A new infectious disease has arrived in Australia. The effects of the disease are mild, but it is highly infectious. Epidemiologists have developed a model for the spread of the disease. They classify the population in 3 groups: ut the number of people that have never been infected by day t, st the number of people sick on day t, and rt the number of people that have recovered from one (or more) previous bouts of the disease by day t. They estimate that each day 20% of uninfected people fall sick, as well as 10% of those that have recovered from a previous bout. Each day 70% of those currently sick (st), recover. The disease arrives on day 0.

Find a matrix A in such a way that (u_t+1, s_t+1, r_t+1) = A(u_t, s_t, r_t)

To find the matrix \( A \) such that the system of equations governing the evolution of the populations \( u_t \), \( s_t \), and \( r_t \) is modeled as:

\[
\begin{pmatrix}
u_{t+1} \\
s_{t+1} \\
r_{t+1}
\end{pmatrix}
= A \begin{pmatrix}
u_t \\
s_t \\
r_t
\end{pmatrix}
\]

We will break the problem into the evolution of each group—uninfected (\( u_t \)), sick (\( s_t \)), and recovered (\( r_t \))—and express each equation in terms of the values at time \( t \).

### Evolution Equations:
1. **Uninfected Group (\( u_t \)):**
   - Each day, 20% of uninfected people (\( u_t \)) fall sick.
   - Therefore, the number of uninfected people the next day is:
   \[
   u_{t+1} = 0.8u_t
   \]
   (because 20% fall sick, 80% remain uninfected).

2. **Sick Group (\( s_t \)):**
   - On day \( t \), 20% of uninfected people become sick.
   - In addition, 10% of those who have recovered (\( r_t \)) become sick again.
   - So, the total number of new sick people is \( 0.2u_t + 0.1r_t \).
   - Also, 70% of those currently sick recover, so 30% of \( s_t \) remains sick on the next day.
   - Therefore, the number of sick people the next day is:
   \[
   s_{t+1} = 0.2u_t + 0.3s_t + 0.1r_t
   \]

3. **Recovered Group (\( r_t \)):**
   - Every day, 70% of the sick people recover, so the number of people who recover in one day is \( 0.7s_t \).
   - Therefore, the number of recovered people the next day is:
   \[
   r_{t+1} = r_t + 0.7s_t
   \]

### Constructing the Matrix \( A \):
Now, we express these equations in matrix form. The system of equations is:

\[
\begin{pmatrix}
u_{t+1} \\
s_{t+1} \\
r_{t+1}
\end{pmatrix}
=
\begin{pmatrix}
0.8 & 0 & 0 \\
0.2 & 0.3 & 0.1 \\
0 & 0.7 & 1
\end{pmatrix}
\begin{pmatrix}
u_t \\
s_t \\
r_t
\end{pmatrix}
\]

Thus, the matrix \( A \) is:

\[
A = \begin{pmatrix}
0.8 & 0 & 0 \\
0.2 & 0.3 & 0.1 \\
0 & 0.7 & 1
\end{pmatrix}
\]

This matrix represents the transition dynamics of the system from day \( t \) to day \( t+1 \).

---

Would you like more details or have any questions on this model?

Here are 5 related questions for further exploration:
1. How would the model change if a vaccine reduced the infection rate for recovered individuals?
2. What is the long-term behavior of the population (i.e., equilibrium values of \( u_t \), \( s_t \), and \( r_t \))?
3. How can we calculate the total number of sick people over time?
4. What happens if the recovery rate from sickness changes to a different value, say 50% per day?
5. How would introducing immunity after the first infection affect the model?

**Tip**: Understanding how the model evolves daily can help predict the overall impact of a disease in the long term and assess intervention strategies.

A new infectious disease has arrived in Australia. The effects of the disease are mild, but it is highly infectious. Epidemiologists have developed a model for the spread of the disease. They classify the population in 3 groups: ut the number of people that have never been infected by day t, st the number of people sick on day t, and rt the number of people that have recovered from one (or more) previous bouts of the disease by day t. They estimate that each day 20% of uninfected people fall sick, as well as 10% of those that have recovered from a previous bout. Each day 70% of those currently sick (st), recover. The disease arrives on day 0. Find a matrix A in such a way that (u_t+1, s_t+1, r_t+1) = A(u_t, s_t, r_t)

Explore how a matrix model can be used to predict the spread of a highly infectious disease in Australia. Learn the equations governing the populations of uninfected, sick, and recovered individuals, with step-by-step guidance. Suitable for university-level students in mathematics and epidemiology.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation