Math Modeling: Day 24

1. Decades:
   

   What we're expecting, in the absence of some systematic effect, is a uniform distribution (all bins the same height). These distributions are not uniform: here are some chi-square results (Statcrunch) which show that they are not uniform (right-click to view the image in order to enlarge it).

   

   ---

   Some of you seemed to think that we wanted to see normal distributions -- not at all! There's nothing special about the middle decades. They should be just as high as any others.

   ---

   They don't exactly meet our expectations, however: in the face of climate change, assuming that the temperature distribution were just shifting toward higher temperatures (with no other change to the distribution) we would have expected the following (histogram below; a "q-q" type plot above, showing how a uniform would be distributed versus how the data was actually distributed -- if it were uniform, the two graphs would agree):

   MaxMax	MinMax	MaxMin	MinMin
   We'd expect to see more max maxes in more recent decades.	We'd expect to see more min maxes in earlier decades.	We'd expect to see more max mins in more recent decades	We'd expect to see more min mins in earlier decades.

   In terms of decades, I would say that things look fairly muddy at the moment. While the results are clearly not uniform, they also do not seem to conform as well to our guesses about what we would expect under climate change as I had hoped they would.

   A couple of observation:

   1. The early years are heavily under-represented. It would be reasonable to suspect that the data from the mid-1800s would not be as reliable as the data from later decades. Who was taking hourly measurements of temperature, and with what equipment? It looks like those first four decades are not behaving.
   2. Max maxes are certainly not behaving as we would expect.
   3. Min mins are pretty much dead on (with the exception of those early decades)
   4. Min maxes are also pretty much dead on (with the exception of those early decades)
   5. Max mins are pretty ambiguous.

2. Now here it is by climate, where I used the later "climates" (30-year stretches, ending in 1992):
   

   All of these are plainly not uniform. So that suggests that the climate has changed: it's simply a matter of understanding how. A couple of observation:

   1. Min mins are behaving as expected.
   2. Max mins are leaning as we expect.
   3. Min maxes are leaning as we expect.
   4. Max maxes are most numerous in the middle period, unexpectedly.

Please, please: put MAT375 (as one word (i.e. not "MAT 375"), as well as assignment info) in the subject line of your message -- I'll be routing all messages using the tag MAT375. It will really help.

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. This Covid-19 situation is probably more suited to a SIRDs model (with disease-induced death).

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

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

8. 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.

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

10. 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."