Problem 1:
The SIR epidemic model assumes that on average, an infected (and infectious) individual encounters a
people per unit time. The susceptible population diminishes as they are infected (i.e.

Question

Problem 1:
The SIR epidemic model assumes that on average, an infected (and infectious) individual encounters a
people per unit time. The susceptible population diminishes as they are infected (i.e. as they encounter
infectious people).
ds
Erie —aSI.
Note that quarantining would decrease the constant a, which is needed to control the spread of the
disease. Also, the infected population increases with rate a, and decreases as people recover with the
recovery rate k.
dr
— =aSI —kI.
a=
Finally, the recovered population increases with the recovery rate k.
dR
— =kI.
dt

(a) Show that I(t) does not increase if S(0) is less than or equal to k/a.

(b) Find and interpret the equilibria.

(c) Set parameters a = 0.003,k = 0.5,1(0) = 1,S(0) = 700, R(0) = 0. Use Euler's method to plot
S.R, I on the same set of axes. Estimate the time at which the number of infected people is at a
maximum, and the time at which the number of infected people decreases to zero. Make a plot
of the trajectories (5,1).

VIPSolutions · Accepted Answer

Let's break down the **SIR Model Problem** step by step:

---

## **Given Equations**
$$
\frac{dS}{dt} = -aSI
$$
$$
\frac{dI}{dt} = aSI - kI
$$
$$
\frac{dR}{dt} = kI
$$

---

## **(a) Show that $I(t)$ does not increase if $S(0) \leq k/a$.**

From the infected equation:

$$
\frac{dI}{dt} = aSI - kI = I(aS - k)
$$

For $I(t)$ **not to increase**,
$$
\frac{dI}{dt} \leq 0 \implies aS - k \leq 0 \implies S \leq \frac{k}{a}
$$

So if $S(0) \leq \frac{k}{a}$, then $\frac{dI}{dt} \leq 0$, and $I(t)$ does **not increase**.

---

## **(b) Find and interpret the equilibria**

At equilibrium, $\frac{dS}{dt} = 0$, $\frac{dI}{dt} = 0$, $\frac{dR}{dt} = 0$:

From $\frac{dI}{dt} = 0$:
$$
aSI - kI = 0 \implies I(aS - k) = 0
$$
So either $I = 0$ or $S = \frac{k}{a}$.

- If $I = 0$: No infected individuals, disease dies out, and all equations are satisfied (since $dS/dt = dR/dt = 0$).
- If $S = \frac{k}{a}$: The disease is at a threshold, and infection neither increases nor decreases (endemic equilibrium).

**Interpretation:**
- **Disease-free equilibrium:** $I = 0$.
- **Endemic equilibrium:** $S = \frac{k}{a}$ when $I \neq 0$.

---

## **(c) Euler’s Method Simulation**

**Given parameters:**
- $a = 0.003$
- $k = 0.5$
- $I(0) = 1$
- $S(0) = 700$
- $R(0) = 0$

### **Euler’s Method Update Rules**
Let $h$ be the time step:

$$
S_{n+1} = S_n + h \left( -a S_n I_n \right)
$$
$$
I_{n+1} = I_n + h \left( a S_n I_n - k I_n \right)
$$
$$
R_{n+1} = R_n + h \left( k I_n \right)
$$

### **Instructions**
- Choose a time step (e.g., $h = 0.1$).
- Simulate for enough steps until $I \approx 0$.
- Plot $S, I, R$ vs time (and optionally $S$ vs $I$).

### **Estimate:**
- **Maximum $I$:** Occurs when $dI/dt = 0 \implies S = k/a \implies S = 0.5/0.003 \approx 166.67$. Track when $S$ reaches this value, and record $I$.
- **Time when $I \to 0$:** When $I$ drops below a small threshold (e.g., 0.01).

### **Python pseudocode for simulation:**
```python
import numpy as np
import matplotlib.pyplot as plt

# Parameters
a = 0.003
k = 0.5
S0 = 700
I0 = 1
R0 = 0
h = 0.1
steps = 200

S = [S0]
I = [I0]
R = [R0]
t = [0]

for n in range(steps):
    S_new = S[-1] - h * a * S[-1] * I[-1]
    I_new = I[-1] + h * (a * S[-1] * I[-1] - k * I[-1])
    R_new = R[-1] + h * k * I[-1]
    t_new = t[-1] + h

S.append(S_new)
    I.append(I_new)
    R.append(R_new)
    t.append(t_new)

S = np.array(S)
I = np.array(I)
R = np.array(R)

plt.plot(t, S, label='Susceptible')
plt.plot(t, I, label='Infected')
plt.plot(t, R, label='Recovered')
plt.xlabel('Time')
plt.ylabel('Population')
plt.legend()
plt.show()

# Trajectory plot (S, I)
plt.figure()
plt.plot(S, I)
plt.xlabel('S')
plt.ylabel('I')
plt.title('Trajectory (S, I)')
plt.show()

# Estimates
I_max = np.max(I)
t_max = t[np.argmax(I)]
t_end = t[np.where(I < 0.01)[0][0]]

print(f"Maximum infected: {I_max} at time {t_max}")
print(f"Time when infected approaches zero: {t_end}")
```

---

Let me know if you’d like help coding this or further explanation!

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