ACADEMIC TALK
Watch the lecture by Dr Tom Crawford below and don’t forget to take a look at the extra resources and have a go at the activity at the end.
Disease modelling
Dr Tom Crawford
During this lecture, you will see how two different areas of maths can be used to model the spread of diseases through a population. This lecture illustrates how what we experience in life (people who become ill, people who recover, and how easily a disease is spread) can be written as mathematical equations to explain what has happened, and (perhaps more importantly) predict what will occur. You will see what the ‘R0’ number means in terms of disease modelling, why it matters, and how we can try to reduce it.
In this lecture, Dr. Crawford goes through how we model disease spread through a population.
This is the model famous for tracking disease spread through the COVID-19 Pandemic. We will come back to this model later. First, let’s look at some questions:
What is the difference between a polynomial sequence and an exponential sequence?
Use Desmos to show this difference:
Desmos | Graphing Calculator
Why do we need to find 3x = xk ?
To solve this equation, why do we need to find k?
Further resources
If you want to go deeper and find out what logarithms are, here is a resource: Introduction to Logarithms.
Here are some practice questions on it!
Q1. Calculate the smallest whole number value of x for the exponential function to ‘win’ when:
k=5
k=6
Q2. If we now instead consider the exponential function 2x, what is the general formula for calculating when it will ‘beat’ the polynomial function xk for some whole number k?
Q3. What is the smallest whole number value of x for the exponential 2x to ‘win’ when k=4?
Now let’s return to the SIR Model:
What does change in susceptibles stand for?
What does change in infected stand for?
What does change in recovered stand for?
Here is the way to calculate the R0 (reproduction) number and some questions:
Let r signify the rate of transmission, a signify the rate of recovery/death, and S0 signify the initial susceptible population.
How can we reduce the value of R0?
If we take R0 = 3 as was originally estimated for the initial spread of COVID-19, and assume that the entire population is initially susceptible, what values could r and a take?
What happens to the graphs when you increase the ‘incubation time’ (bottom slider) to a value above zero?
What do you think this represents?
Using the values you calculated in Q2, plot the appropriate graphs at mathigon.org/pandemic and use them to answer the following questions:
What is the maximum % of the population that is infected at any one time?
What happens to the graphs if you increase the value of the transmission rate r?
What happens if you decrease the value of r?
What happens to the graphs if you increase the value of the recovery rate a?
What happens if you decrease the value of a?
Answers
