and update rule
\begin{align*}
S_{t+1}  S_t &= b S_t I_t\\
I_{t+1}  I_t &= b S_tI_t  aI_t\\
R_{t+1}  R_t &= a I_t.
\end{align*}

Set the parameters to be $a=0.1$ and $b=0.00005$ and the initial conditions to $S_0=20,000$, $I_0=100$, and $R_0=0$.
First some simple questions to make sure you are awake. Given these initial conditions, how many people were sick on day zero?
How many were susceptible on day zero?
OK, now you have a simple calculation. How many people were susceptible on day zero and then became sick on the first day?
How many people were sick on day zero and then recovered on the first day?
(We are being cheerful and saying that people in the removed class have recovered from the disease.)
Hint
This question is not asking you for the number of sick or recovered people on day 1. Instead, it asking about what changes occurred. If you viewed the SIR model as having an arrow from S to I and another arrow from I to R, the question is asking about the numbers associated with those arrows.
How many people changed from being susceptible to being sick? If all sick people went to the hospital, the question would be asking how many new people were admitted to the hospital.
How many people changed from being sick to being recovered? Again, if all sick people went the hospital, the question would be asking how many people were released from the hospital.
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After the first day, what is the total number of people who are susceptible? $S_1 =$
What is the total number of infectives? $I_1 =$
How many have recovered from the disease? $R_1=$
Hint
If you answered the first part correctly, then this answer is easy. Just subtract off how many people got sick from $S_0$ to get $S_1$. Just add those same people who got sick to $I_0$ and subtract off those who recovered to get $I_1$. And, add those same people who recovered to $R_0$ to get $R_1$.
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Determine the values of the three state variables ($S_t$, $I_t$, and $R_t$) after 30 days and after 60 days. (The applet from the SIR model page might be useful here.)
$S_{30} = $
, $I_{30} = $
, $R_{30} =$
$S_{60} = $
, $I_{60} = $
, $R_{60} =$
(Include at least one decimal in your answer, though of course, we can't have fractional people.)
What happens to the number of susceptible people as time passes? Therefore, how severe was the disease?

What does the parameter $b$ mean? It controls the rate that
people
.
Therefore, if we decrease $b$, what should happen to the course of the disease? The outcome of the disease should be
severe.
Keeping $a=0.1$ and the initial conditions as specified above, find a value of $b$ where the outcome of the disease is better than with the case above. By better, we mean that, after a long time, the number of people that are still susceptible should
.
Hint
The parameter $b$ controls the rate that a certain class of folks (susceptible, infected, or recovered) transition to another class of folks. If we decrease $b$, that transition will happen more slowly. How will that affect the disease? (You can always play with the
applet to see what happens.)
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The state variables represents numbers of people. Therefore, to be reasonable, these values should be $\ge$
.
This model, however isn't perfect. Can you find a value of $b$ where this reasonable condition is violated? Oops. Like all mathematical models, this model isn't perfect. If you get results like this, stay alert enough to toss out the results of the model.
Ignoring that problem, it still doesn't make sense for $b$ to any number. What is a reasonable condition on $b$? We should require that $b \ge$
. We need to have sanity checks on our models.
Hint
It doesn't make sense if we have negative numbers of people. How can the model give a negative number for a quantity like the number of susceptible individuals?
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What does the parameter $a$ mean? It controls the rate that
people
.
Therefore, if we increase $a$, what should happen to the course of the disease? The outcome of the disease should be
severe.
Keeping $b=0.00005$ and the initial conditions as specified above, find a value of $a$ where the outcome of the disease is better than with the first case above. By better, we mean that, after a long time, the number of people that are still susceptible should
.
Hint
How does $a$ determine the recovery rate? How does recovery rate influence the rate at which susceptible people get infected?
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If a drug helped healthy people resist getting infected, it would
parameter
.
If a drug helped sick people recover more quickly, it would
parameter
.
A policy that quickly quarantined sick people is trying to
parameter
.
One goal educating people to avoid practices that spread infection is to
parameter
.
All else being equal, a disease in which infected people died quickly (and were promptly buried or cremated) would correspond to a disease with a
value of the parameter
. Therefore, it would likely lead to a
spread of the disease among the population, compared to a disease where infected people died more slowly.

Another way to control the spread of a disease is through vaccinations. When you vaccinate people, you decrease the number of
people.
In the original scenario (with $a=0.1$, $b=0.00005$ and the initial conditions to $S_0=20,000$, $I_0=100$, and $R_0=0$), everyone got the disease and recovered (or, if we aren't feeling so cheerful, we could say everyone was wiped out by the disease). Let's imagine, instead, that at time zero, we vaccinated $V$ people, where $V$ is some number. In this case, we have effectively reduced $S_0$ to a smaller value, $S_0 = 20,000  V$.
If we vaccinated all the people, i.e., set $V=20,000$, then the initial number of susceptible people would be $S_0 = $
. In this case, we would stop the disease in its tracks, as there would be no one new to infect. One question is: do we need to vaccinate everyone to make the disease stop before it infects almost everyone?
Experiment with the applet to see the effect of vaccination. If you vaccinate half the people, i.e. set $V=10,000$ so that $S_0 = $
, about how many people avoid getting sick in the end (i.e., how many susceptibles are left at the end)?
(Round to the nearest integer.) What percentage of the original susceptible folks ($S_0$) remain healthy?
%
If you vaccinate $V=15,000$ individuals so that $S_0 =$
, about
people avoid getting sick in the end, which represents about
% of the original susceptibles.
If you vaccinate $V=19,000$ individuals so that $S_0 =$
, about
people avoid getting sick in the end, which represents about
% of the original susceptibles. In this last example, the disease hardly spread at all, demonstrating that you don't need to vaccinate all people in order to eradicate the disease.

Without vaccinations and $S_0=20,000$, the disease only stopped when it ran out of susceptibles to infect (i.e., $S_t$ went to zero for large time). When you vaccinated enough people, the disease should have stopped without the susceptibles going to zero. Instead, what caused the disease to stop? The number of
individuals went to
.
Hint
What condition would allow a susceptible person to be around and not risk getting infected?
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