After some 104 days of summer vacation, Max and Nora are back in school. One of the big things that my wife Allison and I are concerned about is the fact that most schools are “breeder reactors” for coughs, sniffles, and stomach bugs. Case in point: Max’s third day of school this year was a sick day. As I was putting Max back to bed on Thursday evening, the question hit me: just how “contagious” would a stomach bug have to be to impact a majority of students in Max’s class?
For purposes of this puzzle, Max’s class has 18 students, and they sit at six tables of three students each. Each table is a “work unit,” where the kids share schoolwork, ideas, and basic biologicals. There is no table-to-table conversation or sharing. Kids sit at random tables for the morning session, and then pick brand new tables in the afternoon. Any given child has the same odds of picking something up from the outside and bringing it to the classroom, and each kid at a table has the same chance of picking it up as any other child at that table. Kids are “contagious” for 2 days before they actually succumb to the bug, and are out just one day when it hits. Absent students’ seats are randomly assigned, and both completely empty tables and having single students at their own table is possible.
If we pick a child-to-child transmission rate of 30% and give any individual child a 10% chance of picking something up from the outside and bringing it into the classroom, what are the odds that more than half of the class will be out during a given day? If it can’t happen, how would these rates need to be change to make it happen within a reasonable timeframe?