Probability theory and Bayes' theorem show - a single test for COVID-19 is meaningless and the introduction of strict quarantine is inevitable

Although this information is irrelevant for the island of Hainan and for China, where I am currently and where there have been no COVID-19 patients since February 2020, the question is still interesting when the borders with China will be opened, and for this it is necessary for the epidemic to end. what is possible in In the case of the creation of effective means of antivirus control, which has already been written on the site earlier as it happened in China, everything develops differently in Russia.


Based on open data for the United States, we have the following conditions for the calculation (statistics for the United States looks more plausible, so we calculate using this country for a sample, others will be similar): the frequency of spread of COVID-19 among the US population is currently 0.037, diagnostic methods with a probability of 0.95 reveal a COVID-19 patient, but at the same time the probability of a false positive and false negative test results are, according to various sources, from 30 to 70%, we will accept these as a probability, therefore, with a value of 0.5, such large values ​​are due to the fact that often tests are taken from the subjects incorrectly, without strict adherence to the instructions. Thus, our task is reduced to finding the probability that a person is sick if he was recognized as healthy when tested for COVID-19.


Let's denote through W - the event that the person is sick with COVID-19 in reality, "W" - the event that the test showed that the person is sick, through Z - the event that the person is healthy in reality, and through "Z" - the event that the test showed that the person is healthy. Then the given conditions can be described as follows: conditional probabilities:


P ("W" | W) = 0.95;


P ("W" | Z) = 0.5;


P (W) = 0.037;


P (Z) = 0.963


The probability of being sick, if it was recognized as healthy when tested for COVID-19, is equal to the conditional probability P (W | "Z"), to find it, we calculate the total probability of being recognized as healthy:


P ("Z") = 0.963 × 0.95 + 0.037 × 0.5 = 0.91485 + 0.03705 = 0.9519


The probability that he is sick if he was recognized as healthy when tested for COVID-19 is equal to the conditional probability P (W | "Z"): P (W | "Z") = 0.037 × 0.5 / 0.9519 = 0.039


That is, 4% of the so-called "healthy" with a certificate that they do not have COVID-19 will actually be sick with COVID-19, and after these people are admitted to truly healthy people, these 4% will be enough to infect everyone the rest, calculations for Russia and European countries give A similar situation - performing a single test for COVID-19 - is completely pointless and only misleading and contributes to the spread of COVID-19. Let's now see what happens if we perform several tests for COVID-19, and also decide how many tests for COVID-19 need to be performed in order to more or less consider a person healthy and allow social contacts with him?


Bayes' theorem from the theory of probability gives an answer to this, the probability that a person is sick after receiving a repeated result "healthy" can be calculated using the Bayes formula: P (W | "Z", "Z") = 0.037 × 0.5 × 0 .5 / (0.963 × 0.95 × 0.95 + 0.037 × 0.5 × 0.5) = 0.00925 / (0.8691075 + 0.00925) = 0.0105


That is, even after the second test for COVID-19, the situation remains dangerous for others! At least one person out of 100 with two certificates of absence of COVID-19 will be sick and the epidemic will continue! If the calculations continue in the same way, then at least only three negative tests for COVID-19 more or less guarantee the absence of patients among people with negative tests for COVID-19.


Thus, only the introduction of triple tests for COVID-19 can stop the continuation of the epidemic, taking into account the isolation of patients and the widespread use of anti-epidemic measures, otherwise the introduction of strict quarantine will be inevitable.


#coronavirus, # COVID-19


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