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## instructions

A pathogen is spreading through a small population. The contact chart shows this population, with each person represented by a dot, and the daily interactions among the population, represented by the connections between the dots. This particular pathogen takes one day to infect all those that come into contact with an infected person, unless they are vaccinated against the pathogen.

This example shows how the infection spreads through a population over two days. Notice that those who are vaccinated (blue dots) do not become infected, nor do those who are more than two “contact days” away from the initial infection.

In each puzzle, you are given a contact chart that shows who is infected (green dots) and who is not infected (grey dots). You also know how many days the pathogen has had to spread, and how many people have been vaccinated. You need to determine who brought the pathogen into the population (in other words, who was the first to be infected).

## example

Here is another example of a contact chart, this time showing those who are infected. The pathogen has had two days to spread, and only one person was vaccinated.

Because only four people have been infected, we can run through the possibilities of each person having been the first to be infected. In each case, we need to consider how many people must be vaccinated to produce the resulting infected population.

If the first person infected is in the center, four people must be vaccinated in order to stop the spread of the pathogen, because all but one person is two contact days (or less) away from that center person. Two people must be vaccinated if the first person infected is along the periphery of the contact chart, unless it is the person on the right. Notice that person only comes in contact with two people, instead of three (like all the rest of those infected).

## solving tips

In most cases, it is too time consuming to trace the spread of the pathogen starting from every single person infected in the contact chart. Since you know the number of days the pathogen has had to spread, you can try counting down from those along the periphery of those infected. However, this may result in some contradictions.

Remember that some people along the periphery may not have been infected on the most recent day. Instead, they may have come in contact with someone who was vaccinated and were thus unable to spread the pathogen any further. In addition, it may not always be clear who infected whom when working from the periphery of the chart towards the first person to be infected. Try working from both the outside-in and inside-out.

## get involved

As you can see in each puzzle, the pathogen spreads freely through a population until it encounters a person who is vaccinated against it. While vaccination (just like everything in life) is not 100% safe, establishing herd or community immunity in a population is important to stopping dangerous diseases. Many countries have laws, backed by mathematics and science, requiring vaccinations. For your health and the health of those around you, do your part by following the vaccination requirements where you live.