In the face of scarce resources, we are sometimes forced to make very hard choices. Analytics can help us make most of what we have. Can you help this municipality make vaccine allocations to its different communities?
The municipality consists of three towns, one hamlet, and one village. Each community has different population sizes. In total, the municipality received 50,000 doses of an oral vaccine from the state for a total population of 120,000.
To get close to herd immunity, each community needs to vaccinate about 60% of its population. The likelihood of reaching herd immunity is calculated as
where T is 0.6 * population size and v is the number of people vaccinated. The population sizes and values of T are
How would you allocate the available doses of oral vaccine to the different communities?
[Scroll down for a hint.]
- Consider that when you vaccinate a person in a community that does not achieve herd immunity, then that person still benefits from the vaccination.
- Do you trust the experts to have determined the 0.6 parameter in the herd-immunity likelihood function 100% accurately?
[Scroll down for the solution.]
Let us consider 3 different models. In the first, we assume that the 0.6 parameter is 100% accurate. In the second and third model, we assume that the true phase-transition parameter for the sigmoid function is normally distributed around 0.6 with standard deviation 0.01 and 0.02, respectively.
Seeker provides the following solutions:
In Solutions 1 and 2, we aim to achieve herd immunity in Phlegmus, Bronchitisburgh, and Hack Hollow. We then use any surplus vaccines in Snottington-on-Sea or Wheezewick to protect a few more people, whereby under Model 2, we have less of those surplus vaccines as we prepare for the fact that herd immunity may well only become more likely at a 61% or 62% vaccination rate. At standard deviation 0.01, we already need to vaccinate almost 25,000 people in Phlegmus to be on the safe side.
At standard deviation 0.02, where the phase transition can easily occur at 62%-64%, it is no longer advisable to aim for herd immunity in the largest town. Instead, we provide very sufficient levels to all other communities.
Now let us evaluate these three solutions under all three models.
We see that the first solution, which aggressively aims to get extra points for individuals vaccinated in Snottington-on-Sea, performs very poorly if the experts were a bit off. We should therefore discard this solution. Both Solutions 2 and 3 are justifiable, whereby Solution 3 maximizes the worst-case.