To solve the given IVP (Initial Value Problem) using Euler's method with a step size of $h = 0.2$, we need to implement the method for the differential equation:

$I'(t) = kI(t)(L - I(t))$

Given the parameters:

• $L = 25,000$ (total population)
• $k = 0.00003$ (infection rate constant)
• Initial condition: $I(0) = 250$
• Time interval: $t \in [0, 60]$
• Step size: $h = 0.2$

Euler's method approximates the solution by iterating with the following formula:

$I(t_{n+1}) = I(t_n) + h \cdot I'(t_n)$

Where $t_n$ is the current time step, $I(t_n)$ is the current value of the infected population, and $I'(t_n) = kI(t_n)(L - I(t_n))$ is the rate of change at that time.

Let’s break down the steps:

1. Start with $t_0 = 0$ and $I(0) = 250$.
2. Use Euler's method to compute $I(t_n)$ for each $n$ by iterating from $t_n$ to $t_{n+1} = t_n + h$ until $t_n = 60$.
3. Update $I(t)$ using the formula:
   $I(t_{n+1}) = I(t_n) + h \cdot k \cdot I(t_n) \cdot (L - I(t_n))$
4. Repeat until we reach $t = 60$.

Python Implementation for Euler's Method

Here's how we can implement Euler's method in Python:

pythonCopy code
import numpy as np
import matplotlib.pyplot as plt

# Parameters
L = 25000  # Total population
k = 0.00003  # Infection rate constant
I0 = 250  # Initial number of infected individuals
h = 0.2  # Step size
t_max = 60  # Maximum time (days)

# Initialize arrays
t_values = np.arange(0, t_max + h, h)  # Time steps from 0 to 60 with step size h
I_values = np.zeros(len(t_values))  # Array to store infected population values
I_values[0] = I0  # Initial infected population

# Euler's method to approximate the solution
for n in range(1, len(t_values)):
    I_values[n] = I_values[n - 1] + h * k * I_values[n - 1] * (L - I_values[n - 1])

# Plot the results
plt.plot(t_values, I_values, label="Euler's Approximation")
plt.xlabel('Time (days)')
plt.ylabel('Infected Individuals I(t)')
plt.title("Euler's Method Approximation of Epidemic Spread")
plt.grid(True)
plt.legend()
plt.show()

Explanation:

• Time Steps: The time vector $t_{\text{values}}$ is created using np.arange, starting from $0$ to $60$ with a step size $h = 0.2$.
• Euler's Update: For each time step, the infected population $I(t)$ is updated using Euler's formula.
• Plot: The values of $I(t)$ are then plotted against time.

Let me know if you'd like me to run this or if you have any questions about the implementation!