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.

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.

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%

Day	New	Recovered	Patients
0	100	0	100
10	77	0	584
20	217	47	1,594
30	561	146	4,107
40	1,413	387	10,321
50	3,518	979	25,687
60	8,726	2,444	63,699
70	21,610	6,069	157,734
80	53,482	15,037	390,357
90	132,327	37,222	965,821
100	327,372	92,104	2,389,401
110	809,876	227,870	5,911,049
120	2,003,493	563,728	14,622,895

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

Infection rate 14% : Vaccination 0%

Day	New	Recovered	Patients
0	100	0	100
10	68	0	550
20	176	43	1,374
30	415	122	3,225
40	950	297	7,367
50	2,146	684	16,625
60	4,820	1,551	37,326
70	10,799	3,489	83,611
80	24,168	7,823	187,105
90	54,061	17,512	418,521
100	120,903	39,178	935,967
110	270,361	87,624	2,092,984
120	604,552	195,950	4,680,089

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%

Day	New	Recovered	Patients
0	100	0	100
10	39	0	425
20	68	27	722
30	110	55	1,145
40	167	92	1,743
50	248	142	2,576
60	361	212	3,748
70	520	311	5,387
80	742	450	7,688
90	1,054	644	10,912
100	1,491	916	15,433
110	2,104	1,298	21,774
120	2,963	1,834	30,660

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

Infection rate 20% : Vaccination 0%

Day	New	Recovered	Patients
0	100	0	100
10	134	0	783
20	558	71	3,206
30	2,192	321	12,553
40	8,493	1,274	48,608
50	32,793	4,944	187,669
60	126,519	19,105	724,033
70	488,031	73,726	2,792,833
80	1,882,413	284,404	10,772,366
90	7,260,657	1,097,006	41,550,085
100	28,004,990	4,231,280	160,262,292
110	108,017,601	16,320,434	618,145,102
120	416,632,864	62,949,300	2,384,237,013

These numbers are more pessimistic than reality.

Infection rate and number of patients with vaccination

Infection rate 15% : Vaccination 20%

Day	New	Recovered	Patients
0	100	0	100
10	77	0	584
20	217	47	1,594
30	561	146	4,107
40	1,413	387	10,321
50	3,518	979	25,687
60	6,981	2,444	63,699
70	13,527	6,069	119,411
80	24,157	10,600	213,483
90	43,375	19,052	383,309
100	77,930	34,301	688,611
110	140,009	61,653	1,237,131
120	251,517	110,763	2,222,419

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%

Day	New	Recovered	Patients
0	100	0	100
10	77	0	584
20	217	47	1,594
30	561	146	4,107
40	1,413	387	10,321
50	3,518	979	25,687
60	4,363	2,444	63,699
70	5,547	6,069	73,443
80	5,228	5,382	69,551
90	5,047	5,196	67,147
100	4,897	5,095	65,111
110	4,757	4,960	63,232
120	4,619	4,815	61,397

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

Infection rate 15% : Vaccination 80%

Day	New	Recovered	Patients
0	100	0	100
10	77	0	584
20	217	47	1,594
30	561	146	4,107
40	1,413	387	10,321
50	3,518	979	25,687
60	1,745	2,444	63,699
70	1,317	6,069	39,264
80	406	1,701	12,290
90	137	561	4,152
100	50	206	1,505
110	20	75	623
120	10	29	320

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

Infection rate 20% : Vaccination 20%

Day	New	Recovered	Patients
0	100	0	100
10	134	0	783
20	558	71	3,206
30	2,192	321	12,553
40	8,493	1,274	48,608
50	32,793	4,944	187,669
60	101,216	19,105	724,033
70	295,119	73,726	2,037,623
80	799,948	196,141	5,527,061
90	2,178,216	534,832	15,049,439
100	5,933,566	1,458,696	40,993,870
110	16,163,474	3,974,138	111,669,811
120	44,029,859	10,825,594	304,192,480

Infection rate 20% : Vaccination 50%

Day	New	Recovered	Patients
0	100	0	100
10	134	0	783
20	558	71	3,206
30	2,192	321	12,553
40	8,493	1,274	48,608
50	32,793	4,944	187,669
60	63,260	19,105	724,033
70	113,737	73,726	1,173,300
80	155,630	95,771	1,610,764
90	217,483	134,312	2,250,635
100	304,902	189,756	3,154,037
110	427,600	266,464	4,422,937
120	599,544	373,571	6,201,493

Infection rate 20% : Vaccination 80%

Day	New	Recovered	Patients
0	100	0	100
10	134	0	783
20	558	71	3,206
30	2,192	321	12,553
40	8,493	1,274	48,608
50	32,793	4,944	187,669
60	25,304	19,105	724,033
70	24,834	73,726	573,430
80	10,479	28,802	244,348
90	4,748	13,104	110,687
100	2,206	6,397	51,143
110	1,035	3,023	23,961
120	486	1,407	11,258

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.