Oxford Mathematician explains SIR Travelling Wave disease model for COVID-19 (Coronavirus)

The SIR model is one of the simplest ways to understand the spread of a disease such as COVID-19 (Coronavirus) through a population. Allowing the movement of populations makes the model slightly more realistic and results in ‘Travelling Wave’ solutions.

In this video, Oxford University Mathematician Dr Tom Crawford explains how including population migration modifies the original SIR model. He then goes on to use the results of the model to answer two important questions:

  1. How fast will the disease spread?
  2. How severe will the epidemic be?

The answers to these questions are discussed in the context of the current COVID-19 (Coronavirus) outbreak. The model tells us that to reduce the impact of the disease we need to lower the ‘contact ratio’ as much as possible – which is exactly what current social distancing measures are designed to do.

Watch the first video on the basic SIR model here.

Abel Prize 2020

Congratulations to Gregory (Grisha) Margulis and Hillel (Harry) Furstenberg on being awarded the 2020 Abel Prize. The prize is one of the most prestigious in mathematics and is presented annually by the Norwegian Academy of Science and Letters.

The official announcement states that Margulis and Furstenberg were awarded the prize “for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics” and their work is described by Hans Munthe-Kaas, chair of the Abel committee, as “bringing down the traditional wall between pure and applied mathematics”. So, who are they?

Gregory Margulis

Born in Moscow in 1946, Margulis gained international recognition aged only 16 when he received a silver medal at the International Mathematical Olympiad. He began his academic career at Moscow State University and began working towards his PhD under the supervision of 2014 Abel Prize Laureate Yakov Sinai. At the age of 32, he was awarded the 1978 Fields Medal for his work on the ‘arithmeticity and superrigidity theorems’, but was unable to travel to Finland to receive the medal as the soviet authorities refused to provide him with a visa.

Another major result followed in 1984 with his proof of the Oppenheimer Conjecture – a problem in Number Theory first stated in 1929. The ideas he introduced here centred on what is known as ‘ergodic theory’ (more on this later), and have since been used by three recent Fields Medallists: Elon Lindenstrauss, Maryam Mirzakhani and Akshay Venkatesh. In 2008, Pure and Applied Mathematics Quarterly ran an article listing Margulis’s major results which ran to more than 50 pages.


Hillel Furstenberg

Originally thought to be a pseudonym for a group of mathematicians due to the vast range of ideas published in his early work, Furstenberg is a mathematician with a deep technical knowledge of countless areas of mathematics. He published his first papers as an undergraduate in 1953 and 1955, with the latter giving a topological proof of Euclid’s famous theorem that there are infinitely many prime numbers.

One of his key results came in 1977 when he used methods from ergodic theory to prove a celebrated result by 2012 Abel Prize Laureate Endre Szemerédi on arithmetic progressions of integers. The insights that came from his proof have led to numerous important results, including the recent proof by Ben Green and Terence Tao that the sequence of prime numbers includes arbitrarily large arithmetic progressions.

Screenshot 2020-03-24 at 15.36.43.png

So, what is ergodic theory?

Ergodic theory relates to probability and what we call ‘random walks’, best explained by thinking about a dog trying to find some treats buried in a garden…

If you hide some treats in your garden and let your dog try to to find them, it will most likely start sniffing in what seems to be an apparently random pattern. However, after a short period of time, the dog will more often than not successfully find the treats. This method of search might not seem to be systematic, but yet the dog is following its instinct telling it to randomly change its direction at regular intervals to maximise its chance of success. You can think of it as moving one step forwards, then flipping a coin to decide whether you next go left or right for one step, and repeating this indefinitely.

In maths, the dog’s behaviour is encoded in the concept of a random walk. A random walk is a mathematical object that describes a path consisting of a succession of random steps in some mathematical space. There are numerous examples of physical systems that are modelled by random walks: the behaviour of gas molecules, stock markets, the statistical properties of neurons firing in the brain… But, random walks can also be seen as a tool to explore a mathematical object, in the same way that the dog tries to understand the garden. Of course, Hillel Furstenberg and Gregory Margulis are not using random walks to find treats in a garden, they do random walks on graphs or on groups in order to reveal the secrets of these objects.

If the trajectory of the dog is ergodic, this means that the dog will eventually get close to the treat in the long term. In fact, if we were to draw a circle around the treat, of any size (even as small as you can possibly imagine), after some finite amount of time the dog will be sniffing inside the circle, and therefore will probably discover the treat. This is ergodic theory in a nutshell.

More information on the Abel Prize announcement can be found on the website of the Norwegian Academy of Sciences and Letters here or in the official citation here.

Exponential Growth explained for the COVID-19 (Coronavirus) Epidemic

Oxford University Mathematician Dr Tom Crawford explains exponential growth in the context of an epidemic such as that for COVID-19/Coronavirus. Beginning with one primary infection we see how the number of cases increases dramatically over a period of only 30 days to more than one thousand. All sources are referenced below.

The value of the reproductive number R0 = 3 is taken from current World Health Organisation estimates. Please see here for more information.

The data for the UK population is from 2018 and is sourced from Statista here.

The data for the COVID-19/Coronavirus death rate is from the Chinese Centre for Disease Control and Prevention. Please see here for more information.

Oxford Mathematician explains SIR disease model for COVID-19 (Coronavirus)

The SIR model is one of the simplest disease models we have to explain the spread of a virus through a population. I first explain where the model comes from, including the assumptions that are made and how the equations are derived, before going on to use the results of the model to answer three important questions:

  1. Will the disease spread?
  2. What is the maximum number of people that will have the disease at one time?
  3. How many people will catch the disease in total?

The answers to these questions are discussed in the context of the current COVID-19 (Coronavirus) outbreak. The model tells us that to reduce the impact of the disease we need to lower the ‘contact ratio’ as much as possible – which is exactly what the current social distancing measures are designed to do.

Produced by Dr Tom Crawford at the University of Oxford.

Visiting Students at St Edmund Hall

Calling all US-based students, if you have ever thought you would like to have me as your college professor, now is your chance. I am currently in charge of the visiting student mathematics programme at St Edmund Hall, which means anyone accepted onto the programme will have weekly tutorials with yours truly. Information on the course specifics and how to apply can be found on the St Edmund Hall website here.

Courses available include (but are not limited to):

Michaelmas Term (Autumn)

  • Linear Algebra
  • Geometry
  • Real Analysis: Sequences and Series
  • Probability
  • Introductory Calculus
  • Differential Equations
  • Metric Spaces and Complex Analysis
  • Quantum Theory

Hilary Term (Winter)

  • Linear Algebra
  • Groups and Group Actions (continues next term)
  • Real Analysis: Continuity and Differentiability
  • Dynamics
  • Fourier Series and PDEs
  • Multivariable Calculus
  • Differential Equations
  • Numerical Analysis
  • Statistics
  • Fluid and Waves
  • Integral Transforms

Trinity Term (Spring)

  • Constructive Mathematics
  • Groups and Group Actions (continued)
  • Real Analysis: Integration
  • Statistics and Data Analysis
  • Calculus of Variations
  • Special Relativity
  • Mathematical Biology

The detailed course synopses, as well as some course materials can be found here.

If you have any questions please get in touch with Tom via the contact form, or the admissions office at St Edmund Hall via admissions@seh.ox.ac.uk.

Photo: Flemming, Heidelberg Laureate Forum

How are humans affecting bird populations in the UK?

Over the past 60 years since bird feeders first became commercially available, humans have been changing bird populations across the UK. The overall effect has generally been positive, with an increase in the prevalence of Wood Pigeons, and a shift in the migration pattern of Eurasian Blackcaps, but as with most changes, there is a word of warning… Live interview with BBC Radio Oxford.

How long is a lightning bolt?

A new record flash stretching all the way from Texas to Kansas was discovered recently in data from the GOES-16 spacecraft, though the record may soon be broken… Live interview with BBC Radio Oxford.


Up ↑