Table of Contents

• Did Harry Potter ever get the measles?
• Week 1 and the Search for a Crystal Ball
• 1: Week 0: The crystal ball
• 2: Compartmentalizing the students
• 3: A Muggle Interlude: Models
• Week 2: The plot thickens...
• 4: ...for a week
• 5: Muggle Interlude: Compartments
• 6: A really simple model
• 7: Iterating the Simple Model
• 8: Getting Better at Recovering
• 9: Iteration vs. Closed Forms
• 10: Don’t get them confused!
• Week 2.142057149: Crisis at Hogwarts
• 11: Students dropping like flies
• 12: Getting Better at Getting Sick
• 13: Writing Equations for Contact
• 14: Week 2.142057149…: Crisis at Hogwarts
• 15: The Whole Enchilada
• 16: Comparing Rate-of-Change to Closed Form Equations (again)
• 17: When will the pox peak? And will the tournament go on?
• Week 3: Exploring the model
• 18: Week 3: The mathemagician gets busy
• 19: How many people got sick in the end?
• 20: Do initial conditions matter?
• 21: Does recovery time matter?
• 22: Does transmission rate matter?
• 23: Does quarantine matter?
• 24: Review

Writing Equations for Contact

What we need is equations for how often a sick kid comes in contact with an uninfected kid – in other words, the probability of an S meeting an I.

Just when you thought you had left genetics in the dust … probability rears its ugly head again…

As a review of terms,

• P(kid=S), or P(S) for short, is the probability that any given kid is susceptible at the time of interest. We can calculate P(S) by taking the total number of susceptibles and dividing by the total number of kids, which is S/600.
• Likewise, P(I), is the probability that the kid is infected, calculated as the number of susceptibles divided by total kids, or I/600.

In any given meeting, the probability of having both an S and an I kid is P((kid1=S and kid2=I) OR (kid1=I and kid2=S)), or

P((S and I) or (I and S))

What is this probability?

(QHA – help reviews and and or – answer 2SI/600²)

So now we know the chance that a meeting between 2 kids will contain the ingredients for infection – namely, an S and and I. However, infection will not necessarily happen EVERY TIME an S kids coughs in an I kid’s face. Instead, there’s also some probability of infection, which in the case of pigeon pox happens to be 1 in 167. And we need to know about how many contacts Hogwarts kids make per day, which is about 30,000 (each kid making contact with 100 other kids = 60,000 pairwise contacts, or 30,000 total contacts).

Whew! We are finally ready to figure out the rate of infection…

QHA: What is the actual rate of infection?

What is the probability of that S and I meet? : as above, 2SI/600²

What is the probability that infection occurs at a single meeting? : this is P(S and I and infection successful.) The probability that infection is successful is 1:167, or about 0.006. So P(S and I and success) = P(S and I) * P(success) = 3 x 10^-8 x SI.

Given 30,000 meetings a day, how many infections will occur? : 30,000 x 3 x 10^-8 x SI, or 0.001 SI

Answer: 0.0001 SI

That was a lot of rigmarole to go through to get a simple number like 0.0001 SI. My point in going through it was first, to show that the number 0.0001 makes sense in terms of events and probabilities that can be measured, and secondly, to show why the SI term appears. EVERY simple SIR model will contain the SI term, and the coefficient (0.001 in this case) has already been worked out for a variety of diseases and population sizes. For example, 0.0001 is a reasonable value for measles in a medium-sized town.