**need sections 1 and 4**

The SIR model is a famous compartmental model used to study the spread of infectious disease. For this model, we set:

S(t) number of susceptible people at time t,

I(t) number of infected people at time t, and

R(t) number of recovered people at time t.

The diagram for this compartmental model is as follows:

SI I SIR

Here, and are two parameters we can choose. We assume the disease spreads at too fast a rate for birth rates and death rates (at least, the rate of deaths not due to the disease) to have much of an effect. Thus, we also say that the total population S(t) + I(t) + R(t) is a constant, which we call N. In particular, we can deduce R(t)once we know S(t) and I(t).

In this project, you will be asked to numerically solve the differential equations in the SIR model, as well as to come up with modifications to this model.

Namely, your project should include four sections addressing the following:

1. Section 1: write down the system of differential equations represented by the above compartmental diagram. Choose some values for the parameters and and numerically solve the system for S(t) and I(t) using Eulers method. You should do this part using computer softwareeither write some code or use a spreadsheet. Describe how varying and changes your solutions. You should include at least two graphs in this section to illustrate your point.

2. Section 2: What are the equilibrium points of your system of equations from section 1? Are they stable or unstable?

3. Section 3: Draw a new compartmental model that includes a new compartment, Q, of people who are socially isolated and much less likely to come in contact with an infected person. Write down the system of equations for this model. Use Eulers method to solve this system for some choice of parameters. How does social isolation change your solution to I(t) compared to your solutions from section 1? Again, include at least two graphs to illustrate your point.

4. Section 4: Come up with some other modification to the SIR model. For instance, you might come up with a model that includes two separated populations of people that are geographically separated from each other. Or maybe you have a model that assumes infected people choose to self- isolate once they start showing symptoms, but not before. Its up to you! Whatever you choose, draw the associated diagram and write down the associated system of differential equations.