# A Covid-19 case mortality rate without time delay systematics

TL;DR: In the absence of detailed knowledge of the time delay distribution of (a), the true case mortality rate is obtained by pursuing method (b) at the end of the outbreak when the fate of every case has decisively been rendered.

Abstract: Concerning the two approaches to the Covid-19 case mortality rate published in the literature, namely computing the ratio of (a) the daily number of deaths to a time delayed daily number of confirmed infections; and (b) the cumulative number of deaths to confirmed infections up to a certain time, both numbers having been acquired in the middle of an outbreak, it is shown that each suffers from systematic error of a different source. We further show that in the absence of detailed knowledge of the time delay distribution of (a), the true case mortality rate is obtained by pursuing method (b) at the end of the outbreak when the fate of every case has decisively been rendered. The approach is then employed to calculate the mean case mortality rate of 13 regions of China where every case has already been resolved. This leads to a mean rate of 0.527 ± 0.001 %.

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### Summary

- The authors further show that in the absence of detailed knowledge of the time delay distribution of (a), the true case mortality rate is obtained by pursuing method (b) at the end of the outbreak when the fate of every case has decisively been rendered.
- The approach is then employed to calculate the mean case mortality rate of 13 regions of China where every case has already been resolved.
- This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice.
- The usual definition, on the other hand, is the ratio of the cumulative number of deaths to confirmed infections, both being counted to the date of interest.
- Thus, if e.g. if they both increase with time but the 15 former more steeply, the method of [1] will yield a larger result because the ratio involves a smaller denominator as a result of the smaller number of confirmed infections at an earlier time.
- The authors results for these regions do not corroborate 25 the two high case mortality rates quoted above.
- To begin with, the authors enlist the three quantities which are relevant to the calculation of the case mortality rate.

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A Covid-19 case mortality rate without time delay

systematics

Richard Lieu

a,∗

, Siobhan Quenby

b

, Ally Bi-Zhu Jiang

c

a

Department of Physics, University of Alabama, Huntsville, AL 35899, USA

b

Division of Reproductive Health, Warwick Medical School, The University of Warwick, UK

c

Shenzhen RAK wireless Technology Co., Ltd., China

Abstract

Concerning the two approaches to the Covid-19 case mortality rate published

in the literature, namely computing the ratio of (a) the daily number of deaths

to a time delayed daily number of conﬁrmed infections; and (b) the cumulative

number of deaths to conﬁrmed infections up to a certain time, both numbers

having been acquired in the middle of an outbreak, it is shown that each suﬀers

from systematic error of a diﬀerent source. We further show that in the absence

of detailed knowledge of the time delay distribution of (a), the true case mortal-

ity rate is obtained by pursuing method (b) at the end of the outbreak when the

fate of every case has decisively been rendered. The approach is then employed

to calculate the mean case mortality rate of 13 regions of China where every

case has already been resolved. This leads to a mean rate of 0.527 ± 0.001 %.

In a recent correspondence to Lancet [1], the global case mortality rate of

the coronavirus Covid-19 ([2]) was re-calculated after correcting for the ﬁnite

time delay between diagnosis of the disease and death, which led to a higher

estimate of the rate, namely 5.7 ± 0.2 % for the date of March 1, 2020, to be

compared to the global mean value of 3.43 ±0.01 % as computed from the data5

in [3, 4] for the same day. The reason for the higher value in [1] is the authors’

deﬁnition of the case mortality rate, as the ratio of the number of case-related

∗

Corresponding author

Email address: lieur@uah.edu (Richard Lieu)

Preprint submitted to Journal of L

A

T

E

X Templates March 31, 2020

. CC-BY-NC-ND 4.0 International licenseIt is made available under a

is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted April 6, 2020. ; https://doi.org/10.1101/2020.03.31.20049452doi: medRxiv preprint

NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice.

deaths for the day of interest to the number of new conﬁrmed infections for

the same 1-day period two weeks earlier (obviously, this assumes distribution

of time delay between infection and death is peaked at 14 days with a spread10

of less than 1 day). The usual deﬁnition, on the other hand, is the ratio of

the cumulative number of deaths to conﬁrmed infections, both being counted

to the date of interest. If the daily number of conﬁrmed infections and deaths

are constants, the two deﬁnitions will give the same answer. This is no longer

so if both numbers vary. Thus, if e.g. if they both increase with time but the15

former more steeply, the method of [1] will yield a larger result because the ratio

involves a smaller denominator as a result of the smaller number of conﬁrmed

infections at an earlier time.

The purpose of this paper is to point out the respective limits of validity

of the two approaches above, and under what circumstance would one be able20

to infer the true case mortality rate without assuming any details about what

could happen after an infection is conﬁrmed. We then apply our formalism

to calculate the mean case mortality rate of several parts of China, which is

completely free from the systematic error arising from the uncertain time delay

between diagnosis and death. Our results for these regions do not corroborate25

the two high case mortality rates quoted above.

To begin with, we enlist the three quantities which are relevant to the cal-

culation of the case mortality rate. First is the number of deaths per unit time

N(t), second is the number of conﬁrmed infections per unit time n(t), and third

is the probability per unit delay time p(t) of a person dying at time t after she30

was diagnosed as a conﬁrmed infection. More precisely N(t)dt and n(t)dt are

respectively the number of deaths and the number of conﬁrmed infections be-

tween the times t and t +dt from some arbitrary time origin before the outbreak

of the disease, and p(t)dt is the probability of a person dying between the times

t and t + dt from the time of diagnosis.35

Evidently the three quantities N, n, and p must obey the relation

N(t) =

∫

t

0

n(t −t

′

)p(t

′

)dt

′

. (1)

2

. CC-BY-NC-ND 4.0 International licenseIt is made available under a

is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted April 6, 2020. ; https://doi.org/10.1101/2020.03.31.20049452doi: medRxiv preprint

But since

n(t), p(t) = 0 for t < 0; and n(t) = 0 for t > t

m

, (2)

where t

m

is the time beyond which no new infections are reported, it is possible

to rewrite (1) as

N(t) =

∫

∞

−∞

dt

′

n(t −t

′

)p(t

′

)dt

′

. (3)

The case mortality rate is then given by40

P

0

=

∫

∞

−∞

p(t)dt. (4)

It can also be seen from (3) that N ( t) is just the convolution integral between

the two functions n(t) and p(t). We now explore two scenarios.

The ﬁrst scenario is when the time delay between diagnosis and death has

the unique value t = t

0

, so that p(t) = p

0

δ(t − t

0

) where δ ( t) is the Dirac delta

function and (from (4)) p

0

= P

0

is the case mortality rate being sought. In this45

limit, the integral (3) may readily be evaluated to yield

N(t) = p

0

n(t − t

0

), (5)

which means p

0

is exactly the ratio deﬁned by [1].

Under the second scenario, suppose one computes the cumulative number of

deaths throughout the entire epidemic, as

∫

∞

0

N(t)dt =

∫

∞

−∞

N(t)dt =

∫

∞

−∞

dt

∫

∞

−∞

dt

′

n(t − t

′

)p(t

′

)dt

′

, (6)

where use was made of (1), (2), and ( 3). One can readily change the variable of50

the t integration from t to τ = t − t

′

to show that

∫

∞

−∞

N(t)dt =

∫

∞

−∞

n(τ )dτ

∫

∞

−∞

p(t

′

)dt

′

= P

0

∫

∞

−∞

n(τ )dτ, (7)

where use was made of (4).

Although (7) vindicates the conventional method of taking the ratio of the

cumulative number of deaths to conﬁrmed infections as the case mortality rate,

beware that (7) applies only to the late time scenario when every case is recorded55

and its outcome (in terms of recovery versus death) is accounted for. In this

3

. CC-BY-NC-ND 4.0 International licenseIt is made available under a

is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted April 6, 2020. ; https://doi.org/10.1101/2020.03.31.20049452doi: medRxiv preprint

Existing conﬁrmed Cumulative conﬁrmed Deaths Recovered

Hunan 1018 4 1014

Anhui 990 6 984

Jiangxi 935 1 934

Jiangsu 631 0 631

Chongqing 576 6 570

Fujian 296 1 295

Guizhou 146 2 144

Tianjin 136 3 133

Shanxi 133 0 133

Jilin 93 1 92

Xinjiang 76 3 73

Ningxia 75 0 75

Qinghai 18 0 18

Xizang 1 0 1

Table 1: Number of Covid-19 related cases in each of 3 categories, and for 13 provinces of

China[2, 6, 7] where every infected person has either recovered or died.

respect, [1] is correct in asserting that the ratio does not yield the true prob-

ability of death if it is evaluated in the middle of an outbreak. Unfortunately,

since the Covid-19 pandemic is far from over, and the assumption of a unique

14 day time delay between diagnosis and death is unrealistic, [5], neither [1] nor60

the conventional method would yield the true value of the case mortality rate.

Nevertheless, even at the current stage of the Covid-19 outbreak there are

regions of China in which the verdict of every conﬁrmed case of infection has

been delivered by the date of 16 March, 2020. The data for these regions (and

their sources) are tabulated below.65

Tallying the numbers, one obtains the case mortality rate P

0

in accordance

with (7) as P

0

= 0.527 ± 0.001 %, where the error is due to random Poisson

√

N counting uncertainties in the cumulative death count N = 27. This is the

4

. CC-BY-NC-ND 4.0 International licenseIt is made available under a

true mortality rate for the regions of concern, which is free from the uncertainty

in the time delay between diagnosis and death, but ignores the asymptomatic70

cases of infection. Evidently the rate is considerably lower than the two less

accurate approaches quoted at the beginning of the paper, although the reason

could be the smaller number of conﬁrmed cases in these regions, which allows

each patient to receive more attentive and higher quality health service.

5

. CC-BY-NC-ND 4.0 International licenseIt is made available under a

##### References

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TL;DR: This poster presents a poster presented at the 2016 International Conference of the Association for the Advance Study of Childbirth and Materno-fetal and Obstetrics entitled “Advances in Maternity and Childbirth Education and Research: Foundations of Pediatric Infectious Diseases.”

Abstract: Materno-fetal and Obstetrics Research Unit, Department Woman-Mother-Child, Lausanne University Hospital, 1011 Lausanne, Switzerland (DB, LP, GF); CHESS Center, The First Hospital of Lanzhou University, Lanzhou, Gansu, China (XQ); Division of Pediatric Infectious Diseases, David Geffen School of Medicine at UCLA, Los Angeles, CA, USA (KN-S); Aix Marseille Université, Institut de Recherche pour le Développement, Assistance Publique–Hôpitaux de Marseille, Service de Santé des Armées, Vecteurs— Infections Tropicales et Méditerranéennes (VITROME), Institut Hospitalo-Universitaire– Méditerranée Infection, Marseille, France (DM); and Laboratoire Eurofins—Labazur Guyane, French Guiana, France (DM)

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### "A Covid-19 case mortality rate with..." refers background in this paper

...(4) It can also be seen from (3) that N(t) is just the convolution integral between the two functions n(t) and p(t)....

[...]

...limit, the integral (3) may readily be evaluated to yield N(t) = p0n(t− t0), (5) which means p0 is exactly the ratio defined by [1]....

[...]

...0 N(t)dt = ∫ ∞ −∞ N(t)dt = ∫ ∞ −∞ dt ∫ ∞ −∞ dt′n(t− t′)p(t′)dt′, (6) where use was made of (1), (2), and (3)....

[...]

...(3) The case mortality rate is then given by 40...

[...]

••

TL;DR: The elevated death risk estimates are probably associated with a breakdown of the healthcare system, indicating that enhanced public health interventions, including social distancing and movement restrictions, should be implemented to bring the COVID-19 epidemic under control.

Abstract: Since December 2019, when the first case of coronavirus disease (COVID-19) was identified in the city of Wuhan in the Hubei Province of China, the epidemic has generated tens of thousands of cases throughout China. As of February 28, 2020, the cumulative number of reported deaths in China was 2,858. We estimated the time-delay adjusted risk for death from COVID-19 in Wuhan, as well as for China excluding Wuhan, to assess the severity of the epidemic in the country. Our estimates of the risk for death in Wuhan reached values as high as 12% in the epicenter of the epidemic and ≈1% in other, more mildly affected areas. The elevated death risk estimates are probably associated with a breakdown of the healthcare system, indicating that enhanced public health interventions, including social distancing and movement restrictions, should be implemented to bring the COVID-19 epidemic under control.

159 citations

### "A Covid-19 case mortality rate with..." refers background in this paper

...limit, the integral (3) may readily be evaluated to yield N(t) = p0n(t− t0), (5) which means p0 is exactly the ratio defined by [1]....

[...]