rain and colds -- 4/20/22
Today's selection -- from Everyday Calculus by Oscar E. Fernandez. Why we get more colds when it rains:
"As children we were all told by our mothers to avoid getting rained on. To this day my mother insists that getting rained on will give you a cold; turns out there's some truth to this, but not for the reasons my mother cites. We now know that the common cold spreads through contact with infected individuals. This doesn't depend on whether it's raining, so why am I still worried about the soaking wet people sitting next to me? The reason is that on rainy days more people stay indoors, increasing the likelihood of bumping into someone who already has a cold. I'm not sure if anyone at my meeting has a cold, but if someone does, how likely is it that I'll catch it before the meeting ends?
"We can start by dividing the 20 people at my meeting (including myself) into two groups: those who are infected -- denoted by I -- and those who are susceptible to infection -- denoted by S. Both these numbers may change during the course of the meeting, so both depend on time; let's incorporate this into our analysis by promoting I and S into the functions I(t) and S(t), measuring t in hours. The group size tells us that
I(t) + S(t) = 20. (23)
"But how do we describe the spread of the infection? Well, when the infection spreads, I(t) is changing, so cue our derivatives! To make things concrete, suppose five people in the room have a cold. As they interact with the susceptible population the disease is likely to spread, and the rate at which individuals get infected, I' (t), will be larger if there are more interactions. This is leading us to consider the model
I'(t) = kI(t)S(t), (24)
where k > 0 is a constant that describes how fast people get infected from these interactions, and the product I(t)S(t) is a measure of how many interactions could result. Using equation (23) we can rewrite equation (24) as
I' = kI(20 - I), or I'= 20kI - kI2. (25)
"This equation is an example of a logistic equation. We can verify that the solution to this equation -- the number of infected individuals -- is
I(t) = 20/1+3e-20kt (26)
"... Using my starting assumptions, the number C of infected individuals at time t* is 10.*5 So our model is telling us that after half the people in my meeting get infected, the infection rate begins to slow. But what about the value L? What does that tells us?
"Well, if by some stroke of (bad) luck I get stuck inside this meeting room for days on end, then we'd expect that everyone would eventually catch the cold. This intuition tells us that L = 20, a fact that we can verify by using limits. This is a general feature of the logistic equation when a and b are positive: solutions eventually approach a limiting value L, called the carrying capacity.
"Luckily, I'm here for only an hour. I've already noticed a few people clearing their throats though, so I'm not in the clear just yet. To get some peace of mind I can calculate the expected number of infected individuals one hour after the meeting starts:
I(1) = 20/1+3e-20k
"Notice that this number depends on k, the number that describes how fast the infection spreads. For a relatively normal k = 0.02, I(1) ≈ 6.64, meaning that by the time the meeting ends almost two more people have caught the cold. The problem is that I can't tell who's infected or not. It could be the soaking wet student to my left, or the perfectly normal-looking guy to my right. But either way, our analysis has helped us narrow down the possibilities. This logistic approach to the spread of disease has connected me -- quite literally, since I'm one of the S(t) -- to the other people in the room."