(b) Ebola (a hemorrhagic fever) is spreading in a small village with a population size of 2000 inhabitants. Consider that the population is divided into three categories: susceptible, infected, and recovered (or removed). Once someone gets the infection this year, they cannot get the infection again. The village has a healthcare system that can facilitate vaccinations to prevent the spread of Ebola. The average length of the disease is $$ \frac{11}{6} $$ weeks, over which time the person is deemed infected and can spread the disease. The time period for the model is $$ t=\{0,1,2, \ldots, n\} $$, number of weeks. (i) Initially at $$ t=0 $$, only 8 cases are reported. Given that $$ a=1 / \bar{t}=\frac{1}{ ext { (average length of disease in weeks) }} $$, and that after one week, the total number of the infected is 10 , use the discrete SIR model to compute the value of $$ r $$. (ii) At $$ t=0 $$, determine the rate of change of S , the rate of change of I , and the rate of change of R.

Question

(b) Ebola (a hemorrhagic fever) is spreading in a small village with a population size of 2000 inhabitants. Consider that the population is divided into three categories: susceptible, infected, and recovered (or removed). Once someone gets the infection this year, they cannot get the infection again. The village has a healthcare system that can facilitate vaccinations to prevent the spread of Ebola. The average length of the disease is $$ \frac{11}{6} $$ weeks, over which time the person is deemed infected and can spread the disease. The time period for the model is $$ t=\{0,1,2, \ldots, n\} $$, number of weeks. (i) Initially at $$ t=0 $$, only 8 cases are reported. Given that $$ a=1 / \bar{t}=\frac{1}{	ext { (average length of disease in weeks) }} $$, and that after one week, the total number of the infected is 10 , use the discrete SIR model to compute the value of $$ r $$. (ii) At $$ t=0 $$, determine the rate of change of S , the rate of change of I , and the rate of change of R.

UpStudy AI · Accepted Answer

Let's analyze the discrete SIR (Susceptible, Infected, Recovered) model based on the given problem.

### Problem Overview:
- **Population (N):** 2000 inhabitants
- **Categories:** 
  - **S(t):** Susceptible
  - **I(t):** Infected
  - **R(t):** Recovered/Removed
- **Disease Duration:** Average length is $$ \frac{11}{6} $$ weeks, hence recovery rate $$ a = \frac{6}{11} $$ per week.
- **Initial Conditions at $$ t=0 $$:**
  - $$ I(0) = 8 $$
  - $$ S(0) = 2000 - 8 = 1992 $$
  - $$ R(0) = 0 $$
- **After one week ($$ t=1 $$):**
  - $$ I(1) = 10 $$

### (i) Computing the Value of $$ r $$:

The discrete SIR model updates the number of infected individuals as follows:
$$
I(t+1) = I(t) + \frac{r S(t) I(t)}{N} - a I(t)
$$
Given:
$$
I(1) = 10 = 8 + \frac{r \cdot 1992 \cdot 8}{2000} - \frac{6}{11} \cdot 8
$$
Simplifying:
$$
10 = 8 + \frac{r \cdot 15936}{2000} - \frac{48}{11}
$$
$$
10 = 8 + 7.968r - 4.3636
$$
$$
7.968r = 10 - 8 + 4.3636
$$
$$
7.968r = 6.3636
$$
$$
r = \frac{6.3636}{7.968} \approx 0.798 \text{ per week}
$$

**Answer for (i):**
$$
r \approx 0.798 \text{ per week}
$$

### (ii) Determining the Rate of Change at $$ t=0 $$:

Using the discrete SIR model equations:

1. **Change in Susceptible ($$ \Delta S $$):**
$$
\Delta S = S(t+1) - S(t) = -\frac{r S(t) I(t)}{N}
$$
$$
\Delta S = -\frac{0.798 \cdot 1992 \cdot 8}{2000} = -6.3636
$$

2. **Change in Infected ($$ \Delta I $$):**
$$
\Delta I = I(t+1) - I(t) = \frac{r S(t) I(t)}{N} - a I(t)
$$
$$
\Delta I = 6.3636 - 4.3636 = 2
$$

3. **Change in Recovered ($$ \Delta R $$):**
$$
\Delta R = R(t+1) - R(t) = a I(t)
$$
$$
\Delta R = \frac{6}{11} \cdot 8 = \frac{48}{11} \approx 4.3636
$$

**Answer for (ii):**
$$
\begin{align*}
\frac{dS}{dt} & \approx -6.364 \text{ individuals per week} \
\frac{dI}{dt} & = 2 \text{ individuals per week} \
\frac{dR}{dt} & \approx 4.364 \text{ individuals per week}
\end{align*}
$$

### Summary:
- **(i)** The rate $$ r $$ is approximately **0.798 per week**.
- **(ii)** At $$ t=0 $$:
  - **$$ \frac{dS}{dt} \approx -6.364 $$**
  - **$$ \frac{dI}{dt} = 2 $$**
  - **$$ \frac{dR}{dt} \approx 4.364 $$**

Answer:
Problem (ii) Answer

At time t = 0, the discrete SIR model gives

• the rate of change of S is –70⁄11 ≈ –6.364 people per week,

• the rate of change of I is 2 people per week,

• and the rate of change of R is 48⁄11 ≈ 4.364 people per week.

Therefore,

dS/dt = –70/11,

dI/dt = 2,

dR/dt = 48/11.
At time $$ t=0 $$, the rates of change are: - $$ \frac{dS}{dt} \approx -6.364 $$ people per week, - $$ \frac{dI}{dt} = 2 $$ people per week, - $$ \frac{dR}{dt} \approx 4.364 $$ people per week. So, $$ \frac{dS}{dt} = -\frac{70}{11}, \quad \frac{dI}{dt} = 2, \quad \frac{dR}{dt} = \frac{48}{11} $$

Real Tutor Solution

Answer

Solution

Extra Insights

Related Questions

Latest Biology Questions