Let’s solve the problem step by step: --- ### **Step 1: Identify Parameters and Their Values (Part a)** **Parameters needed:** - **β (beta):** Infection rate (vampire bites per susceptible per day) - **γ_v (gamma_v):** Rate at which bitten people become vampires (per day) - **γ_r (gamma_r):** Rate at which bitten people recover (per day) - **p_g:** Proportion of people with garlic (who recover) **Values from the problem:** - A vampire bites **3 people per day**: β = 3 - Becomes vampire **5 days after bitten**: γ_v = 1/5 per day - Garlic recovery in **2 days**: γ_r = 1/2 per day - **40%** of population prepared (has garlic): p_g = 0.4 **End of Step 1 Explanation:** We identified the rates and proportions that control the flow between the populations: susceptible, bitten, vampire, and recovered. --- ### **Step 2: Write Differential Equations (Part b)** Let: - S = Fraction of susceptible people - B = Fraction of bitten people - V = Fraction of vampires - R = Fraction of recovered people **Flows:** - S → B (via bites, at rate β * S * V) - B → V (those who don’t have garlic, at rate (1 - p_g) * γ_v * B) - B → R (those with garlic, at rate p_g * γ_r * B) **Equations:** \[ \frac{dS}{dt} = -\beta S V \] \[ \frac{dB}{dt} = \beta S V - (1-p_g)\gamma_v B - p_g\gamma_r B \] \[ \frac{dV}{dt} = (1-p_g)\gamma_v B \] \[ \frac{dR}{dt} = p_g\gamma_r B \] **End of Step 2 Explanation:** These equations describe the rate of change for each population based on the given scenario. --- ### **Step 3: Plot Population Fractions Over Time (Part c)** **Initial conditions:** - S(0) = 0.999 - V(0) = 0.001 - B(0) = 0 - R(0) = 0 **MATLAB/Python Example (using scipy):** ```python import numpy as np from scipy.integrate import odeint import matplotlib.pyplot as plt # Parameters beta = 3 gamma_v = 1/5 gamma_r = 1/2 p_g = 0.4 # Differential equations def deriv(y, t, beta, gamma_v, gamma_r, p_g): S, B, V, R = y dSdt = -beta * S * V dBdt = beta * S * V - (1 - p_g) * gamma_v * B - p_g * gamma_r * B dVdt = (1 - p_g) * gamma_v * B dRdt = p_g * gamma_r * B return [dSdt, dBdt, dVdt, dRdt] # Initial conditions S0 = 0.999 B0 = 0 V0 = 0.001 R0 = 0 y0 = [S0, B0, V0, R0] # Time grid t = np.linspace(0, 100, 1000) # Integrate equations ret = odeint(deriv, y0, t, args=(beta, gamma_v, gamma_r, p_g)) S, B, V, R = ret.T # Plot plt.figure(figsize=(10,6)) plt.plot(t, S, label='Susceptible') plt.plot(t, B, label='Bitten') plt.plot(t, V, label='Vampire') plt.plot(t, R, label='Recovered') plt.xlabel('Time (days)') plt.ylabel('Fraction of Population') plt.title('Vampire Apocalypse SIR-SEIR Model') plt.legend() plt.show() ``` **End of Step 3 Explanation:** This code numerically solves the system and plots the evolution of the four population fractions over time. --- ### **Final Answer:** **Parameters:** β = 3, γ_v = 1/5, γ_r = 1/2, p_g = 0.4 **Equations:** \[ \frac{dS}{dt} = -\beta S V \\ \frac{dB}{dt} = \beta S V - (1-p_g)\gamma_v B - p_g\gamma_r B \\ \frac{dV}{dt} = (1-p_g)\gamma_v B \\ \frac{dR}{dt} = p_g\gamma_r B \] **Initial conditions:** S(0) = 0.999, B(0) = 0, V(0) = 0.001, R(0) = 0 **Plot:** Use the provided code to see the time evolution of each population. Let me know if you want a plot or explanation of the results!