# Does COVID-19 have a case fatality rate of 41%? Is this formula correct?

According to the data on the Johns Hopkins Coronavirus Tracker, as of 3rd February 2020 there were 17491 confirmed cases of COVID-19 globally, 536 total recoveries and 362 deaths. From my non-expert calculation this implies a mortality rate of:

`(Nd / (Nd + Nr)) * 100 = 41%`

where:

Nd is the total number of deaths, Nr is the total number of full recoveries.

This leaves 16593 people still suffering from the disease who have neither recovered or died.

This is in stark contrast to the publicly disseminated value of ~2% mortality, so have I made a mistake in my calculation or assumptions, or is COVID-19 much more dangerous than commonly claimed?

[After a helpful discussion in the comments, 'mortality rate' is not the correct term to use here, instead I should say 'Case Fatality Ratio'.]

• Now (as of 2020-02-04) it shows 697 recoveries and 427 fatalities (out of 20679 confirmed cases), so the calculated mortality rate dropped to about 38%, and it confirms that it's too early to evaluate it, this value is very short-term. Feb 4, 2020 at 10:53
• @Jan I don't really understand why that is considered a sensible calculation. If someone is still diseased then by definition you don't know if they will a) die or b) recover so it doesn't make sense to include them in the method. Feb 4, 2020 at 12:20
• I think there is a problem with your formula : 41% of 17491 would mean around 8000 deaths ! The actual rate is (Ndeaths / (total contaminated living + deaths) )*100 , leading to around 2%
– Benj
Feb 4, 2020 at 12:30
• @Jan I agree that you cannot include "diseased (not-yet-recovered) persons into mortality rate", which is why I haven't. The calculation is just a comparison between those who had the disease and died and those who had the disease and recovered. Feb 4, 2020 at 12:41
• @Jan But that is not a sensible way of calculating mortality, by using exactly your argument I could claim that only 0.536% of people will recover, which is clearly nonsensical. Feb 4, 2020 at 13:12

The definition of mortality rate that you've given does not match any practical definition I'm familiar with.*

When people talk about the mortality rate of a disease, what they usually mean is the case fatality rate or the death-to-case ratio, which is simply defined as Nd / Ni, where Nd is the number of deaths attributed to the disease over a given time period and Ni is the total number of new cases of the disease observed during the same time period. By this definition, the current case fatality rate of 2019-nCov according to your quoted figures is 362 / 17491 ≈ 2.07%.

(The tracker seems to have been updated since you asked your question, and now lists a total of 20679 confirmed cases and 427 deaths, for a CFR of 427 / 20679 ≈ 2.06%.)

*) As a theoretical definition of the mortality rate in the long run, when all infected patients have either died or recovered, it can sort of make sense. But then it becomes equivalent to the usual definition of the case fatality rate.

To compare this with your definition of "mortality rate" (as Nd / (Nd + Nr), where Nr is the number of individuals who have recovered from the disease), we need to start by observing that there's no single universal and unambiguous definition of what "recovering from a disease" means. Commonly used definitions tend to be something like "no symptoms for X days" and/or "viral load below N particles per mL for X days" or simply "whenever a doctor declares that you're healthy again and lets you out of the hospital".

Now, let's say that we're using a (somewhat) objective definition of recovery like "no detectable symptoms for two days". The first observation is that any epidemic first observed less than two days ago would, according to your definition, inevitably have a mortality rate of 100% simply because none of the people infected so far would have had time to be considered definitely recovered yet. (That is assuming that at least one person had died from the infection; otherwise both the numerator and the denominator would be zero, and the rate thus undefined.)

Further, even after some of the earliest cases have been symptom-free long enough to be counted as recovered, your definition would still yield a highly upwards biased estimate of the "true" long-term fatality rate during the early phase of the epidemic, when the number of new cases per day is still increasing. This is because, for most infectious diseases, any deaths typically occur when the disease is at its most severe state, whereas those who survive the disease will then experience a gradual decline in symptoms as their immune system succeeds in halting and reversing the progress of the infection.

For an illustrative example, let's consider a hypothetical disease with a theoretical 1% long-term average CFR — that is to say, exactly 1% of all (recognizably) infected patients will die of the disease. Let's further assume that this disease typically takes two days to progress from the initial onset of recognizable symptoms to the state of maximum severity, which is when most of the deaths occur. After this, assuming that the patient survives, the symptoms gradually decline over the following three days. As remission is possible (but rare), doctors will generally consider a patient recovered only after showing no symptoms for at least two days. Thus, a typical case would progress as follows:

onset of symptoms → increasing symptoms (2 days) → peak severity → declining symptoms (3 days) → no symptoms → observation (2 days) → officially recovered (total time: approx. 7 days from onset)

or, for the 1% of patients for whom the disease is fatal:

onset of symptoms → increasing symptoms (2 days) → death (total time: approx. 2 days from onset)

Now, let's assume that, during the early period of an epidemic when the infection is still spreading exponentially, the number of new cases increases by a factor of 10 every three days. Thus, during this period, the number of new cases, recoveries and deaths per day might grow approximately as follows (assuming for the sake of the example that exactly 1%, rounded down, of the patients diagnosed on each day will die two days later):

``````    |     cases     |   recovered   |     deaths    |         |            |
day |   new | total |   new | total |   new | total | Nd / Ni | Nd/(Nd+Nr) |
----+-------+-------+-------+-------+-------+-------+---------+------------+
1 |     1 |     1 |     0 |     0 |     0 |     0 |   0.00% |        N/A |
2 |     2 |     3 |     0 |     0 |     0 |     0 |   0.00% |        N/A |
3 |     5 |     8 |     0 |     0 |     0 |     0 |   0.00% |        N/A |
4 |    10 |    18 |     0 |     0 |     0 |     0 |   0.00% |        N/A |
5 |    20 |    38 |     0 |     0 |     0 |     0 |   0.00% |        N/A |
6 |    50 |    88 |     0 |     0 |     0 |     0 |   0.00% |        N/A |
7 |   100 |   188 |     0 |     0 |     0 |     0 |   0.00% |        N/A |
8 |   200 |   388 |     1 |     1 |     0 |     0 |   0.00% |       0.0% |
9 |   500 |   888 |     2 |     3 |     1 |     1 |   0.11% |      25.0% |
10 |  1000 |  1888 |     5 |     8 |     2 |     3 |   0.16% |      27.3% |
11 |  2000 |  3888 |    10 |    18 |     5 |     8 |   0.21% |      30.8% |
12 |  5000 |  8888 |    20 |    38 |    10 |    18 |   0.20% |      32.1% |
``````

As you can see from the table above, naïvely calculating the case fatality rate as (total number of deaths) / (total number of cases) during this exponential growth period does underestimate the true long-term CFR by a factor of (in this case) about 5 due to the two-day lag time between infection and death. On the other hand, using your formula of (total deaths) / (total deaths + recovered) would overestimate the true CFR by a factor of about 30!

Meanwhile, let's assume that, after the first 12 days, the growth of the epidemic saturates at 10,000 new cases per day. Now the total numbers will look like this:

``````    |     cases     |   recovered   |     deaths    |         |            |
day |   new | total |   new | total |   new | total | Nd / Ni | Nd/(Nd+Nr) |
----+-------+-------+-------+-------+-------+-------+---------+------------+
13 | 10000 | 18888 |    50 |    88 |    20 |    38 |   0.20% |      30.2% |
14 | 10000 | 28888 |    99 |   187 |    50 |    88 |   0.30% |      32.0% |
15 | 10000 | 38888 |   198 |   385 |   100 |   188 |   0.48% |      32.8% |
16 | 10000 | 48888 |   495 |   880 |   100 |   288 |   0.59% |      24.7% |
17 | 10000 | 58888 |   990 |  1870 |   100 |   388 |   0.66% |      17.2% |
18 | 10000 | 68888 |  1980 |  3850 |   100 |   488 |   0.71% |      11.2% |
19 | 10000 | 78888 |  4950 |  8800 |   100 |   588 |   0.74% |       6.3% |
20 | 10000 | 88888 |  9900 | 18700 |   100 |   688 |   0.77% |       3.5% |
21 | 10000 | 98888 |  9900 | 28600 |   100 |   788 |   0.80% |       2.7% |
``````

As you can see, the two measures of mortality rate do eventually start converging as the growth of the epidemic slows down. In fact, in the long run, as the majority of patients either recover or die, they do both end up converging to the "true" long-term case fatality rate of 1%. But by then, the epidemic will be basically over.

There are various ways to obtain a more accurate estimate of the long-term fatality rate even during the early exponential growth phase of an epidemic. One such method would be to look at the outcomes of a single cohort of patients diagnosed at the same time. For our hypothetical example epidemic, looking e.g. at just the 1000 patients diagnosed on day 10, we could get an accurate estimate of the CFR by day 12 simply by dividing the 10 deaths within that cohort by the total number of patients in the cohort. Furthermore, observing multiple cohorts would give us a pretty good idea of how long after diagnosis we would need to wait before the estimated case fatality rate for each cohort gets close to its final true value.

Unfortunately carrying out this kind of cohort analysis for 2019-nCov would require more detailed information than the tracker you've linked to provides. Even the time series spreadsheet the tracker links to doesn't directly provide such detailed cohort data, although it might be possible to obtain better estimates from it by making some more or less reasonable assumptions about the typical progress of the disease.

Addendum: A few preliminary cohort studies of the kind I describe above do appear to have already been published for 2019-nCoV.

In particular, "A novel coronavirus outbreak of global health concern" by Wang et al. and "Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China" by Huang et al., both published on January 24 in The Lancet, note that, out of the first 41 patients diagnosed with 2019-nCoV before Jan 2, 2020 in Wuhan, six had died (and 28 had been discharged, leaving seven hospitalized) by Jan 22, giving a case fatality rate of 14.6% in this cohort.

However, they do advise treating this figure with due caution, noting a number of reasons (besides just the small number of cases examined) why it may not fully reflect the eventual long-term CFR:

"However, both of these [CFR] estimates [of 14.6% from the 41 patient cohort and of 2.9% from all 835 cases confirmed at the time of writing] should be treated with great caution because not all patients have concluded their illness (ie, recovered or died) and the true number of infections and full disease spectrum are unknown. Importantly, in emerging viral infection outbreaks the case-fatality ratio is often overestimated in the early stages because case detection is highly biased towards the more severe cases. As further data on the spectrum of mild or asymptomatic infection becomes available, one case of which was documented by Chan and colleagues, the case-fatality ratio is likely to decrease."

There's also a later paper titled "Epidemiological and clinical characteristics of 99 cases of 2019 novel coronavirus pneumonia in Wuhan, China: a descriptive study" by Chen et al., published on Jan 30, that examines a cohort of 99 patients diagnosed between Jan 1 to Jan 20 and reports a CFR of 11% within this cohort. However, the study only followed these patients up to Jan 25, by which time more than half of them (57 out of 99) still remained hospitalized.

• Thanks! I really appreciate the modelling that you have done here, it really makes the point clear. Feb 4, 2020 at 17:28
• So the upshot of this is that we don't really know what the mortality rate is right now, though from what you showed it looks like it could be quite a bit higher than current estimates of 2.5%, but also could mercifully be quite a bit lower than OP's estimate?
– bob
Feb 6, 2020 at 20:40
• @bob: Exactly. FWIW, I did manage to find a couple of early cohort studies for 2019-nCoV and added them to my answer above. Those studies suggest a CRF between 10% to 15% among the earliest cases in Wuhan, although they also mention various reasons why this ratio may still be biased. Feb 6, 2020 at 21:31
• WHO current given date is 3,4%, higher than previously 2% estimates cnbc.com/amp/2020/03/03/… Mar 4, 2020 at 0:13
• @Pablo I think this 3,4% could vary depending on where you are (country). I was thinking that, for instance if your country doesn't have enough meds or enough precautions, then this could affect the percentage. Also the weather is another fact that one should take into account. If you live in country where there's a "lot of sun", then that could stop the virus at least a little. Mar 4, 2020 at 2:58

The equation you use for mortality is only really useful in the very long term for a known disease, when most cases have resolved.

It's not very informative in the short-term, when the vast majority of total cases are neither deaths nor recoveries.

Right now, the vast majority of people diagnosed have a mild illness and are very unlikely to die, but it takes a long time for them to be considered in the "recovered" category. Additionally, many of those who have died are particularly vulnerable. From WHO:

As with other respiratory illnesses, infection with 2019-nCoV can cause mild symptoms including a runny nose, sore throat, cough, and fever. It can be more severe for some persons and can lead to pneumonia or breathing difficulties. More rarely, the disease can be fatal. Older people, and people with pre-existing medical conditions (such as, diabetes and heart disease) appear to be more vulnerable to becoming severely ill with the virus.

Estimates for mortality that you see in the news might instead be based on deaths/cases, or are based on expert comparisons to past epidemic coronavirus strains and knowledge of the typical course of the illness.

Additionally, we don't know how accurate the numbers are, especially for cases. There may be many more mild cases that go unreported.

There won't be good estimates of the actual mortality rate until more time has passed, and even in that case it is unlikely that a single number will be very informative. Instead, the risk will vary by age and other factors. Good sources for information, like the WHO, don't report mortality rates: they are only reporting cases and deaths at this time.

Some good sources for further information:

https://www.who.int/emergencies/diseases/novel-coronavirus-2019

https://www.cdc.gov/coronavirus/2019-nCoV/summary.html

https://www.nhs.uk/conditions/wuhan-novel-coronavirus/

• Comments are not for extended discussion; this conversation has been moved to chat. Feb 4, 2020 at 17:13

I'd like to chime in with an explanation of what exactly is wrong with the calculation offered in the question, rather than just saying "it's a wrong formula". Understanding the "whys" of the fallacy is important. So I'll try to answer your question from the math point of view.

TL;DR: The root cause of the fallacy is that recovery takes much longer that death.

`(Nd / (Nd + Nr)) * 100 = 41%`
where: Nd is the total number of deaths,
Nr is the total number of full recoveries.

That formula (and the logic behind it) is correct as long as `Nd` and `Nr` both refer to the same fixed group of people. That is, if we had picked `N` infected people, waited for them all to reach the final state (recovery or death), and put those `Nr` and `Nd` to that formula above - then yes, it would give the statistical mortality rate in that group.

However, the current counts of recovery/death outcomes do not refer to the same group. `Nd` in each WHO report refers to the group of all people infected thus far since the start of the outbreak. But the final outcome of all people in that group is yet unknown. Daily `Nr` refers only to a subgroup of all those infected (excluding those unknowns), see? So you can't take `Nd` and `Nr` from a WHO report and put those numbers to that formula - that would be apples and oranges...

To illustrate this point, consider a grossly simplified imaginary situation:
there's a disease which may lead to death on the 3rd day, while the rest of infected people will fully recover on the 15th day. In that case, `Nd` in the official report would encompass all people infected 3 days ago and before, while `Nr` would encompass all people infected 15 days ago and before. Given the high flow of new confirmed cases coming each day, the difference between those two groups is huge: it is all those people infected in 12 days!

In our real case that difference is far greater than `Nr` and `Nd` combined, which means the error from ignoring that difference renders the calculation totally useless. (Well, it's useful as an absolute upper limit, but no more).

• What is the official name of the thing this formula calculates? It's surely not "mortality rate" as it is named in the question.
– Jan
Feb 4, 2020 at 15:43
• @Jan, it calculates the probability of one given outcome of two possible outcomes for a given set of experiments. Feb 4, 2020 at 15:45
• OK, does this have any more fixed official name?
– Jan
Feb 4, 2020 at 15:47
• I'm not really proficient in English official words (even less so with medical ones). Let me scan Wikipedia article on discrete probability theory to see if it has some. Feb 4, 2020 at 15:51
• This answer explains why medical research typically refers to things like "5 year survival rate" for many diseases and syndromes (e. g. most cancers). That way you don't have to worry about when to count people as survivors-it's "five years after diagnosis." Of course, we can't wait 5 years to start collecting stats on 2019-nCoVa. Feb 4, 2020 at 16:20

According to earlier answers, in this early phase of 2019-nCoV, Nd/(Nd+Nr) is an overestimator, and Nd/Nc is an underestimator.

Since the currently bantered about rate matches the underwestimator Nd/Nc, you are correct that 2019-nCoV is more 'dangerous' than commonly claimed. I used quotes because dangerous a squirmy term.

Noting that Nd/Nc equals Nd/(Nd+Nr) after the epidemic is over, a better estimate would be to track the two quotients over time, and extrapolate their curves to the point they meet. That would still be a biased estimator, but less so than either on it's own. I'm guessing there are more sophisticated estimators with less bias, and I've posted that question here:

What is a sophisticated estimate of the 2019-nCoV fatality rate?

• Though, Nc is likely to be under-estimated itself (people with mild symptoms do not get diagnosed), so Nd/Nc is not strictly an underestimator.
– jpa
Feb 6, 2020 at 13:28
• @Mick: see my answer below for a possibly better estimator.
– Fizz
Mar 28, 2020 at 13:31

I understand what you're trying/hoping to do here, but the correction method you try to apply is unsuitable. You need to explicitly account for the time delays to deaths and consider a confined population of cases or try to infer from a closed sample a correction factor to apply to the open-ended/ongoing epidemic. Such a study was recently published based on the Diamond Princess (cruise ship) cases, using the information collected therein to correct (in this paper) the data on China.

In real time, estimates of the case fatality ratio (CFR) and infection fatality ratio (IFR) can be biased upwards by under-reporting of cases and downwards by failure to account for the delay from confirmation to death. Collecting detailed epidemiological information from a closed population such as the quarantined Diamond Princess cruise ship in Japan can produce a more comprehensive description of asymptomatic and symptomatic cases and their subsequent outcomes. Our aim was to estimate the IFR and CFR of coronavirus disease (COVID-19) in China, using data from passengers of the Diamond Princess while correcting for delays between confirmation and death and for the age structure of the population.

During an outbreak, the so-called naive CFR (nCFR), i.e. the ratio of reported deaths date to reported cases to date, will underestimate the true CFR because the outcome (recovery or death) is not known for all cases, assuming all cases are detected. We can estimate the true denominator for the CFR (i.e. the number of cases with known outcomes) by accounting for the delay from confirmation to death. We assumed that the delay from confirmation to death followed the same distribution as the estimated time from hospitalisation to death, based on data from the COVID-19 outbreak in Wuhan, China, between 17 December 2019 and 22 January 2020, accounting for underestimation in the data as a result of as-yet-unknown disease outcomes [...]

To adjust the CFR to account for delay to outcome, we use the method developed in Nishiura et. al (2009) where case and death incidence data are used to estimate the number of cases with known outcomes, i.e. cases where the resolution, death or recovery, is known to have occurred:

where ct is the daily case incidence at time t, (with time measured in calendar days), ft is the proportion of cases with delay t between onset or hospitalisation and death; ut represents the underestimation of the known outcomes and is used to scale the value of the cumulative number of cases in the denominator in the calculation of the cCFR. Given that asymptomatic infections are typically not reported, especially during an ongoing outbreak of a novel infection, this correction is normally used to calculate the cCFR. However, because of the high level of testing on the cruise ship, we were able to use this correction to calculate the corrected IFR (cIFR). After that, we used the measured proportions of asymptomatic to symptomatic cases on the Diamond Princess to scale the cIFR to estimate the cCFR. [...]

We estimated that the all-age cIFR on the Diamond Princess was 1.3% (95% confidence interval (CI): 0.38–3.6) and the cCFR was 2.6% (95% CI: 0.89–6.7). However, as the age distribution on the ship was skewed towards older individuals (mean age: 58 years), we also report age-stratified estimates. Using the age distribution of cases and deaths on the ship to estimate for only individuals 70 years and older, the cIFR was 6.4% (95% CI: 2.6–13) and the cCFR was 13% (95% CI: 5.2–26). The 95% CI were calculated with an exact binomial test, with death count and either cases or known outcomes (depending on whether it was an interval for the naive or corrected estimate).

Using an approach similar to indirect standardisation, we used the age-stratified nCFR estimates reported in a large study in China to calculate the expected number of deaths of people on board the ship in each age group, (assuming this nCFR estimate in the standard population was accurate). This produced a total of 15.15 expected deaths, which corresponds to a nCFR estimate of 5% (15.15/301) for the Diamond Princess, which falls within the top end of our 95% CI. As our cCFR for Diamond Princess was 2.6% (95% CI: 0.89–6.7), this suggests we need to multiply the nCFR estimates in China by a factor 52% (95% CI: 14–100) to obtain the correct value. As the raw overall nCFR reported in the data from China was 2.3%, this suggests the cCFR in China during that period was 1.2% (95% CI: 0.3–3.1) and the IFR was 0.6% (95% CI: 0.2–1.7). Based on cases and deaths reported in China up to 4 March 2020, the nCFR calculation was considerably higher than the cCFR we estimate here (based on data taken from [8], nCFR = 2,984/80,422 = 3.71% (95% CI: 3.58–3.84)). The confidence intervals calculated for China using an indirect standardisation method reflect the uncertainty in the Diamond Princess estimates, as it is carried forward in the scaling.

As you can see, if one does this correction properly, the "death rate" (cCFR) for Covid-19 is actually lower (than the nCFR).

If the above is too dense/technical of an explanation, the Nature news coverage of it:

Another team used data from the ship to estimate that the proportion of deaths among confirmed cases in China, the case fatality rate (CFR), was around 1.1% — much lower than the 3.8% estimated by the World Health Organization (WHO).

The WHO simply divided China’s total number of deaths by the total number of confirmed infections, says Timothy Russell, a mathematical epidemiologist at the London School of Hygiene and Tropical Medicine. That method does not take into account that only a fraction of infected people are actually tested, and so it makes the disease seem more deadly than it is, he says.

By contrast, Russell and his colleagues used data from the ship — where almost everyone was tested, and all seven deaths recorded — and combined it with more than 72,000 confirmed cases in China, making their CFR estimate more robust. [...]

The group also estimates that the infection fatality rate (IFR) in China — the proportion of all infections, including asymptomatic ones, that result in death — is even lower, at roughly 0.5%. The IFR is especially tricky to calculate in the population, because some deaths go undetected if the person didn’t show symptoms or get tested.

(Nature news says the [latter] paper had not been peer-reviewed/published, but in the meantime it has been published by Eurosurveillance, the same journal that had published the 1st Diamond Princess paper.)

I should also noted that an 8th death was reported much later (March 20) in relation to the Diamond Princess. It probably doesn't substantially change the conclusions of that paper (which included only the 7 reported deaths you see in the graph.)