# Math Insight

### A discrete SIR infectious disease model

Math 1241, Fall 2020
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Due date: Sept. 18, 2020, 11:59 p.m.
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Total points: 3
1. Initial exploration of model.
• $S_t$ = the number of susceptibles in day $t$
• $I_t$ = the number of infectives in day $t$
• $R_t$ = the number of removed in day $t$.

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*}

1. 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.)

2. 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? $R_1=$
3. 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?

4. 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
.

5. 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.

6. 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
.

7. 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.

8. 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 $S_0 =$
people avoid getting sick in the end, which represents about
% of the original susceptibles.

If you vaccinate $V=19,000$ individuals so that $S_0 =$
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.

9. 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
.

2. Mathematical condition for stopping the spread of the disease.
1. From the previous exploration, you should have found two conditions on the state variables that made the disease stop. Now, it's time to find those conditions (and one more obvious condition for the disease stopping) using mathematics!

The SIR model involves two transitions: one from S to I and one from I to R. Which of these transitions is the one that determines how fast people are getting sick? The
transition.

The rate of this transition is a function of the state variables $S_t$, $I_t$, and $R_t$ and parameters $a$ and $b$ (possibly not involving all of these). What is the expression involving these quantities that is equal to the number of people that get sick in one day?
(When entering your answer online, enter S_t for $S_t$, etc., and use * (or space) for multiplication.)

If the disease isn't spreading, then this expression must be zero. Let's call this the no-spread condition. Write the formula for the no-spread condition:

= 0.

2. The no-spread condition should be an equation where the product of three factors is zero. If the product of three factors is zero, what do you know about the three factors individually? One of the factors must be =
.

You should end up with three different conditions, or three equations where a single quantity is set to zero. These equations are:

(Online, use S_t for $S_t$, etc.)

Each of these conditions alone is sufficient for the disease to stop spreading.

One of the conditions involves $b$; it is:
. This condition means that if the
rate is zero, the disease won't spread. This is a pretty obvious condition, as it correspond to a disease that isn't contagious (or a population where people don't interact with each other.)

Another condition involves $I_t$; it is:
. This condition means that there are no
individuals to infect any of the
individuals. We like this condition for stopping the spread of the disease and hope we hit this condition before we hit the third condition.

The remaining condition involves $S_t$; it is:
. This condition means that the disease has run out of
individuals to infect. We don't like this condition for stopping the spread of the disease, as it means everyone got sick.

3. Outbreak or extinguish.
1. The no-spread condition is a very strong condition. It means absolutely no more people get sick. However, just because some people are still getting infected (i.e., the no-spread condition is not met), the disease might still be winding down (or extinguishing). A weaker condition that still indicates that the disease is subsiding is the condition that the number of infectives is decreasing.

To determine the net increase in the number of infectives, we have to account for the rate at which susceptibles get sick as well as the rate at which infectives recover. In terms of the state variables and parameters, what is the net gain in the number of infectives over the course of a day?
Change in number of infectives =

(As usual, this expression will involve the state variables and the parameters, which you'll enter online as S_t, etc., with * or space for multiplication.)

We'd like to know if the number of infectives is increasing or decreasing. What is the condition that the number of infectives is decreasing?
We'll call this the extinguish condition. (It's likely your answer will involve either < or >.)

2. Now, imagine that the no-spread condition is satisfied. In this case, one of the terms in the extinguish condition should drop out. What does the extinguish condition become if we know the no-spread condition is satisfied?

What do you know about the signs of the parameters and the state variables? They must be
. If there are no infectives, $I_t=0$, then we can't talk about extinguishing, as the number of infectives cannot decrease any more. But, if $I_t \ne 0$ and the no-spread condition is satisfied, we know that the extinguish condition
satisfied.

On the other hand, if the extinguish condition is satisfied, must the no-spread condition be satisfied?
Can you come up with example values of the parameters and state variables where the extinguish condition is satisfied but the no-spread condition is not satisfied?

This is what we mean when we say the extinguish condition is a weaker condition than the no-spread condition. The no-spread condition implies the extinguish condition (at least if $I_t \ne 0$), but not vice-versa.

3. The extinguish condition you derived should have involved two state variables. We can make it simpler because, in order for the extinguish condition to make sense, we assume some infective people are around ($I_t \ne 0$). In this case, you can eliminate one of the state variables from the extinguish condition. Write a simplified extinguish condition as an inequality involving both parameters but just one state variable.

This means that the extinguish condition does not depend on the number of
individuals but depends only on the number of
individuals.

4. When $S_t=20,000$, $a=0.1$, and $b=0.00005$, show that the extinguish condition is not satisfied. If you keep $S_t$ and $b$ fixed at those values, what is the condition on the recovery rate parameter $a$ so that the extinguish condition is satisfied?

Verify with the SIR applet that the initial increase or decrease of the infective population size is predicted by this condition.