As we all know, nucleic acid detection is the most effective method to find COVID-19 infection as soon as possible, and it plays an indispensable role in epidemic prevention and anti-epidemic. Therefore, how to detect nucleic acid timely, accurately and efficiently has become a major issue that we must pay close attention to. From a mathematical point of view, it is an optimization problem of experimental design, that is, to detect as many people as possible with the least amount of detection and the least cost in the shortest time.
At present, there are two common nucleic acid detection methods: single detection and mixed detection. What is "single detection" means putting the samples collected by everyone into a special test tube for "one-on-one" detection. The test results are accurate. However, if there are many people who need to be tested, such as hundreds of thousands, millions or even tens of millions, the workload of "single inspection" is too large and the cost is too high, so it is difficult to complete the testing work in time.
In actual testing, there are often thousands or more people who need to be tested, and it is difficult to be competent only by "single inspection". At this time, "mixed inspection" is needed, that is, samples collected by several people are mixed and put into the same test tube for "many-to-one" inspection. If the test result is negative, then all these people pass; If the test result is positive, it means that there are virus carriers in these people, and further testing is needed for these people.
Suppose there are 100 "close contacts" in an epidemic area who need to be tested for nucleic acid, and the nucleic acid samples of these 100 people have been collected. If "single detection" is adopted, it needs to be detected 100 times; In low-risk areas, the virus infection rate is lower than 1% (that is, on average, only 1 person may carry COVID-19). In order to detect 100 people, we spent hundreds of times of time and cost. If "mixed inspection" is adopted, efficiency can be improved, time can be shortened and cost can be reduced.
At present, the common "1: 10 unified mixed inspection" in China is as follows: firstly, the collected samples of 10 people are numbered in turn, and each group of 10 people is divided into 10 groups, 1 to 10 groups.
Then, 10 samples of each group are "mixed" respectively (that is, samples of 10 people are mixed together and tested once); Which group of "mixed test" results are negative, which group will pass all at once; Which group of "mixed test" results are positive, indicating that there are virus carriers in this 10 population, and it is necessary to do "single test" for this 10 population respectively.
The most optimistic situation is that all ten groups of "mixed inspection" results are negative, 100 people all pass the inspection, and it only takes one tenth of the time and cost to complete the inspection.
Most likely, one of the ten groups is positive (assuming the second group) and the other groups are negative. Except for the second group 10, all the other 90 people passed the test.
Then the second group with positive "mixed test" will be subjected to one-on-one "single test", which can accurately find the virus carrier. Therefore, for a sample of 100 people, it only takes one fifth of the time and cost of "single inspection" to complete the same inspection. In other words, the biggest possibility is that "mixed inspection" can complete the same inspection in only one-fifth of the time and cost of "single inspection".
I don't know if you still remember that we introduced the dichotomy of "experimental design optimization" in "Xiaoxing said mathematics: the problem of poison detection in mice"
Theoretically, the current "unified mixed inspection" can be further improved from "dichotomy" to "dichotomy mixed inspection" The most likely situation when using "binary mixed inspection" is that it only takes one seventh of the time and cost of "single inspection" to complete the same number of inspections.
The key of "binary mixed test" is to gradually group the samples to be tested into half as much as possible: it is still assumed that the samples to be tested are 100 people, and these 100 people are sequentially numbered and grouped into half, and the numbers from 1 to 50 are A 1 and 565438+.
Then, the groups A 1 and A2 are "mixed" respectively. ...
Ideally, the results of "mixed test" in groups A 1 and A2 were negative, and all 100 people passed the test. The detection work was only completed twice.
The most likely situation is that the results of "mixed test" between A 1 and A2 group, one group is negative and the other group is positive. Assuming that the positive group is A 1 group, A 1 group needs to continue to use "dichotomy mixed test"; All 50 people in negative group A2 and A2 passed the test.
A 1 group is still divided into two groups, with numbers 1 to 25 being B 1 group and numbers 26 to 50 being B2 group.
Then, the B 1 group and B2 group were "mixed checked" respectively:
It is known that group A 1 was positive in the previous test, so the "mixed test" results of group B 1 and group B2 in this step can't show that both groups are negative, and the biggest possibility is one yin and one yang. Assuming that the positive group is B 1 group, B 1 group needs to continue to use "dichotomy mixed test"; The negative group was B2, and all 25 people in B2 passed the test.
Group B 1 continues to adopt "dichotomy mixed inspection", and the group B 1 ~13 is C 1 group, and the group C 14 ~ 25 is C2 group. The negative group was C2, C2 12 people all passed the test.
? C 1 group continues to adopt "dichotomy mixed inspection", and C 1 group is divided into two groups as far as possible, namely, D 1 group from1to 7, D2 group from 8 to 13, and D 1 group and D2 respectively. However, D2 group was negative, and all 6 people in D2 group passed the test.
D 1 group continues to adopt "dichotomy mixed inspection", and D 1 group is divided into two groups as far as possible, namely, E 1 group from1to No.4, and E2 group from No.5 to No.7, and "mixed inspection" is carried out on E 1 group and E2 group respectively. The biggest possibility is still one yin and one yang. The negative group was E2, and all three people in E2 passed the test.
Continue to use "dichotomy mixed inspection" for E 1 group, and divide all four people in E 1 group into two groups, none. 1 and No.2 are F 1 groups, No.3 and No.4 are F2 groups, and F 1 and F2 are "mixed" respectively. The biggest possibility is still one yin and one yang. The negative group was F2, and both of them passed the test.
There are only two people left in the positive group of F 1, so the continuous test can only be "single test", and the biggest test result may still be one yin and one yang. In any case, after six times of "binary mixed inspection" and two times of "single inspection",/kloc-0 * * 1 4 inspection, we have completed the inspection of 100 people, and the time and cost are only one seventh of that of "single inspection".
In fact, the above detection method can be improved, instead of six binary mixed inspections and two single inspections, the fifth binary mixed inspection is followed by four single inspections, and one * * * is also 14 single inspections, thus completing the detection of 100 people.
The above-mentioned "1: 10 uniform mixed detection" and "dichotomy mixed detection" have an important premise, that is, the known virus infection rate is less than1%; If the epidemic situation worsens, the risk level increases and the virus infection rate reaches 5%, how should nucleic acid detection be carried out? . Limited to space, I won't go into details, leaving only one question for everyone to think about:
Or a test of collecting samples from 100 people. Assuming that the positive rate is 5% (that is, there are at most 5 virus carriers per 100 people), it is now "1:5 uniform mixed test", that is, 100 people are divided into 20 groups according to 5 people in each group. How should the "mixed test" be arranged? How many tests do you need at most?
Tip: Only two extreme cases need to be considered: (1)5 positive samples are just assigned to the same group; (2) Five positive samples were divided into five groups.
Look at the world from a mathematical perspective, and look at medicine from a mathematical perspective. Mathematics is everywhere, and nucleic acid detection is no exception! In such an important practical problem as nucleic acid detection, we once again see the powerful power of mathematical methods!