A naive estimate that phase III statistics could show an effective vaccine by November 1.
To start, I have no qualifications to make such an estimate. I am not a doctor, an epidemiologist, a statistician, a modeler, or a big data guy. Clearly, we have to follow the actual data as interpreted by professional and experienced experts.
The scattered articles which I have found on enrollment do not match the assumptions of the analysis. Around mid-August, Pfizer had only enrolled around 23,000. On Sept. 1, Moderna had 21,000 enrolled. AstraZeneca did not start US enrollment until September 1, but had 17,000 in Brazil, South Africa, and the U.K. However, the clinics involved in the trials are chosen in US hot spots, which may help the infection rates. AztraZeneca has paused their enrollment for a few weeks due to a spinal cord case. Update, Sept. 12. AstraZeneca has restarted their UK program which will have 18,000 enrolled. I will include an analysis for half of the number of cases by November 1 as well as for those used below.
I am assuming for simplicity that the Phase III trials have already filled up their 30,000 volunteers, and that they have taken their second shots. Otherwise, the analysis will be reduced in effectiveness by November 1. We also have simply assumed that the vaccine is only tested in the United States, and use the IHME projections of September 3 for the United States cases.
We are going to naively estimate the number of Phase III placebo vaccine testers who might be infected by November 1, and find that it could be as many as 560. This would be a good test for the effectiveness of the vaccine, since a 50% effective vaccine could leave only 280 of the vaccinated infected. We than compare this to the projected situation on January 1, where the number of cases in the trial will be almost tripled, using the IHME case projections for both dates. We will also discuss the detection of complications of the vaccine, down to the 1 part in 2,000, 1 part in 5,000 and 1 part in 10,000 levels.
The longer we wait to decide, the statistical accuracy only increases as the inverse of the square root of the number of cases or the time, while the number of the public infected may continue to grow linearly or faster with time, so time is of the essence. The detection of low level vaccine complications independent of infection may increase linearly with time.
We calculate the number of new cases to November 1 from the IHME projections, take the ratio of this to the US population, and apply it to both placebo and vaccinated groups of 15,000 volunteers each in a 30,000 volunteer Phase III program.
The IHME projects 145,000 new cases a day for Sept. 10, and 318,000 for November 1, so we use their average of 232,000 cases a day for 52 days, giving 12 million new cases by November 1. Taking the uninfected US population as roughly 320 million and dividing 12/320, gives 3.75% of the US population to be newly infected between now and November 1.
Taking 3.75% of the 15,000 vaccine volunteer cohorts gives 563 infected in each. Since the infections are a Gaussian distribution, its standard deviation is the square root of 563 or 24, which is 4.2% of the 563.
If the vaccine reached the minimum limit of 50% effectiveness, only half of 563 who received the vaccine and were exposed would be infected, giving 282, with a standard deviation of 17, or 6.0%.
Even without statistics, it is quite obvious that the vaccine is a success in this example. If we take the difference as 282, its standard deviation is the square root of the sum of the squares of the two groups, or sqrt(563 + 282) = 29. The difference of means of 282 divided by the standard deviation of the difference is then 282/29 = 9.7 standard deviations, a certainty.
Because of the slow start and rate of recruitment, let’s assume that the group results were halved by November 1. So we assume that the placebo group had 280 cases, and the vaccinated group for a 50% effective vaccine had 140 cases. The standard deviation or square root of 280 is 17, and of 140 is 12. The combined standard deviation for the difference of 140 is the square root of the sum of 280 and 140 or 420, which is 20.5. So 140/20.5 = 6.8 standard deviations, still a certainty.
There are several different methods that vaccines work by, which I leave to presentations by the virologists and biologists. However, several vaccines just help us fight off the Coronavirus, without terminating it in its initial exposure. So effectiveness has to be measured in degrees of infection, from simple symptoms, to those requiring hospital visits, oxygen, ICU’s or ventilators, and leading to complications or death. There also have to be comparisons between volunteers with different ages, sexes, comorbidities, races, jobs, and exposures. If we have several successful vaccines, demographics may determine which ones work best for which cohorts. For all of these subcategories, we need as large a sample of cases as possible, so we extend our test estimates to January 1.
There are another 61 days between November 1 and January 1, 2021. The averages of 318,000 and 328,000 on these dates is 323,000 new cases per day. That gives about 20 million more cases, out of approximately 300 million uninfected Americans giving another 6.7% of cases. Multiplying by 15,000 cohorts gives another 1,005 cases. So on Jan. 1, two more months after Nov. 1, the placebo group moves to 1,570 cases, and the vaccinated group gains to 785 effective cases. This is almost three times the number of cases to examine demographic and comorbidity breakdowns. By waiting, however, another 20 million more Americans have gotten sick.
The IHME projected number of new US deaths between Nov. 1 and Jan. 1 is 155,000. If we divide that by 300 million uninfected Americans at that time, it is about 1 in 2,000. In the vaccinated group of 15,000, it would match a lethal problem that would show up with 7.5 cases on average. The Poisson probability that 0 cases would show up is only 0.055%.
One pessimist expert says that it may take two years to vaccinate everybody. The distribution network is complex, but early vaccination will be prioritized to those who need it most, saving lives and health care workers.
For finding any side effects of vaccination, which are independent of infection, the sample is the entire 15,000 vaccinated. For side effects at 1 in 5,000, three cases are expected on average. The Poisson distribution of probabilities of different number of cases for a mean of 3 are then:
0 0.056 This means that there is a 5.6% chance that such an effect would not be seen in even one case.
1 0.149
2. 0.224
3. 0.224
4. 0.168
5 0.101
6. 0.050
7 0.022
8 0.008
9 0.003
10 0.001
For side effects of 1 in 10,000, with a mean of 1.5, the Poisson distribution of cases is:
0. 0.223 This means that there is a 22% chance that such an effect would not be seen.
1 0.335
2 0.251
3 0.126
4 0.047
5 0.014
6. 0.004
7 0.001
One question which medical trial experts could answer, is if those who volunteer are more likely to be exposed than the general public, and if they are more likely to expose themselves even with a 50% probability of being vaccinated. If so, this would lead to a greater number of cases more than our estimates based on the general public. Volunteers, however, may be more conscientious, and observe social distancing and masking, leading to fewer cases. Time will tell.
