Math Modeling: Day 30

1. Flattening the curve with social distancing
2. The impact on death rates
3. Surprising implications of doing too good a job....

Hopefully you've got that one in hand. It's due tomorrow, Thursday.

I want to give you some idea of direction, and let you know about some developments which, under ordinary circumstances, I might have shared in class. But now that's not quite as easy, and it's a little easier, perhaps, to do in theses chunks. I want to get you back up to speed.

So here are three things that I'd like you to do by Monday, 4/13:

1. I would like you to read starting at section 3 "Analysis": pp. 8-12 (through to the conclusion). There are several tables and figures (figures are at the end, unfortunately, rather than embedded in the text). If you have questions, please ask.

2. Now here is a special development, that we want to investigate (mentioned in those sections of the paper):
   • There is a fair amount of evidence that the diurnal temperature range (DTR) -- the difference between maximum and minimum temperatures -- has actually decreased over much of the land surface of the globe.

     This is because minima are increasing faster than maxima, so the two are getting closer together.

     In some cases, it appears that maxima are not increasing much at all, perhaps because of cloudier conditions.

     Minima may be increasing faster in part because of night-time warming (when it is typically cooler).

   • So this changes our initial sense about what we would expect under climate change.

   Now, for something to share with you all -- and to get you back into the game. I have some new data -- modern temperature data -- from a small town (Custar) with a good weather station not far from Bowling Green (county seat of Wood County, and the station from which most of our modern Fletcher data derived).

   • The Custar data (csv format -- years 1982--2020 -- more than a climate period's worth!)
   • Bowling Green Climate Normals (csv format)

   Here's how you might import these files into Mathematica:

   Custar = Import["http://ceadserv1.nku.edu/longa//classes/mat375/days/docs/Fletcher/data/Custar.csv", "CSV"];
   Custarheader = Custar[[1]]
   BGNormals = Import["http://ceadserv1.nku.edu/longa//classes/mat375/days/docs/Fletcher/data/BowlingGreenDailyClimateNormals.csv", "CSV"];
   BGNormalsheader = BGNormals[[1]]
   

   But you could use any software. The headers tell you what variables each data set contains.

   I'd like you to find the DTR for the Custar data (you have columns for the MaxAirTemp and MinAirTemp -- the DTR would be the difference of those two variables -- DTR=max-min).

   Regress the DTR, modeling it as a linear function: i.e. \[ DTR(t)=\beta t + \alpha \]

   The question is whether the parameter $\beta$ is significantly different from 0, and whether in fact it's negative -- so that the DTR in Custar has been decreasing over the 38 years or so of data.

   You'll need to create a variable for "year". You could just construct this as the sequence of integers from 1 to length(MaxAirTemp), then divide by 365.25 (to make up for leap years -- kind of an odd thing -- but we need to realize that years are actually 365.25 days long...!).

   So carry out this experiment to see if Custar is experiencing the decrease in DTR that others are seeing around the world.

3. If you're really getting your regression feet under yourself, then you won't be at all bothered by the next task:
   1. to compute one of those sinusoidal models to the climate normals (max and min temperatures) of Bowling Green: e.g. \[ Max(t) = \alpha + A\sin{\left(2\pi (t - t_0)\right)} \]

      It's important that $t$ be in years for this regression -- otherwise you'd haveadd a period term, \[ Max(t) = \alpha + A\sin{\frac{2\pi(t - t_0)}{T}} \] to handle the unit conversion.

      It's not hard, and you've certainly done a fair number of these. You'll perhaps be interested and surprised to see how sinusoidal the normals actually are.

      Here are some Mathematica commands to get started with the normals (creating the important variables).

   2. But the interesting twist is to fit a model of the form \[ Max(t) = \beta t + \alpha + A\sin{\left(2\pi(t - t_0)\right)} \] (for example), and to see what is happening in $\beta$ -- is it significant?

      If there were no climate change, then we would expect the climate normals to be periodic functions -- December wrapping right back into January. However, if, as we expect, there's global warming going on, then we'll see that normals at the end of the year are actually a little higher than the normals at the beginning of the year.

      And, if our hypothesis is right that the minima are changing more rapidly than the maxima, then the coefficient of the minimum will be larger than the coefficient of the maximum.

      See what you discover!

Note, however, that we've had to move beyond the Fletcher data to get there....

Some good links that I might recommend (a few of which we'll focus on):

1. A great intro to many of the most important question: "When a new virus emerges, no one is immune. A highly transmissible virus, like the coronavirus behind the current pandemic, can spread like wildfire, quickly burning through the dry kindling of a totally naive population. But once enough people are immune, the virus runs into walls of immunity, and the pandemic peters out instead of raging ahead. Scientists call that the herd immunity threshold."

2. How epidemics like covid-19 end (and how to end them faster): A coronavirus causing a disease called covid-19 has infected more than 70,000 people since it was first reported in late 2019. To predict how big the epidemic could get, researchers are working to determine how contagious the virus is.

3. The SIR Model for Spread of Disease - Introduction (from the MAA).

   In particular, we will implement The SIR Model for Spread of Disease - The Differential Equation Model in InsightMaker.

4. Mathematics of the Corona outbreak, with Professor Tim Britton

   (An excellent introduction to SIR models, from both the infectious disease and mathematical sides)

   Questions:

   1. Where is the "calculus moment" in this video?

   2. How does Britton suggest breaking $R_0$ down into "actionable" pieces?

5. An on-line model developed by Ashleigh Tuite and David Fisman, Dalla Lana School of Public Health, University of Toronto

6. This one incorporates space into the model. These models are called "agent-based models".

7. Could Coronavirus Cause as Many Deaths as Cancer in the U.S.? Putting Estimates in Context

   This on-line estimator (i.e., a model!) allows one to estimate deaths, as well as death by age-category.

8. A nice R-based introduction to infectious diseases and nonlinear differential equations.

9. Use of a log-scale (which they take pains to explain) to illustrate deaths by country, updated daily. There is even a separate page at the New York Times to explain log plots: "A Different Way to Chart the Spread of Coronavirus: Those skyrocketing curves tell an alarming story. But logarithmic graphs can help reveal when the pandemic begins to slow."