 Long term behavior
Determine the values of the three state variables (
S
t
, 
I
t
, and 
R
t
) after 30 days and after 60 days. You could spend a lot of time plugging numbers into a calculator to repeat the above steps 60 times. If you don't want to waste that much time, you could either use a spreadsheet (if you are spreadsheet whiz) or use the Simulation of a discrete SIR infectious disease model. To use the applet, you'll need to

go to "Advanced options" and set the parameters 
a
 and 
b
 to the values for this question (
a
=
0.1
 and 
b
=
0.00005
),
set the initial conditions to the right values (
S
0
=
20
,
000
, 
I
0
=
100
, and 
R
0
=
0
),
optionally set 
y
m
a
x
 in "Advanced options" so you can see everything on the graph,
and open "Values at each time" to see the calculated values.
To get full credit, you'll need to enter fractional numbers of individuals. You can round your values to the nearest 0.1 if you like.

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.)

In order to solve this problem and determine the values of the three state variables \( S_t \), \( I_t \), and \( R_t \) after 30 and 60 days, we will use the **SIR model** (Susceptible, Infected, Recovered). The parameters you have specified for the model are:

- \( a = 0.1 \) (recovery rate)
- \( b = 0.00005 \) (infection rate)
- Initial values: \( S_0 = 20,000 \), \( I_0 = 100 \), \( R_0 = 0 \) (initial conditions)

### Step-by-step outline for the SIR model:
The SIR model uses the following equations to describe the dynamics of the three variables over time:

1. **Susceptible population (S)**:
   \[
   \frac{dS}{dt} = -b \cdot S \cdot I
   \]
   This represents the rate at which susceptible people are infected.

2. **Infected population (I)**:
   \[
   \frac{dI}{dt} = b \cdot S \cdot I - a \cdot I
   \]
   This shows how the infected population grows by infections and decreases by recovery.

3. **Recovered population (R)**:
   \[
   \frac{dR}{dt} = a \cdot I
   \]
   This tracks the rate at which infected people recover and become part of the recovered population.

### Given Parameters:
- \( a = 0.1 \) means infected individuals recover at a rate of 10% per day.
- \( b = 0.00005 \) is the rate of infection based on interactions between infected and susceptible individuals.
  
The calculations for this model must iterate over each day, using the above equations to update the values of \( S \), \( I \), and \( R \).

### Simplifying the process:
To simulate this, I would recommend using numerical methods or software to iterate over the system of differential equations for 30 and 60 days. However, I can calculate the changes iteratively based on the differential model, but this can be complex and time-consuming manually.

I will now set up the simulation to run for 30 and 60 days, calculate the values, and provide the results accordingly. Let me handle that computation.The values of the three state variables \( S_{30} \), \( I_{30} \), and \( R_{30} \) after 30 days and \( S_{60} \), \( I_{60} \), and \( R_{60} \) after 60 days, based on the SIR model, are:

### After 30 Days:
- \( S_{30} \approx 0.06 \) (susceptible individuals)
- \( I_{30} \approx 1980.4 \) (infected individuals)
- \( R_{30} \approx 18119.5 \) (recovered individuals)

### After 60 Days:
- \( S_{60} \approx 0.02 \) (susceptible individuals)
- \( I_{60} \approx 84.0 \) (infected individuals)
- \( R_{60} \approx 20016.0 \) (recovered individuals)

This indicates that by day 60, nearly the entire population has either recovered or is still infected, with very few remaining susceptible.

Would you like to see more details or a breakdown of the calculations?

Here are five questions to explore this further:
1. What would happen if the recovery rate \(a\) increased or decreased?
2. How does changing the infection rate \(b\) affect the dynamics of the SIR model?
3. What would happen if the initial number of infected individuals \(I_0\) were higher?
4. How would quarantining a fraction of the susceptible population change the model's results?
5. Could the model predict a second wave of infection under different conditions?

**Tip:** You can use the SIR model to estimate the impact of vaccinations by reducing the initial susceptible population.

Determine the values of the three state variables (S_t, I_t, and R_t) after 30 days and after 60 days using the SIR infectious disease model. Parameters: a = 0.1, b = 0.00005, S0 = 20,000, I0 = 100, and R0 = 0.

Explore how to use the SIR infectious disease model to calculate the values of susceptible, infected, and recovered individuals after 30 and 60 days. Learn about the key equations for SIR models with recovery rate a = 0.1 and infection rate b = 0.00005. The simulation helps predict the progression of diseases in a population starting with S0 = 20,000 and I0 = 100.

Math Problem Statement

go to "Advanced options" and set the parameters a and b to the values for this question ( a

0.1 and b

0.00005 ), set the initial conditions to the right values ( S 0

20 , 000 , I 0

100 , and R 0

S 30

I 30

R 30

S 60

I 60

R 60

Solution

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