The pathogen has had 5 days to spread after the initial infection, and 3 people have been vaccinated.

Because there are not enough vaccinated people to stop the pathogen at every edge of the contact chart, at least some of the people at the edge will have been infected on the fifth and final day.

Let's start by trying to align those people along the bottom right edge and working backwards towards the center of the infected population. (Note that the dotted line in each chart connects other possible patient zeros, with preference given to those nearest the center of the chart.)

Filling out the rest of the chart, this requires 8 vaccinated people to explain the infected population.

Let's try moving those infected on the fifth day to the top right edge.

Turns out, this is worse, as it now requires 9 vaccinated people, and there is a person who is not infected until the sixth day.

What if we make a small adjustment, so that person is now infected on the fifth day (instead of the sixth)?

This increases the required number of people vaccinated up to 10.

We're clearly going in the wrong direction here, so let's switch things around. Now let's align those infected on the fifth day to the lower left corner.

This is an improvement, as now only 7 vaccinated people are required.

If we move patient zero down the diagonal dotted line, this results in both those in the lower and upper right corners having been infected on the fifth day.

And only the available 3 vaccinated people are necessary to stop the infection in the lower right corner.