The SIR Model of Infectious Diseases and the COVID-19 Exerperience

 

(Graphic: Creative Commons — Attribution 3.0 Unported — CC BY 3.0)

At the time of this post, we are over two years into the COVID-19 pandemic. Since the virus’s worldwide spread of early 2020, we’ve seen various experts and agencies project the duration and severity of the disease. The public was confused throughout the pandemic as it saw both underestimates of the spread of the disease and overestimates of it. Frustrating for all was how we experience several waves of the disease as incidents increased to a peak and then suddenly decrease only to be followed by another wave in a few months.

This chart is the cases in the United States (1/31/2022) as tracked by the Johns Hopkins University Coronavirus Resource Center: New COVID-19 Cases Worldwide - Johns Hopkins Coronavirus Resource Center (jhu.edu)

The last two peaks of the chart reflect the “delta” and “omicron” variants of COVID-19.

 

To understand the rise and fall of the disease, it helps to begin with a very simple model. Let’s assume there is a very large population of susceptible victims. To begin, we’ll assume that the population is infinite (not realistic, but just a way to begin modeling the spread of the disease). Next there will be one initial patient who has the disease and introduces it to the general population. If over the course of that initial patient’s experience with the disease, the patient infects two other people, then the basic reproduction number, R0, of the disease is 2. In this very simplified example, the first patient infects two others who go on to infect four more and the disease spreads exponentially. 

 

This first model isn’t realistic because while the initial population may be very large, it is not infinite. However, the pattern seen early in the disease does match with the relatively low number of infections followed by a very steep increase in the number of cases.

The SIR mathematical model was introduced in the 1920s to better describe and predict how a disease spreads through a population. The acronym SIR stands for Susceptible, Infected, and Recovered (sometimes R also stands for Removed in the cases where a subject dies). At each point in time every member of the population can be divided into one of the three groups, S, I, and R.

The next iteration of developing the mathematical model, is the assumption that members of the recovered group acquire immunity. In that case, the portion of the population in the susceptible group (S) decreases as the disease spreads.

As the size the susceptible population decreases, the effective rate at which each infected person spreads the disease decreases. This continues to a point where initially the disease was spreading exponentially, the rate of spread decreases and eventually the number of current cases (the Infection group) drops.

 

A link to the spreadsheet to produce the above graphic is given here – one can change the initial parameters, such as the R0 and population size: SIR Model

The graph shown above resembles the various peaks we’ve seen in the COVID-19 spread. To explain the recurrence of peaks, if members of the “recovered” group, R, lose their immunity over time or if the virus mutates so former members of the R group have less immunity, then the population of the susceptible group increases after initially decreasing, therefore allowing the caseloads to increase again in successive waves. This is what we experience with the annual flu. Within a given season, if one catches the flu, the person generally has immunity but not immunity for the next subsequent seasons as the flu mutates. 

 

Suggested References:

Waves of disease: What Makes a “Wave” of Disease? 

SIR Model Wikipedia

My post was a simplified SIR model. Those wishes to explore the differential equation SIR model are referred to: Extending the basic SIR Model in R | by Najma Ashraf | Towards Data Science or The SIR Model for Spread of Disease - The Differential Equation Model | Mathematical Association of America (maa.org)

2022 Anniversaries

Science News just celebrated their 100th anniversary and posted the top 10 anniversaries for 2022.

Of their top ten, my favorite is the 400th anniversary of William Oughtred's invention of the slide rule (see prior post also using this photo of my slide rule). Another is the 50th anniversary of the Pioneer 10 launch. It was designed for a 21-month mission to fly pass Jupiter. Pioneer 10 ended up sending back information about our solar system for 30 years. The last signal was received from Pioneer 10 twenty years ago this year. One other anniversary that is very important to me is the 100th anniversary of Frederick Banting's first human trial of injecting insulin to treat a 10-year-old boy with diabetes. 

How Many Black Holes are there in the Universe?

 

In a recent post, we explored the estimated number of planets in the universe which could be as high as 10²⁸ including rogue planets (not orbiting a star). The existence of black holes has been predicted for over a hundred years, but it wasn’t until 1964 when a black hole was actually detected (Cygnus X-1).

A black hole cannot be directly observed, but clear evidence for its presence can be detected as material is accelerated toward the black hole and is heated or by a black hole’s large mass can deflect light from stars. In the Astrophysical Journal this month, a paper provided an estimate of the black holes in the universe to be 40 x 10¹⁸ (40 quintillion) and they may make up 1% of the matter of the universe.

Image: NASA Goddard Media Studio - GMS: Black Hole Accretion Disk Visualization (nasa.gov)

Also see possible black hole image at M87.

Number of Planets in the Universe

The first exoplanets were detected in 1992 and since then nearly 5000 been detected since then. The pace of these discoveries will accelerate greatly once the James Webb Space Telescope (JWST) becomes operational later this year (as of the time of this writing, the JWST was just three days from reaching its final position at the Lagrange Point L2. (Related post from book with profile of 20 exoplanets).

The site Big Think has a nice estimate of planets in the universe. It estimates that there are over a trillion galaxies and each galaxy has a trillion to ten trillion planets orbiting its starts putting the total at 10²⁵. The article brings up the fact that there are also rogue planets that do not orbit starts which can put the total number of planets 10 to 10,000 times higher. These rogue planets are also described in the book, Imagined Life: Math Vacation: Book Review: Imagined Life by James Trefil and Michael Summers (jamesmacmath.blogspot.com).

Also see this post regarding number of black holes in the universe: Math Vacation: How Many Black Holes are there in the Universe? (jamesmacmath.blogspot.com)

Update 1/24/2022: The JWST successfully entered into its halo orbit at L2 - Where Is Webb? NASA/Webb.

Update 2/12/2022: The JWST took its first image of a star. This part of the long process to align the 18 primary mirrors and the secondary mirror to produce a single image. The image (below) and the alignment process are described at the NASA JWST site.

Image Credit: NASA - Post: Edit (blogger.com)

Update 2/22/2022
The alignment of the JWST mirrors continues as the image above has been adjusted to the hexagonal arrangement of the individual mirrors: JWST alignment continues, light-points now in hexagonal formation - Space Explored

Furthermore, considering a prior post about other civilizations in our galaxy, the JWST may be able to gain evidence of remote civilizations by detecting their pollution or by detecting evidence of photosynthesis on distant planets: Could Nasa's James Webb Space Telescope detect alien life? - BBC News.

Update 2/26/2022

The JWST made another step toward its mirror alignment. It stacked the eighteen individual images into one. Before and after images below (from Webb Mirror Alignment Continues Successfully – James Webb Space Telescope (nasa.gov)). 

Before:

After:

Images Credit: NASA/STScI

Book Review: The Math of Life & Death by Kit Yates

In the book, The Math of Life & Death, Kit Yates walks the reader through seven mathematical principals that shape our lives. British readers will be happy to know the book was originally published in Great Britain as The Maths of Life & Death.

Kit Yates website: The Maths of Life and Death | Seeing the world through a mathematical lens (kityates.com)

Kit Yates twitter: (18) Kit Yates (@Kit_Yates_Maths) / Twitter

I learned something new immediately in the introduction where he describes the simple method of capture-recapture for estimating wildlife populations (although this method may have proved difficult in my past effort on completing a worm census).

Yates avoid including equations and relied on describing clear examples. For instance, in explaining exponential decay, he gives simple example using M&M candies. One chapter (each chapter is dedicated to one of the seven principles) focuses on misuses of statistics and mathematical reasoning in the courtroom. Another clear example is given instructing how to measure your lung capacity using just a plastic bag and a container of water. Another chapter is dedicated to tragic mistakes made in aviation, space exploration and medicine that occurred from decimal point errors, the use of wrong units (metric vs. imperial units) and rounding errors.

Update 1/22/2022 - At the time of this post we are approaching a turning point in the COVID-19 pandemic. As Yates was finishing this book, the pandemic had just started, and he included a very timely chapter on the math used by epidemiologists for tracking and predicting the course of a disease. 

What are all those different numbers on U.S. Currency?

 

Every day when you handle currency, you probably pay the most attention to the numbers in the four corners in the front and back which indicate the bill’s denomination. There are other numbers found on currency that we’ll review in this post.

Denomination

The denomination is also spelled out on the front and back. With denominations, five dollars and higher, there will also be field of small cluster of “05”s, “10”s etc. found on either the front or the back. With the five, the cluster is found on the front, left of Lincoln’s head. With the twenty, the cluster is found on the back, left and right of the White House. For those with exception near vision or with the aid of a magnifying glass, the denomination is also microprinted in various locations as a security feature. Some examples of the microprint are in the security thread (can be viewed when held up to light) and other locations depending on the denomination such as “20 USA 20” found between Jackson’s shoulder and the signature of the Treasurer of the United States. Bills of denomination five and higher, have a security watermark. With the five, it is a large 5 on the right-hand side; with the twenty, the watermark is a portrait of Jackson on the right-hand side. The locations of some of these features are shown below.

Serial Number

The serial number of U.S. currency is printed twice on the front. It consists of one or two letters, followed by eight numbers and ending in a final letter. See my prior posts about the eight-digit serial numbers. If the bill has two leading letters, the first letter corresponds to the “series” of the currency – the series is also listed just right of the portrait, near the signature of the Secretary of the Treasury. When the bill’s serial number has only one leading letter, the letter corresponds to one of the twelve Federal Reserve Banks (this would be the second letter on newer bills having serial numbers beginning with two letters). In the example note shown above, the letter N corresponds to the 2017 Series and the letter L corresponds to the 12th Federal Reserve Bank - San Francisco. Older notes would list the bank by name left of the portrait, but newer notes use a common United States Federal Reserve System symbol left without listed the specific bank of the portrait. On the front of the bill, there should also be number printed two to four times that matches the Federal Reserve Bank serial number letter designation – 1 for A, 2 for B, etc. Newer bills, this number is also found with a repeat of the corresponding letter, for instance A1, B2…L12.

 

Plate numbers

Modern currency is not printed one bill at a time, but in sheets of 32 or more. Each note’s position on the sheet is designated a small letter and number. The plate position designation is found on the left-hand side (G3 in the above example). For the sheets of 32, the sheet is divided into four quadrants designated by the numbers 1 through 4 and the position within the quadrant is denoted by the letters A through G. The plates that print the bills are also serialized. Their number is printed on the right-hand side and consists of letters and numbers (FW G 3 in the example). In slightly larger font, the plate position letter designation is repeated in the middle of the plate serial number (G in the example). The back of the bill will also have a plate serial number and is typically just a one-, two- or three-digit number.

Star numbers

My favorite find is a bill that has a serial number ending in a star instead of a letter. The star notes are used when a bill is removed from a printing run because it was misprinted, damaged or otherwise did not meet quality standards. When the bill is removed, the physical count of bills for that run no longer reconciles with the serial numbers. Re-printing the removed bill is too costly, so it is replaced with a star note. The star notes serial number will be different than its neighbors. I have found star notes in multiple denominations. A sample is shown below. 

Here are some excellent resources for learning more about U.S. currency:

The United States Bureau of Printing and Engraving.

Further information on currency serial numbers.

Evaluate the value of potential rare currency.

Book Review: Mathematics - From creating the pyramids to exploring infinity by Anne Rooney

 

Anne Rooney's book covers mathematics most important concepts: numbers and numbering systems, geometry, infinities and infinitesimals, statistics, set theory, and many more topics. I particularly liked how each chapter was supplemented with vignettes about famous math problems and mathematicians. I also liked revisiting concepts familiar to me but explained in new ways. As a challenge, I invite readers to learn about chronograms and the famous magic square made by architect, Antoni Gaudi, included in the beautiful Sagrada Familia in Barcelona.

Math's True Source - Man or Nature?

The British Journal for the Philosophy of Science recently accepted for publication a manuscript by Sam Baron titled Mathematical Explanation: A Pythagorean Proposal. A summary of Baron's paper is given here: Pythagoras’ Revenge: Humans Didn’t Invent Mathematics, It’s What the Physical World Is Made Of (scitechdaily.com).

Baron proposes that our world has an inherent mathematical part in addition to its matter. In this framework, mathematics provides matter with its form while matter gives mathematics its substance.

I believe Baron's proposal is similar to Stephen Wolfram's proposal in A New Kind of Science.

A prior post in this blog wrote about the many examples of fractals found in nature.

Linking nature and mathematics is a common theme in teleological arguments for the existence of God. See prior post on Rev. Bayes.