The Covid-19 pandemic
This Project is due by 9:30 a.m. on Thursday, May 6 (our last class meeting).
In this project, you will be creating a system of differential equations to model the Covid-19 pandemic,
and then analyzing your model to make predictions about the course of the pandemic.
You will be creating an S-I-R system of differential equations that models the spread of the Coronavirus
through the population. As you hopefully recall from the worksheet we had on this type of model (it is
still on BB!), this divides the population into subgroups who are (S)usceptiable, (I)nfected, or
(R)ecovered. A system of differential equations models the movement of people into and out of each of
these groups. Once you have set up your model, you will analyze it and use it to make predictions about
the spread of the virus, and the impact that different measures can have on its spread. While the end of
the pandemic is hopefully near, the latest news suggests that we might be in for another wave, so it is
still important to understand the dynamics of its spread.
The system will be coupled and non-linear, and so you will likely not be able to solve for explicit
formulas for the different group sizes. As we’ve seen all semester, however, this doesn’t mean you
won’t be able to analyze the behavior of the solutions. Qualitative methods, including graphical
approximations (like slopefields) and numerical (like Euler’s Method or more sophisticated numerical
approximation algorithms such as MATLAB’s ODE45), are perfectly suited for situations when we cannot
explicitly solve the problem.
The project description will provide guidance on how to build your model, but this is intended an outline
to guide you and not step-by-step instructions. You will need to supplement what I provide with some
research on your own – there will be some helpful mathematics in our book, useful data online, etc.
And you may well think of improvement to the model beyond what I suggest – this outline will produce
“good” results, but not “great”– if you can make it better, you should do so!
You may seek help, and you may collaborate with others in this class, but ultimately what you submit
must be your own work, not just a copied version of someone else’s. And I definitely am not interested
in seeing someone else’s model (published or not) for how the virus is spreading – it is the act of
creating the model that is important here.
Step 1: The first thing you will need to do is select a time and place to model. Choose a focus that will
not be chosen by others in the class – that way you can discuss your work with others with no worry
that you will be getting the same answers (remember, working together is allowed, but multiple people
submitting the same work is not!). As an example of what I mean, you may choose to look at how the
virus first spread in Maryland last spring, but you could also concentrate on a different state, or different
country – if you are from someplace not around here, or have family in a different part of the country or
world, pick a place that is meaningful to you. Then decide whether you are going to work with data
from when the pandemic first struck your region, or maybe a later wave. You have a lot of flexibility
here, but please remember that the emphasis will be on the mathematical modeling, not public policy –
so while it is great that Australia has pretty much eliminated the virus, and Israel has succeeded in
vaccinating the majority of its population, these countries’ efforts should be admired (and possibly
emulated), but would not make for a very interesting mathematical model at this point!
As you build your model, you will need to make some simplifying assumptions, but want this to be at
least somewhat realistic. Record the assumptions you make, and why you chose to make them, and be
sure to discuss these in your analysis.
Start your model simple, and then build it up to make it more realistic – don’t try to do everything at
once! I suggest you start by modeling the spread in the absence of the vaccine (if you are modeling a
time period before the past few weeks it is unlikely that many people were vaccinated anyway), and
then go back and see what the impact would have been if different levels of the population were
Questions to think about:
1) What is the population size of the region you are modeling? Over a few week time period during
the wave that you are studying, what was the size of I? Based on these values, estimate values
of the derivative of I during that time period.
2) Each interaction between a member of S and a member of I has x% chance of spreading the
disease (and moving that person out of S and into I). Recall that interactions between
populations are proportional to the product of the population sizes. Based on your data and
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