Book Review – Instant Mathematics by Paul Parsons and Gail Dixon

 

(Image from Simon & Schuster: Instant Mathematics | Book by Paul Parsons, Gail Dixon | Official Publisher Page | Simon & Schuster (simonandschuster.com)

In Instant Mathematics, Key Thinkers, Theories, Discoveries and Concepts, Paul Parsons and Gail Dixon put together a very easy to read collection of 160 lessons, each explained on a single page. The lessons cover both mathematical concepts and mathematicians. The book is chronologically starting with the discovery of the oldest known mathematical artifact, the Lebombo bone (found with counting tally marks) and ending with modern day analytics.

The illusrations are clear and supplement the concepts reviewed. As a material science engineer, I particularly liked the lesson on Kepler’s Conjecture which asserts the most efficient way to pack spheres in a volume. Also see the post in this blog: Math Vacation: Atomic interstitial sizes in higher dimensions (jamesmacmath.blogspot.com).

A few of the lessons cover cryptography, including examples from ancient Mesopotamia and Greece and ending with modern quantum algorithms. In a prior post I propose a much less sophisticated method - Math Vacation: One-Time Pad Coding and a Proposal for Improvement (jamesmacmath.blogspot.com).

The book is part of Simon & Schuster series called Instant Knowledge.

Instant Mathematics | Book by Paul Parsons, Gail Dixon | Official Publisher Page | Simon & Schuster (simonandschuster.com)

Estimating Populations by the Capture-Recapture Method

The capture-recapture method is used to estimate population sizes when it is impractical to physically count each member of the population. For instance, estimating the population of a species in a large National Park. The method is also known as the capture-mark-recapture method. I learned about this method in Kit Yates' book - see Math Vacation: Book Review: The Math of Life & Death by Kit Yates (jamesmacmath.blogspot.com). This method may not be practical for some species. For instance, consider taking a census of worms where marking them may be difficult. A worm farm census was completed by a different sampling method found in this prior post: worm census.

The method is simple. First, a researcher captures alive a sample of the target species. Each captured animal is marked using a tag, a collar or other method, and then is released. At a future date, the researcher returns to the same area and again captures the target species. During this visit, the researcher counts how many animals in the second sample had also been marked in the first sample (repeated captures).

The ratio of the animals marked (m) in the first visit to the total population (N) should be approximately equal to the ratio of repeated captures (r) to the size of the second sample (n).

Therefore, N can be approximated by:

The method is subject to a number of assumptions. First, one needs to assume that the animals released after the first sample have sufficient time to mix with the full herd. Also, one has to assume the samples are representative of the herd. For example, if the captured animals just happen to be the animals that are most easily captured, then those specific animals are more likely to be recaptured and the ratio above would not be true. 

I tried a tabletop experiment using acorns. I picked up a large number of acorns fallen from a tree in my yard. They are pictured below.

I took 30 of the acorns and marked them by cutting off a small portion of the acorns (m=30) - unmarked and marked specimens shown below. 

Unmarked acorns:

Marked acorns:

I returned the 30 marked acorns to the original group and scrambled all of them. Then, I scooped out a cup of the acorns to represent my second sample. I counted the total number of the sample (n=47) and the number found within this sample that were marked (they were also from the original 30). In this case there were 7 repeats. The estimated population of all acorns is therefore:

I repeated this twice. After returning the prior sample and remixing, I had a scoop with n=42 and r=4 (repeats). This estimated N to be approximately 315. Returning the sample and remixing, the next sample had n=44 and r=3 estimating N to be 440. 

The actual count of the full group of acorns was 300. The middle estimate was very close while the other two yielded estimates that were about 33% lower and higher than the actual count. For samples (m or n) that are roughly 10% of the size of the population (N), one should expect errors of this magnitude. 

In understanding possible errors, I like looking at extreme cases. Consider the case when in the first sample, the researcher actually captures the full population (so m=N). When returning, no matter what new sample size (n) is captured, all the second sample will be marked so n=r. 

In the equation above, if n=r then the estimate of N will always be m, which is the actual population N because we stated for this extreme case the researcher marked the full population. Another extreme is when N is very large compared to the sample size m. In this case it is possible that the researcher finds no marked (r) animals in the second sample (m). This would lead to N being undefined as we have equation with division by zero. The researcher would recognize this error and return to increase the second sample until at least one repeat was found. This would result in N being very large. At this larger sample size, m, the researcher might find 2 repeats. The new estimate of N would be half of the original so a small difference of 1 repeat in the sample leads to 50% to 100% change in the estimate of N.

Some resources on the capture-recapture method are given below:

Mark and recapture - Wikipedia

A Review of Capture-recapture Methods and Its Possibilities in Ophthalmology and Vision Sciences (tandfonline.com)

Spatial Capture-Recapture Models to Estimate Abundance and Density of Animal Populations | U.S. Geological Survey (usgs.gov)

https://www.newscientist.com/article/mg26635490-700-the-maths-hack-that-can-help-you-count-things/

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)