Lately the focus in public health has been getting people to get a vaccine for Covid-19. There's been advertising around it. Also an incentive game of "if this percentage, 70,80, etc. gets vaccinated by this date, then some of these restrictions might be considered to be lifted". Which is also pressure and punishment if the goal is failed to be reached, to identify and deal with the wreckers.
I got thinking about ways to increase the vaccinated rates for a population. Two basic tactics come to mind. Somehow persuade the unvaccinated holdouts to get a vaccine. Or remove the unvaccinated. I thought about it purely from which approach is mathematically more effective at increasing overall vaccination rates.
At first I thought this might be a hard to describe mathematically or set up an equation. However after a bit I realized if you describe the groups the right way, then it is easy to understand and the more effective strategy is clear.
So assume you have some kind of population, P. Within this population are vaccinated and unvaccinated persons. So the starting percentage S, is more than 0 and less than 100 percent.
The trick is to treat a target group of unvaccinated as "outside" the main population. Then apply each strategy and determine the effect.
With the conversion strategy, people in the outside target group choose to get vaccinated. In this case a subgroup of 100% vaccinated is added to the initial population P, which is under 100%. it should be clear that adding a 100% vaccinated group to P will increase the overall vaccinated rate of the combined group from S to a bit higher than S.
Now with the removal strategy, then the holdouts are just somehow removed, eliminated. In this case P is the same, and S is also the same. So the overall vaccinated percentage does not increase when the groups are set up this way.
So it is then mathematically clear that motivating holdouts to get vaccinated is a more efficient way to increase the overall vaccinated percentage than, ahem, removing the holdouts.
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An example will demonstrate. This could actually be used as a junior high or high school math question on fractions, setting up a model, and reasoning.
Suppose there is a group of 10 people and 6 are vaccinated. Is it more effective to cast a holdout from the group, or to convince a holdout to get a vaccine. Which would be more effective to increase the overall vaccinated percentage.
In this case we would treat P as 9, with 6 vaccinated. So the starting point S is 6 of 9 vaccinated, or 67%
If you add a vaccinated to P then it becomes
6 of 9 + 1 of 1 = 7 of 10 = 70%
if you remove a holdout then it just remains as P
6 of 9 = 67%
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So it should be clear that converting holdouts to get vaccinated is a more effective strategy than removing them to increase the overall population vaccination percentage.
