A BRIEF FOREWORD
This is an exercise for math students, using back-of-the envelope calculations to make a rough prediction. The spread of the viruses between humans, while displaying exponential growth characteristics, is strongly influenced by factors that are far more difficult to model (movement patterns of humans and goods, hygeine, etc), and case data is limited by the use and accuracy of testing regimes.*
Over the next couple of weeks, as we wait for the effects of the latest measures to show up, the effects of countless other measures that were implemented much earlier will also show up, hopefully resulting in overall less cases.*
Please stay up-to-date with the latest updates and health advice, and before ignoring this advice, take a minute to consider the impact on your friends and family and the wider community. Also please consider that even if you are not personally worried about the risks, you could already be infected and contagious, and should act accordingly.*
This article aims to provide a step-by-step problem solving strategy for exponential growth/decay scenarios using a real-life example that as of recently, most people should be able to relate to.
At midnight on 2020-03-29, the Australian Government implemented new measures to restrict the spread of the COVID-19 virus. Because symptoms of the virus can take up to 14 days to manifest, it will be around 14 days after the measures were implemented before the effects become apparent in the number of positive cases.
Using the below data (source) and an exponential function, model the number of cases between 2020-03-11 and 2020-04-03, then predict the number of cases on 2020-04-12 (two weeks after the implementation of the new measures).
1. Establish the analysis framework
The number of cases can be estimated using the below exponential growth/decay function:
N = N0·e(k·t)
- N = the number of cases at a given point in time
- N0 = the number of cases at the start of a time period
- k = the exponential growth rate constant
- t = time elapsed since N = N0
2. Copy down key information from the question; re-write in terms of analysis framework
- when t = 0 days, N = 113
- when t = 23 days, N = 5454
3. Rewrite the question itself in terms of analysis framework
"Predict the number of cases on 2020-04-12 (two weeks after the implementation of the new measures)"
This step is key, and is often where students get stuck. Over a few steps, lets translate the questions from English into Maths:
In Part 1 we established a general equation, and in Part 2 we determined two "pieces on information" that can be plugged into the general equation. Let's first plug them in, then see if we can solve for any of the unknown. After this we will take stock of where we are and decide the next steps.
Using the first piece of information (@ t = 0 days, N = 113):
(113) = N0·ek·(0)
Because e0 = 1 (explanation here), this simplifies to:
N0 = 113
Using the second piece of information (@ t = 23 days, N = 5454):
(5454) = N0·ek·(23)
We now know N0, so plug that in as well:
(5454) = (113)·ek·(23)
We now have 1 equation with 1 unknown (k), which means it is ready to solve (rearrange) for k. Solving for k gives:
k = 0.16855... (step-by-step solution here)
Hint: It is good practice to use more significant figures than usual for exponential problems - your final result will is very sensitive to small changes in the exponential growth constant!
5. Take stock
Originally we had a general equation with four unknowns:
N = N0·e(k·t)
But after plugging in two data points, there are now only two unknowns:
N = 113·e(0.16855·t)
Perfect - using known data, we have taken the general model and turned it into a specific model for Australian cases, and we can graph the Number of Cases (N) vs. Time (t), to give us the predicted number of cases for any time, or simply plug in any time value to get the predicted Number of Cases.
6. Use the specific model to draw a graph
We enter the specific model into a graphing calculator to obtain the below graph (edit in desmos):
Number of Cases N vs. Time (t) [days after 2020-03-11]
Alternatively, you could simply "plug in" the relevant time value to the specific model find the number of cases:
when t = 32 days,
N = 113·e(0.16855·)
N = 24,866 cases
Hopefully you now have a better understanding of the math behind exponential growth, and more importantly, a useful problem solving strategy that can be applied to any problems in this area. If you have any further questions, feel free to drop me a message, or if you are keen to go over a question in depth as soon as possible, book a session now.
Teachers and Professors
Teachers and professors are welcome to use this as lesson material provided the below attribution is included:
Chris Palmieri (2020), Modelling Coronavirus using Exponential Equations - Brisbane Math and Engineering Tuition, source: https://brisbanemetuition.com.au/blog/post-2/