To solve Exercise 3(e), we need to apply Euler’s Method with a step size $h = 1$ to approximate $S(t)$, the number of infected students, over time. Let’s begin by summarizing the problem:

The differential equation provided is:

$\frac{dS}{dt} = 0.434S \left(1 - \frac{S}{800}\right),$

where $S$ is the number of infected students, and $t$ is the number of days.

Euler's Method Steps:

1. Start with the initial condition $S(0) = 10$.
2. Use the formula: $S_{n+1} = S_n + h \cdot \frac{dS}{dt}(S_n, t_n),$ where $\frac{dS}{dt} = 0.434S \left(1 - \frac{S}{800}\right)$.
3. Continue iterating until the desired $t$ value is reached.

I'll compute the results for the first few days. Define the initial conditions and parameters h = 1 # Step size S0 = 10 # Initial number of infected students t0 = 0 # Initial time total_time = 20 # Maximum time for the calculation

Parameters of the logistic growth

r = 0.434 # Growth rate K = 800 # Carrying capacity

Implement Euler's Method

steps = int(total_time / h) t_values = [t0] S_values = [S0]

Iterate using Euler's method

for i in range(steps): S_current = S_values[-1] t_current = t_values[-1] dS_dt = r * S_current * (1 - S_current / K) S_next = S_current + h * dS_dt t_next = t_current + h

S_values.append(S_next)
t_values.append(t_next)

Prepare results for analysis

list(zip(t_values, S_values))Using Euler's Method with a step size of $h = 1$, we approximate the number of infected students $S(t)$ over time as follows:

The population approaches the carrying capacity $K = 800$ as $t \to 20$.

Comparison:

Euler's Method provides an approximate solution. To compare, you could solve the logistic differential equation analytically to check for deviations. Let me know if you'd like to see that process or any other part in more detail.

Would you like details about:

1. How Euler's Method was applied step-by-step?
2. A visualization of $S(t)$?
3. Solving the differential equation analytically for comparison?
4. Practical significance of this logistic growth model?
5. Adjusting step size $h$ for improved accuracy?

Tip:

Smaller step sizes in Euler’s Method improve accuracy but increase computation time.