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Simple Predictive Model: does vaccination reduce the number of coronavirus infections?

The idea of mass immunity is gaining attention as a means of controlling the explosive increase in the number of people infected by coronaviruses. People who are infected with the virus become immune, so that even if they become infected again, the virus is immediately eliminated by the immune system and cannot be transmitted to other people. In this way, collective immunity prevents the spread of infection.

There is no doubt that mass immunity is the ultimate means by which mankind can overcome the problem of coronaviruses, but it should be achieved by vaccines, not by leaving mass infections unchecked. We can achieve mass immunity by using vaccines without causing severe disease or death. However, it is difficult to say to what extent vaccination can prevent the spread of infection.

Let’s try to calculate the change in vaccination rates and the number of people infected by the virus.

It is impossible to simulate a real virus such as coronavirus, unless you are an expert. In this article, we will consider a hypothetical virus, virus A, and perform the calculations using the simplest model.

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Model

There are 100 people infected with virus A. Each infected person has a 15% chance of transmitting the virus to another uninfected person. For example, there are 100 infected people on day 1, and 15 of them will transmit the virus to another 15 uninfected people. Thus, on the second day, there will be 115 infected persons, and the same will be true after the second day. After 14 days, the infected persons are removed from the total number of infected persons by recovery or death. Once a person has recovered, he or she will not become infected again. The number of infected persons is then determined if the vaccine is not introduced. We also assume that after 60 days, 20% of the uninfected are vaccinated and have antibodies. In this case, the infection has a 20% chance of failing and not being added to the infected. Then we calculate the number of infected individuals. Also, similar calculations will be made for various vaccination rates for comparison.

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Infection rate and number of patients without vaccination

First, here are the changes in the number of people infected in the absence of vaccines. The change in patients may resemble bank interest. Small differences in rates make a big difference.

Infection rate 15% : Vaccination 0%

DayNewRecoveredPatients
01000100
10770584
20217471,594
305611464,107
401,41338710,321
503,51897925,687
608,7262,44463,699
7021,6106,069157,734
8053,48215,037390,357
90132,32737,222965,821
100327,37292,1042,389,401
110809,876227,8705,911,049
1202,003,493563,72814,622,895

We see that the initial 100 cases have increased to 14 million after 120 days.

Infection rate 14% : Vaccination 0%

DayNewRecoveredPatients
01000100
10680550
20176431,374
304151223,225
409502977,367
502,14668416,625
604,8201,55137,326
7010,7993,48983,611
8024,1687,823187,105
9054,06117,512418,521
100120,90339,178935,967
110270,36187,6242,092,984
120604,552195,9504,680,089

Although the difference in infection rates is 1%, the results may differ from your prediction. These results suggest that masks can make a big difference, even if you believe they have little or no effect on wearing them.

Infection rate 10% : Vaccination 0%

DayNewRecoveredPatients
01000100
10390425
206827722
30110551,145
40167921,743
502481422,576
603612123,748
705203115,387
807424507,688
901,05464410,912
1001,49191615,433
1102,1041,29821,774
1202,9631,83430,660

Taking into account the tragic circumstances of today, we might consider this number a miracle.

Infection rate 20% : Vaccination 0%

DayNewRecoveredPatients
01000100
101340783
20558713,206
302,19232112,553
408,4931,27448,608
5032,7934,944187,669
60126,51919,105724,033
70488,03173,7262,792,833
801,882,413284,40410,772,366
907,260,6571,097,00641,550,085
10028,004,9904,231,280160,262,292
110108,017,60116,320,434618,145,102
120416,632,86462,949,3002,384,237,013

These numbers are more pessimistic than reality.

Infection rate and number of patients with vaccination

Infection rate 15% : Vaccination 20%

DayNewRecoveredPatients
01000100
10770584
20217471,594
305611464,107
401,41338710,321
503,51897925,687
606,9812,44463,699
7013,5276,069119,411
8024,15710,600213,483
9043,37519,052383,309
10077,93034,301688,611
110140,00961,6531,237,131
120251,517110,7632,222,419

On day 60, 20% of non-infected people were vaccinated. The results are better than without the vaccine, but the number of infected people continues to rise. This suggests that if a vaccine is introduced, it will be very difficult to ascertain whether it is effective or not.

However, we can confirm that there is a significant difference in the calculations.

Infection rate 15% : Vaccination 50%

DayNewRecoveredPatients
01000100
10770584
20217471,594
305611464,107
401,41338710,321
503,51897925,687
604,3632,44463,699
705,5476,06973,443
805,2285,38269,551
905,0475,19667,147
1004,8975,09565,111
1104,7574,96063,232
1204,6194,81561,397

A 50% vaccination rate confirms a reduction in the number of infected people.

Infection rate 15% : Vaccination 80%

DayNewRecoveredPatients
01000100
10770584
20217471,594
305611464,107
401,41338710,321
503,51897925,687
601,7452,44463,699
701,3176,06939,264
804061,70112,290
901375614,152
100502061,505
1102075623
1201029320

With a vaccination rate of 80%, it looks like the infection is finally coming to an end.

Infection rate 20% : Vaccination 20%

DayNewRecoveredPatients
01000100
101340783
20558713,206
302,19232112,553
408,4931,27448,608
5032,7934,944187,669
60101,21619,105724,033
70295,11973,7262,037,623
80799,948196,1415,527,061
902,178,216534,83215,049,439
1005,933,5661,458,69640,993,870
11016,163,4743,974,138111,669,811
12044,029,85910,825,594304,192,480

Infection rate 20% : Vaccination 50%

DayNewRecoveredPatients
01000100
101340783
20558713,206
302,19232112,553
408,4931,27448,608
5032,7934,944187,669
6063,26019,105724,033
70113,73773,7261,173,300
80155,63095,7711,610,764
90217,483134,3122,250,635
100304,902189,7563,154,037
110427,600266,4644,422,937
120599,544373,5716,201,493

Infection rate 20% : Vaccination 80%

DayNewRecoveredPatients
01000100
101340783
20558713,206
302,19232112,553
408,4931,27448,608
5032,7934,944187,669
6025,30419,105724,033
7024,83473,726573,430
8010,47928,802244,348
904,74813,104110,687
1002,2066,39751,143
1101,0353,02323,961
1204861,40711,258

There is a difference when compared to a 15% infection rate. When the vaccination rate is 50%, the number of infected people at 15% infection rate is decreasing, but the number of infected people at 20% infection rate is increasing. However, we can confirm that an 80% vaccination rate can eliminate the virus, even if the infection rate is high.

Summary

We compared different simulations and learned that small differences in infection rates can make a big difference in outcomes. We also learned that the majority of people need to be vaccinated to reduce the number of infections.

If you are a teacher, please let your students know this fact.

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