This project will investigate the SIR model and use numeric methods to find solutions to the system of coupled, non-linear differential equations. We will work through the derivation of the model and some assumptions. You will need to use technology, in the form of a spreadsheet (Excel/Google Sheets) or computer code (C, C++, Python, Java, etc) to obtain approximate numeric solutions. Some Background The SIR model is a very useful compartmental model to help understand the spread of disease through a population during some given time. The mathematical model has three compartments: Susceptible members of the population (S), Infected members of the population (I), and Recovered-or Removed-members of the population (R). As with any mathematical model we will make some assumptions that we should be mindful of when using this in a "real-world" context. Lets begin with some initial information. 1. We will assume, for simplicity, that the number of susceptible individuals in a population at some time t can be given by the function S(t), and further that each of the infected and removed individuals will be given by functions I(t) and R(t) respectively. 2. We can also assume that for a population N,S(t)+I(t)+R(t)=N. Explain why this is a reasonable assumption. 3. We will also assume that the population we are studying is closed, that is we never add to the susceptible members and that once removed, you will not be susceptible. Explain why this simplification may not reflect a real-world scenario. Deriving some differential equations: First, lets look at how the susceptible population might be changing. We will fix a model parameter, called , that will be the number of daily contacts each infected person has with a susceptible person resulting in disease infection. We can write it like this: dtdS=NS(t)I(t) Explain why this is reasonable. Why is there an I(t) ? Why do we have NS(t) ? Why is it negative? For reasons of convenience, instead of considering the change in numbers of each compartment we will look at the change in proportion of each compartment, so we can say that: s(t)=NS(t)i(t)=NI(t)r(t)=NR(t) We will get the following changes: s(t)+i(t)+r(t)=1 and the differential equation for change in proportion of susceptible will be: dtds=s(t)i(t) Now consider the change in removed proportion. For this we will need to introduce another parameter, , which you can think of as a fixed proportion of the infected members who will be recovered/removed each day or as the recovery rate: 1/ (days to recover). So we get: dtdr=i(t) We know that s(t)+i(t)+r(t)=1, so it follows that if we take the derivative with respect to t we get: dtd(s(t)+i(t)+r(t))=dtd(1) Write the resulting equation below: Use the equation you found to solve for dtdi and substitute to find dtdi in terms of the model parameters and , and the functions s(t),i(t),r(t). Write it below: Together these three equations give a system of coupled, non-linear differential equations. Fill in the missing e.

1. This project will investigate the SIR model and use numeric methods to find solutions to the
system of coupled, non-linear differential equations. We will work through the derivation of the
model and some assumptions. You will need to use technology, in the form of a spreadsheet
(Excel/Google Sheets) or computer code (C, C++, Python, Java, etc) to obtain approximate
numeric solutions. Some Background The SIR model is a very useful compartmental model to
help understand the spread of disease through a population during some given time. The
mathematical model has three compartments: Susceptible members of the population (S),
Infected members of the population (I), and Recovered-or Removed-members of the population
(R). As with any mathematical model we will make some assumptions that we should be mindful
of when using this in a "real-world" context. Lets begin with some initial information. 1. We will
assume, for simplicity, that the number of susceptible individuals in a population at some time t
can be given by the function S(t), and further that each of the infected and removed individuals
will be given by functions I(t) and R(t) respectively. 2. We can also assume that for a population
N,S(t)+I(t)+R(t)=N. Explain why this is a reasonable assumption. 3. We will also assume that the
population we are studying is closed, that is we never add to the susceptible members and that
once removed, you will not be susceptible. Explain why this simplification may not reflect a
real-world scenario. Deriving some differential equations: First, lets look at how the susceptible
population might be changing. We will fix a model parameter, called , that will be the number of
daily contacts each infected person has with a susceptible person resulting in disease infection.
We can write it like this: dtdS=NS(t)I(t) Explain why this is reasonable. Why is there an I(t) ?
Why do we have NS(t) ? Why is it negative?
For reasons of convenience, instead of considering the change in numbers of each compartment
we will look at the change in proportion of each compartment, so we can say that:
s(t)=NS(t)i(t)=NI(t)r(t)=NR(t) We will get the following changes: s(t)+i(t)+r(t)=1 and the
differential equation for change in proportion of susceptible will be: dtds=s(t)i(t) Now consider
the change in removed proportion. For this we will need to introduce another parameter, , which
you can think of as a fixed proportion of the infected members who will be recovered/removed
each day or as the recovery rate: 1/ (days to recover). So we get: dtdr=i(t) We know that
s(t)+i(t)+r(t)=1, so it follows that if we take the derivative with respect to t we get:
dtd(s(t)+i(t)+r(t))=dtd(1) Write the resulting equation below: Use the equation you found to
solve for dtdi and substitute to find dtdi in terms of the model parameters and , and the functions
s(t),i(t),r(t). Write it below:
Together these three equations give a system of coupled, non-linear differential equations. Fill in
the missing equation from your work above: dtds=s(t)i(t)dtdi=dtdr=i(t) It can be difficult, but not

2. impossible, to find closed-form solutions to these systems of equations, so we will resort to
numeric methods for our solutions. Euler's method: Recall that the formula for Euler's method
that given some differential equation of the form dxdy=f(x,y), we can approximate future values
of the function like this: yn+1=yn+f(xn,yn)h This is really just saying that we can make a linear
approximation for the next value, given our current value, and the slope of a tangent line. I will
also replace h with t since we can consider different lengths of time( t=5 would correspond to 5
days). We will get the following system of difference equations:
sn+1in+1rn+1=sn(snin)t=in+(sninin)t=rn+(in)t Work: 1. Use Euler's method with
t=5,=1/6,=1/3,s(0)=1106,i(0)=106,r(0)=0 and plot the solution curves for the model. Include this
in your submission. 2. Use Euler's method with t=1 and the same initial conditions/parameters
from 1. Plot the solution curves for the model. Include this in your submission. 3. Try a few
different values of and . Describe what behavior they have on the longterm trajectory of the
system. Include a few in your submission with comments on how it affects the long-term
trajectory.
Ihen you are observing an epidemic happening you can estimate some of the model arameters,
specifically, you can estimate as the reciprocal of the days to recover that ras shown above, but
we cannot estimate initially. We will often try to estimate omething called a contact number, we
will call it c and use that to estimate . For this you an look at the equation c=/ which makes
intuitive sense. Since we are looking at the roportion of daily contacts that result in infections
times the number of days infected. ow do you find the value of a contact number? Lets find out.
1. We can find a change in infected with respect to change in susceptible by using the chain rule
from calculus. Find dsdi by using the fact that dsdi=dtdidsdt. 2. Now you can integrate dsdi with
respect to ds to get an expression for i in terms of i and s, as well as some other constants. You
should get that i=s+c1lns+k where k is some constant. Show your work for this. 3. It turns out
that we can estimate c by finding the following: c=tlims(t)1lns(t) We cannot let the time go to
infinity in real life, but we can look at what is happening after a long time of disease spread,
whenever infections are near 0 after some time. Go to the initial problem with =1/6 and =1/3 find
an estimate of the susceptible population after a long time, when infected is near 0 and try to
estimate c. 4. Use your solution from part 3 to try to estimate . Is your answer close to the real
value of 1/3?
A look at heard immunity: When we had the contact number from the part above, it tells us how
many people will be infected by a single individual. We can also use information about contact
numbers to estimate heard immunity numbers and vaccination efficacy. It turns out that the rate
of infection is governed by only a few things. Lets take a look at the equation in more detail.

3. dtdi=s(t)i(t)i(t) Which we can factor and look at like this: dtdi=(s(t))i(t) It follows that i(t) will
always be positive, and the rate of infection will be decided by the terms s(t) 1. Explain why
there will be no disease spread if the initial susceptible amount s(0) is less than 1/c (the
reciprocal of the contact number) 2. Suppose that a disease has a contact number of 4 , and you
have a vaccine with an efficacy 100% (which is not realistic). This means that every person who
receives the vaccine, will be immune from contracting the disease. What percentage of the
population would need to receive the vaccine in order to stop an epidemic from starting? 3.
Suppose that a vaccine is developed with an efficacy of approximately 95%. What proportion of
the population would need to receive the vaccine to prevent an epidemic spread?
Modification for people to become susceptible again. You can also have a case where people
become susceptible to a disease after some time, due to variants or mutations and potential
decaying effects of immunity. This is known as an SIRS model. Suppose that you have a
situation similar to the one in the very first part with the following modification:
dtds=s(t)i(t)+vr(t)dtdi=s(t)i(t)i(t)dtdr=i(t)vr(t) Modify your existing model to make these changes
and use the parameters =1/6,=1/3 and v=1/5 1. Show a printout of the graph for this system. 2.
Describe the behavior of the system. How does this change things?