🧖🏻 🥂 👨🏿‍🔬 Mortality, mortality, coronavirus and matan 🧖🏽 👨🏿‍🔧 🤹🏻

To begin with, let's deal with two important epidemiological concepts: mortality and mortality. Immediately make a reservation that Wikipedia (both Russian and English) gives an erroneous definition of mortality, which is confusing.

Mortality is the probability of dying if a patient is diagnosed with a disease. Here is a quote from a scientific article :

one of the most important epidemiologic quantities to be determined is the case fatality ratio — the proportion of cases who eventually die from the disease.

Mortality is the ratio of the number of deaths from a disease to the size of a population over a period of time . Usually, they count how many deaths per 100 thousand people per unit of time. Mortality is directly related to mortality: this is the product of the probability of getting sick (over a specific time period) and mortality. In fact, in order to die from an illness, she must first become infected, and then, if she’s not lucky ...

High mortality does not automatically mean that mortality is also high. For example, a disease kills with a probability of 1, but affects only 0.1% of the population, say, per year (the Ebola virus behaves in a similar way, for example). Then the mortality rate will be only 1/1000. While a disease with a mortality of one hundred times less (0.01) can have 10 times higher mortality (1/100) if it affects the entire population over the same period.

Mortality clearly depends on time - over time, the number of infected people, as a rule, increases, and therefore mortality increases. Mortality does not depend on time explicitly, but, for example, may decrease over time if a medicine is found / invented.

We can also say that mortality is the conditional probability of death under the condition of the disease, and mortality is the probability of dying from the disease over a certain period of time.

Mortality, in turn, is divided into Case Fatality Ratio (CFR) and Infection Fatality Ratio (IFR) :
CFR is the mortality rate calculated on confirmed cases. This indicator has a pitfall: in the first place, those who have pronounced symptoms are usually tested. Therefore, we can say that in a first approximation, CFR is the probability of death, subject to the presence of the disease and severe symptoms.

IFR- this is mortality, that is, the probability of death in the presence of the disease. This indicator also includes mild and asymptomatic cases of the disease and therefore can be much less than CFR. Accurately calculating this indicator is almost impossible, because few people will test the entire population to take into account asymptomatic carriers, too, but it can be estimated.

In epidemiology it is extremely important to be able to assess mortality at the beginning of an epidemic in order to be able to take measures commensurate with the severity of the disease. Unfortunately, this is extremely difficult to do and now we will find out why.

One of the most popular mortality assessment methods is a simple formula: Deaths / Cases, that is, the number of deaths from the disease divided by the total number of infected by the current moment. Unfortunately, this highly popular assessment (also called the naive method) has a congenital flaw, which is illustrated by the following example:

Let a certain disease kill in exactly 1 month with a probability of 1. Let also the number of cases doubled every 10 days. Suppose x people died in the first month . But there are 7 times more sick people who have not died yet! Just because in a month there will be three doublings of the initial patient population (and this is an increase of 8 times). Therefore, the method, when dividing the number of deaths by the number of those diagnosed, will estimate mortality only in $\frac{x}{x+7x}=\frac{1}{8}=12.5$ %!

This underestimation of the naive method leads to false speculation. For example, during the SARS epidemic, a naive estimate grew over time, generating rumors that the virus was evolving into a more deadly killer. And the reason for this is simple mathematics: the growth in the number of cases slows down, which reduces the underestimation of mortality by a naive estimator.
Thus, it can be said that the naive method underestimates mortality, reducing it in

e^{b t_{d e a t h}}

$e^{bt_{death}}$ times where

t_{d e a t h}

$t_{death}$ Is the time from infection to death, and b is the parameter characterizing the time of doubling the number of infected. But, unfortunately, such an amendment does not work well in real life, because patients do not die strictly after a certain period of time by organized groups, but randomly. Let's take this into account and derive a correction formula that will be applicable in real life.

a little bit of very simple math

, , n- . :

c_{1} P (d a y = j, d e a t h)

$c_1P(day=j, death)$ ,

с_{1}

$_1$ — ,

P (d a y = j, d e a t h)

$P(day=j, death)$ — j .

P (d a y = j, d e a t h)

$P(day=j, death)$ — , j- . :

P (d a y = j, d e a t h) = P (d a y = j | d e a t h) P (d e a t h)

$P(day=j, death) = P(day = j|death)P(death)$ ,

P (d e a t h)

$P(death)$ — ( ,

P (d e a t h | d i s e a s e)

$P(death|disease)$ , ).

n:

d e a t h s_{1} = \sum_{j = 1}^{n} c_{1} P (d a y = j | d e a t h) P (d e a t h)

$deaths_1=\sum_{j=1}^nc_1P(day=j|death)P(death)$

( n ) :

d e a t h s_{t o t a l} = \sum_{i = 1}^{n} \sum_{j = i}^{n} c_{i} P (d a y = j - i | d e a t h) P (d e a t h)

$deaths_{total}=\sum_{i=1}^n\sum_{j=i}^nc_iP(day=j-i|death)P(death)$

c_{i} = N_{0} (e^{b i} - e^{b (i - 1)})

$c_i = N_0(e^{bi} - e^{b(i-1)})$ ( , ). :

\frac{D e a t h s}{C a s e s} = \frac{P (d e a t h) \sum_{i = 1}^{n} \sum_{j = i}^{n} c_{i} P (d a y = j - i | d e a t h)}{N_{0} e^{b n}}

$\frac{Deaths}{Cases}=\frac{P(death)\sum_{i=1}^n\sum_{j=i}^nc_iP(day=j-i|death)}{N_0e^{bn}}$

bias-corrected :

P (d e a t h) = \frac{D e a t h s}{C a s e s} b i a s

$P(death) = \frac{Deaths}{Cases}bias$

b i a s = \frac{N_{0} e^{b n}}{\sum_{i = 1}^{n} N_{0} (e^{b i} - e^{b (i - 1)}) \sum_{j = i}^{n} P (d a y = j - i | d e a t h)}

$bias=\frac{N_0e^{bn}}{ \sum_{i=1}^nN_0(e^{bi} - e^{b(i-1)})\sum_{j=i}^nP(day=j-i|death)}$

= \frac{e^{b n}}{\sum_{i = 1}^{n} (e^{b i} - e^{b (i - 1)}) \sum_{j = i}^{n} P (d a y = j - i | d e a t h)}

$=\frac{e^{bn}}{ \sum_{i=1}^n(e^{bi} - e^{b(i-1)})\sum_{j=i}^nP(day=j-i|death)}$

\frac{D e a t h s}{C a s e s}

$\frac{Deaths}{Cases}$

b i a s

$bias$ .

Now let's try to evaluate this bias to assess mortality in the early period of the development of the epidemic of coronavirus infection in the Chinese city of Wuhan. To do this, we use the following assumptions: the doubling time for the number of cases is 5 days, and the average time from registration to death is 18 days.

substantiation of assumptions

(5 ) (22.3 )
, . , 4.25 . , 18 .

We also assume that the day of death has a Poisson distribution :

P (d a y = j | d e a t h) \sim P o i s s o n (18)

$P(day = j|death) \sim Poisson(18)$

Substituting the values in the formula, we find that the naive method underestimates mortality by about 9 times. Thus, CFR for confirmed cases is about 18%! I emphasize that CFR does not include undocumented patients, the number of which was estimated by Chinese scientists: according to their model, 86% of cases were not registered. This allows us to calculate IFR: IFR = 0.14 * CFR = 2.5%. These estimates are perfectly consistent with the estimates of CFR (18%, 11% -81%), and IFR (1%, 0.5% -4%), which were obtained by specialists of Imperial College London.

It is important to understand that the IFR value should not be used to assess the likelihood of dying from a disease, since the probability of dying from a disease depends on many factors:

age
the presence of concomitant diseases
hospital congestion
viral load
etc.

Then why is it so important to know IFR at least approximately? You need to know this in order to be able to compare with known diseases. For example, the lethality (IFR) of influenza is 0.01%, which is at least ten times lower. Given the fact that coronavirus is more contagious (R0> 2 versus about 1.3 in influenza), this can lead to tens of millions of deaths worldwide, as the flu can take up to 650,000 lives a year. Therefore, in no case should not be considered that "it's just the flu."

This article has the following objectives: to explain the difference between mortality and mortality, to explain what CFR and IFR are (so that people do not look for the difference between Italy and other countries in the level of medicine), to explain that one cannot rely on estimates obtained by the Deaths / method Cases, and for mathematics lovers like me, I also figure out how to fix this method.

Mortality, mortality, coronavirus and matan

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