals is $N_{\text{c}}+N_{\text{u}}={\tilde{f}}_{\text{b}}N=(5/14)\times 54\approx 19$ when the single false positive in this sample is included, and ${\tilde{f}}_{\text{b}}N=(4/14)\times 54\approx 15$ when the false positive is excluded, and allows us to more accurately infer the IFR. Note that CFR is defined according to the tested quantities D_c/N_c which are precisely 2/9 and 2/5 for the blue and green sample, respectively, if false positives are considered. When false negatives are known and factored out CFR = 2/8 and 2/4, for the blue and green samples, respectively.

FIG. 3:. Excess deaths versus confirmed deaths across different countries/states.

The number of excess deaths in 2020 versus confirmed deaths across different countries (a) and US states (b). The black solid lines in both panels have slope 1. In (a) the blue solid line is a guide-line with slope 3; in (b) the blue solid line is a least-squares fit of the data with slope 1.132 (95% CI 1.096–1.168; blue shaded region). All data were updated on December 10, 2020 [20, 27, 29, 36].

FIG. 4:. Different mortality measures across different regions.

(a) The apparent (dashed lines) and corrected (solid lines) IFR in the US, as of November 1, 2020, estimated using excess mortality data. We set ${\tilde{f}}_b=0.093$, 0.15 (black,red), FPR = 0.05, FNR = 0.2, and N = 330 million. For the corrected IFR, we use $\widehat{f}$ as defined in Eq. (6). Unbiased testing corresponds to setting b = 0. For b > 0 (positive testing bias), infected individuals are overrepresented in the sample population. Hence, the corrected IFR is larger than the apparent IFR. If b is sufficiently small (negative testing bias), the corrected IFR may be smaller than the apparent IFR. (b) The coefficient of variation of D_e (dashed line) and IFR (solid lines) with σ_I = 0.02, σ_II = 0.05, and σ_b = 0.2 (see Tab. II).

FIG. 5:. Mortality characteristics in different countries and states.

(a) The values of relative excess deaths r, the CFR, the $\text{IFR}=p{\overline{D}}_{\text{e}}/N_{\text{c}}$ with p = N_c/(N_c + N_u) = 0.1 [31], the confirmed resolved mortality M, and the true resolved mortality $\mathcal{M}$ (using γ = 100) are plotted for various jurisdictions. (b) Different mortality measures provide ambiguous characterizations of disease severeness. (c–g) The probability density functions (PDFs) of the mortality measures shown in (a) and (b). Note that there are only very incomplete recovery data available for certain countries (e.g., US and UK). For countries without recovery data, we could not determine M and $\mathcal{M}$. The number of jurisdictions that we used in (a) and (c–g) are 77, 246, 73, 191, and 21 for the respective mortality measures (from left to right). All data were updated December 10, 2020 [20, 27, 29, 36].

FIG. 6:. Different mortality measures across different regions.

We show the values of M and CFR (a) and $\mathcal{M}$ (using γ = 100) and $\text{IFR}=p{\overline{D}}_{\text{e}}/N_{\text{c}}$ with p = N_c/(N_c+N_u) = 0.1 [31] (b) for different regions. The black solid lines have slope 1. If jurisdictions do not report the number of recovered individuals, R_c = 0 and M = 1 [light red disks in (a)]. In jurisdictions for which the data indicate ${\overline{D}}_{\text{e}}<D_{\text{c}}$, we set $\gamma \left({\overline{D}}_{\text{e}}-D_{\text{c}}\right)=0$ in the denominator of $\mathcal{M}$ which prevents it from becoming negative as long as ${\overline{D}}_{\text{e}}\ge 0$. All data were updated on December 10, 2020 [20, 27, 29, 36].