A Simple Model for Epidemics

This is the second in a series of posts related to teaching math during the Covid-19 pandemic.

Around 15 years ago, when I was teaching Calculus II, I used an ice-breaker on the first day of class that simulated the spread of an epidemic. The activity certainly is not something that we could do now, but the results are instructive. I’ve added a another simulation that illustrates what we’ve all been hearing about the effects of social distancing.

A Mathematica Notebook can be downloaded if you would like to experiment with this model – I’m sure the programming could be streamlined.

I’d be interested to hear if anyone can come up with a way to to adapt this ice-breaker to online teaching.

The Classroom Simulation

Introduction: We’re going to model an epidemic. Here is how it will work. At the start of the process, one of you will be “sick” and we will go through a series of “days” Each day everyone will do the following: you will make contact with exactly one person by tapping him/her on the shoulder, telling whether you are sick or not, and then you can introduce one another. If you are sick and the person you tapped on the shoulder was well, then he/she becomes sick. If you are well and you tapped a sick person on the shoulder, you don’t become sick. You may be tapped on the shoulder more than one time or not at all in one day, but you should tap exactly one person on the shoulder. As the days pass, you may tap the same person on the shoulder more than once — ideally you should choose people randomly.

Before we get started, take out a sheet of paper. Draw what you think might be the graph of the number of people who get sick verses the days. This will actually be a discrete graph, like a bar graph, but you can draw a continuous curve to indicate the shape of the graph

You will get a “Post-it.” If it has the number 1 on it, you are the lucky person who is sick and starts the epidemic. When you get sick, make a note of the day on which you get sick on the Post-it

On the first day, the “patient zero” gets sick and we assume that day 0 has passed; so we start with day 1.

After 10 days, see if anyone is well. Continue until almost everyone is sick.

Stack your Post-it notes in the column marked by the day you got sick.

Computer Simulation

A computer simulation using Mathematica produces a similar result to the classroom activity – the code can be seen in the Mathematica Notebook. Here we assume 500 individual and find the the number of days before everyone is infected doesn’t change all that much with a larger “class.” Here are graphs of the cumulative infections and the daily new cases.

Spread of an epidemic among 500 individuals with no “social distancing.”

A second version of the model, which I added in 2020 allows for selecting a level of social distancing. Here we assume that a random 70% of individuals are quarantined on any day. Here are the results.

Spread of an epidemic among 500 individuals with 70% “social distancing.”

The two sets of plots have the same aspect ratio so you have to pay attention to the scales. Alternatively, here are the daily exposures for the two models plotted together.

Daily new infections with and without social distancing.

Here, you can see the flattening of the curve we’ve all heard about.

Online Math Teaching – Textbooks

This is the first in a series of posts that will discuss issues relating to online teaching during the covid-19 pandemic.

Covid-19 virus – a mathematical object?

There are countless issues that confront your students and one of them is access to textbooks. Although current courses have already adopted texts, student may not have their texts at home. You might want to give your students an alternative reference as you start working online. For mathematics, there are two collections of texts that contain virtually every topic we currently teach. In these sites, the books that are listed all offer free pdf’s and in most cases, online versions. The pdf versions are useful for students who don’t have high-speed internet access. The companion online versions use a bit more bandwidth, but have the advantage of having a variety of interactive features.

  • The American Institute of Mathematics has an Open Textbook Initiative with curated list of open source textbooks.
  • The PreTeXt project has a catalog of texts that have been developed with PreTeXt, an XML application that facilitates publishing a single source in multiple formats, including pdf, html, and Braille.

Even if you intend to keep using your current text, you might want make your students aware of these alternative resources.

The online versions frequently include editable code such as this page from a differential equations by Thomas Judson, which is listed on the PreTeXt site.

New Faculty: Elisa Perrone

We are happy to welcome Dr. Elisa Perrone to the faculty of Mathematical Sciences. Her research spans from mathematical statistics to applied statistics. She is mainly interested in multivariate statistics and dependence modeling, with particular emphasis on copulas and their geometric properties. This work includes the development of copula-based approaches used in optimal experimental design and environmental sciences such as hydrology and weather forecasting.

Dr. Elisa Perrone

Prior to joining UMass Lowell, Dr. Perrone was a postdoc at the Massachusetts Institute of Technology and the principal investigator of the project ‘Geometry of discrete copulas for weather forecasting’, funded by the Austrian Science Fund (FWF). Elisa has a doctoral degree from the Johannes Kepler University Linz (Austria).

Dr. Perrone’s faculty website.

New Faculty: Amanda Redlich

We are happy to welcome Dr. Amanda Redlich to the faculty of Mathematical Sciences.   Her research is in probabilistic combinatorics and randomized algorithms.  In her combinatorial work she looks for patterns in big random structures, like social networks.  In her algorithmic work she looks for efficient ways to solve big problems, like analyzing genetic code.  Her most recent work has been on randomized allocation algorithms and biological random graphs.

Dr. Amanda Redlich

Before coming to UML, Amanda worked at Bowdoin College and Rutgers University.  She has a bachelor’s degree from University of Chicago and a PhD from MIT.

Dr. Redlich’s faculty web page

New Faculty: Daniel Glasscock

Last fall, we were happy to welcome Dr. Daniel Glasscock to the UML Department of Mathematical Sciences.

Image of  Dr. Daniel Glasscock.
Dr. Daniel Glasscock

Daniel’s research lies at the intersection of combinatorics and analysis.  He’s interested in applications of tools and ideas in dynamical systems (a branch of analysis with its origins in celestial mechanics) to combinatorics and combinatorial number theory.  Daniel has degrees in math from Rice University (Bachelors), Central European University (Masters), and The Ohio State University (PhD).  He is interested in teaching and research at all levels, and he organizes a yearly math summer study abroad program in Budapest, Hungary.

His faculty web page

What’s Your Favorite Theorem?

A while ago I discovered a Podcast called My Favorite Theorem and have eagerly listened to each new episode. The basic format is that a mathematician is invited by the two hosts (Kevin Knudson and Evelyn Lamb) to describe his/her favorite theorem and also to pair the theorem with some non-mathematical thing, usually food or music.

What’s your favorite theorem?  Mine is the Chinese Remainder Theorem.  It’s got an obvious pairing, but that isn’t why I picked it.  My reason is that is appears in several courses I’ve taught and also has connections with my dissertation research way back in the 1970’s.

In an abstract algebra course, the Chinese Remainder Theorem says that if two positive integers, m and n, are relatively prime, then the ring of integers mod m \cdot n is isomorphic to the direct product the rings of integers mod m and n.

In number theory, the same fact is framed differently, that the system of congruences x \equiv a \pmod{m} and x \equiv b \pmod{n} always has a unique solution mod m\cdot n as long as m and n are relatively prime.

My dissertation research involved approximation and interpolation of functions, and when you generalize the Chinese Remainder Theorem to Euclidean Domains, one immediate implication is that given n+1 points on the plane (pick any field) with distinct x-values, there is always a unique polynomial of degree n or less that passes through the points.

If you have a favorite theorem feel free to post it in the comments!

UML Participation in the 2019 William Lowell Putnam Mathematics Competition

UML Students working on the Putnam.

What a way to spend your Saturday! Get yourself to campus for 10 AM and work on six math problems for three hours. Then after a two hour break, spend another three hours of six more problems. That’s what thousands of undergraduate students throughout the US and Canada, including 34 UMass Lowell students, did on December 7 to take part in the 2019 William Lowell Putnam Mathematics Competition.

The competition, sponsored by the Mathematical Association of America, took place concurrently throughout the US and Canada. Last year,  4,623 students from 568 institutions participated. There were two 3 hour sessions, each with six problems. As usual, the problems were tough. Here is probably the easiest of them:

Determine all possible values of the expression
A3 +B3 +C3 – 3 A B C,
where A, B, and C are nonnegative integers.

A complete list of problems: 2019 Putnam Problems

Professor Kenneth Levasseur served as supervised competition at UML.   Thanks to the Honors College for providing refreshments for the students on the day of the event.

Results will be announced in late March.

Al Doerr, 1938-2018

Prof. Alan Doerr, Mathematical Sciences

Professor Alan Doerr, one of the longest serving professors at UMass Lowell, passed away on October 14, 2018 at the age of 80. Al retired in 2016, completing a career that spanned the entire history of the UML Mathematical Sciences department.

A Lawrence MA native, Al started his career as a high school mathematics teacher in New York City in 1960. After earning a graduate degree from Hunter College, he was hired as part of the Lowell Technological Institute’s Physics Department. Soon afterward, he was among the physics faculty who became founding members of the Mathematical Sciences Department. He was instrumental in creating both undergraduate and graduate degree programs in the department.

From 1976 to 1987, Al chaired the department as Lowell Tech merged with Lowell State to become the University of Lowell. After completing four terms as chair, he spend another twelve years as associate chair, during which the institution became UMass Lowell. He also was a coordinator in Lowell’s Continuing Education program from 1966 to 2015.

Al’s mathematical interests were primarily in Linear Algebra, Abstract Algebra and Discrete Mathematics. He taught courses in these subjects countless times, but taught virtually every course in the curriculum except statistics (which he admitted to disliking). In 1985, he co-authored Applied Discrete Structures for Computer Science, with his colleague, Ken Levasseur. The book was successful in the 1980’s and has been re-released as an open content text that is currently used at several universities. In 1994, he also coauthored College Algebra and Trigonometry with Leonard Andrusaitis and Ken Levasseur.

The department hopes to establish a scholarship in Al’s name in the near future. As of January 1, 2019, we have raised almost half of what is needed to establish an endowment.   If you’d like to pledge any amount, please contact Ken Levasseur.

UML Participation in the 2018 William Lowell Putnam Math Competition

2018 William Lowell Putnam Math Competition at UML

Forty-six UMass Lowell students participated in the 2018 William Lowell Putnam Mathematics Competition on Saturday, December 1. The competition, sponsored by the Mathematical Association of America, took place concurrently throughout the US and Canada. Last year, 4,638 students from 575 institutions participated. There were two 3 hour sessions, each with six problems. As usual, the problems were tough. Here is one of them:

Find all ordered pairs (a,b) of positive integers for which 1/a + 1/b = 3/2018.

A complete list of problems:  putnam2018probs

Professor Kenneth Levasseur served as supervised competition at UML.   Thanks to the Honors College for providing refreshments for the students on the day of the event.

Results will be announced in late March.