SEIR Model Tutorial

Complete guide to understanding and using the SEIR (Susceptible-Exposed-Infected-Recovered) epidemic model.

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📋 Table of Contents

• What is the SEIR Model?
• Why SEIR Instead of SIR?
• Mathematical Background
• Parameters Explained
• Incubation Period
• Code Examples
• Comparing SIR vs SEIR
• Advanced Usage
• Common Pitfalls

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🧠 What is the SEIR Model?

The SEIR model extends the classic SIR model by adding an Exposed compartment for individuals who have been infected but are not yet infectious.

Compartments

Compartment	Symbol	Description
Susceptible	S	Healthy individuals who can catch the disease
Exposed	E	Infected but not yet infectious (incubation period)
Infected	I	Infectious individuals who can transmit the disease
Recovered	R	Individuals who recovered and are immune

Flow Diagram

Susceptible (S) → Exposed (E) → Infected (I) → Recovered (R)
     ↓               ↓                ↓               ↓
   Healthy       Incubating        Sick/Fatal      Immune
                (not contagious)   (contagious)

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🤔 Why SEIR Instead of SIR?

The SEIR model is more realistic for diseases with a significant incubation period.

Real-World Examples

Disease	Incubation Period	SEIR Needed?
COVID-19	4-7 days	✅ Yes
Ebola	2-21 days	✅ Yes
Measles	10-14 days	✅ Yes
Seasonal Flu	1-4 days	⚠️ Maybe
Common Cold	1-3 days	⚠️ Maybe
Food Poisoning	6-24 hours	❌ No

When to Use SEIR

Scenario	Recommendation
Incubation period > 2 days	✅ Use SEIR
Quarantine of exposed individuals	✅ Use SEIR
Contact tracing is important	✅ Use SEIR
Quick onset disease	❌ SIR is sufficient

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📐 Mathematical Background

Differential Equations

The SEIR model is described by four differential equations:

dS/dt = -β × S × I / N
dE/dt =  β × S × I / N - σ × E
dI/dt =  σ × E - γ × I
dR/dt =  γ × I

Where:

• N = Total population (S + E + I + R)
• β (beta) = Infection rate
• σ (sigma) = Incubation rate (1/incubation period)
• γ (gamma) = Recovery rate

Conservation Law

Total population remains constant:

S(t) + E(t) + I(t) + R(t) = N = constant

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📊 Parameters Explained

Infection Rate (β) - Same as SIR

Value	Meaning
0.0 - 0.3	Low transmission
0.3 - 0.6	Moderate transmission
0.6 - 1.0	High transmission

Incubation Rate (σ)

σ = 1 / (incubation period in days)

Incubation Period	σ Value
1 day	1.0
2 days	0.5
5 days	0.2
7 days	0.143
10 days	0.1
14 days	0.071

Recovery Rate (γ) - Same as SIR

γ = 1 / (infectious period in days)

Infectious Period	γ Value
2 days	0.5
5 days	0.2
7 days	0.143
10 days	0.1
14 days	0.071

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⏰ Incubation Period

What is Incubation Period?

The time between infection and becoming infectious.

Real-World Incubation Periods

Disease	Incubation Period	σ Value
COVID-19	4-7 days	0.14 - 0.25
Ebola	2-21 days	0.05 - 0.5
Measles	10-14 days	0.07 - 0.1
Flu	1-4 days	0.25 - 1.0
Common Cold	1-3 days	0.33 - 1.0

Basic Reproduction Number (R₀)

For SEIR, R₀ is calculated the same way:

R₀ = β / γ

The exposed compartment affects dynamics, not the final R₀.

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💻 Code Examples

Basic SEIR Simulation

from sir_simulator.core_models.seir_model import run_seir_simulation

# Parameters
beta = 0.5    # Infection rate
sigma = 0.2   # Incubation rate (5 days incubation)
gamma = 0.1   # Recovery rate (10 days infectious)
S0 = 990      # Initial susceptible
E0 = 5        # Initial exposed
I0 = 5        # Initial infected
R0 = 0        # Initial recovered
t_max = 100   # 100 days

# Run simulation
df = run_seir_simulation(beta, sigma, gamma, S0, E0, I0, R0, t_max)

print(df.tail())

Calculate Key Metrics

# Calculate R₀
R0 = beta / gamma
print(f"R₀ = {R0:.2f}")

# Find peak metrics
peak_infected = df['Infected'].max()
peak_exposed = df['Exposed'].max()
peak_day = df[df['Infected'] == peak_infected]['time'].values[0]

print(f"Peak exposed: {int(peak_exposed)}")
print(f"Peak infected: {int(peak_infected)} on day {int(peak_day)}")

# Final outbreak size
final_susceptible = df['Susceptible'].iloc[-1]
total_infected = S0 - final_susceptible
print(f"Total infected: {int(total_infected)}")

Calculate Incubation Period

# Convert sigma to incubation period (days)
incubation_days = 1 / sigma
print(f"Incubation period: {incubation_days:.1f} days")

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📊 Comparing SIR vs SEIR

Code to Compare Both Models

import matplotlib.pyplot as plt
from sir_simulator.core_models.sir_model import run_sir_simulation
from sir_simulator.core_models.seir_model import run_seir_simulation

# Common parameters
beta = 0.5
gamma = 0.1
S0 = 990
I0 = 10
t_max = 150

# Run SIR
df_sir = run_sir_simulation(beta, gamma, S0, I0, 0, t_max)

# Run SEIR (with same parameters + exposure)
sigma = 0.2  # 5 days incubation
E0 = 5
df_seir = run_seir_simulation(beta, sigma, gamma, S0, E0, I0, 0, t_max)

# Plot comparison
plt.figure(figsize=(12, 6))
plt.plot(df_sir['time'], df_sir['Infected'], label='SIR - Infected', 
         color='red', linestyle='-', linewidth=2)
plt.plot(df_seir['time'], df_seir['Infected'], label='SEIR - Infected', 
         color='blue', linestyle='--', linewidth=2)
plt.plot(df_seir['time'], df_seir['Exposed'], label='SEIR - Exposed', 
         color='orange', linestyle=':', linewidth=2)

plt.xlabel('Time (days)', fontsize=12)
plt.ylabel('Population', fontsize=12)
plt.title('SIR vs SEIR Model Comparison', fontsize=14)
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

Key Differences

Feature	SIR	SEIR
Compartments	3	4
Incubation period	❌ No	✅ Yes
Peak timing	Earlier	Later (due to incubation)
Peak magnitude	Higher	Lower (spread out)
Realism	Basic	More realistic
Complexity	Simple	Moderate

Visual Comparison Table

┌─────────────────┬──────────────┬──────────────┐
│ Metric          │ SIR          │ SEIR         │
├─────────────────┼──────────────┼──────────────┤
│ Peak Infected   │ Higher       │ Lower        │
│ Peak Day        │ Earlier      │ Later        │
│ Outbreak Length │ Shorter      │ Longer       │
│ R₀              │ Same β/γ     │ Same β/γ     │
│ Final Size      │ Slightly     │ Slightly     │
│                 │ higher       │ lower        │
└─────────────────┴──────────────┴──────────────┘

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🎯 Advanced Usage

Compare Different Incubation Periods

import matplotlib.pyplot as plt

sigma_values = [0.5, 0.2, 0.1, 0.05]  # 2, 5, 10, 20 days incubation
colors = ['blue', 'green', 'orange', 'red']

plt.figure(figsize=(12, 6))

for sigma, color in zip(sigma_values, colors):
    df = run_seir_simulation(0.5, sigma, 0.1, 990, 5, 5, 0, 150)
    incubation_days = 1/sigma
    plt.plot(df['time'], df['Infected'], 
             label=f'Incubation: {incubation_days:.0f} days', 
             color=color)

plt.xlabel('Time (days)', fontsize=12)
plt.ylabel('Infected Population', fontsize=12)
plt.title('SEIR Model - Effect of Incubation Period', fontsize=14)
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

Quarantine of Exposed Individuals

# Modify simulation to quarantine exposed individuals
# This would require custom code or using the scenario comparator

from sir_simulator.advanced_features.scenario_comparison import ScenarioComparator

comp = ScenarioComparator(population=1000, initial_infected=5, beta=0.3, gamma=0.1)
scenarios, metrics = comp.compare_all_scenarios(days=100)

print(metrics)

Parameter Optimization with Real Data

import numpy as np
from sir_simulator.advanced_features.parameter_optimization import ParameterOptimizer

# Load your real data
# data = pd.read_csv('your_data.csv')
# observed = data['cases'].values
# t = data['day'].values

# For demo, use synthetic data
t = np.linspace(0, 100, 100)
observed = 100 * np.exp(-0.1 * t) + 5 * np.random.randn(len(t))

optimizer = ParameterOptimizer(model_type='seir')
results = optimizer.fit(observed, t, [990, 5, 5, 0])

print(f"β = {results['beta']:.3f}")
print(f"σ = {results['sigma']:.3f}")
print(f"γ = {results['gamma']:.3f}")
print(f"Incubation period = {1/results['sigma']:.1f} days")

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⚠️ Common Pitfalls

1. Confusing σ with Incubation Period

Incorrect	Correct
σ = 5 (meaning 5 days)	σ = 1/5 = 0.2
σ = incubation period	σ = 1/incubation period

Remember: σ is a rate, not a duration.

2. Initial Exposed = 0

If E0 = 0 and I0 > 0, the model works but misses the incubation effect.

3. Too Many Parameters

SEIR has 3 parameters vs SIR's 2. More data needed for fitting.

4. Forgetting to Set E0

# Good
df = run_seir_simulation(beta, sigma, gamma, S0, E0=5, I0=5, R0=0, t_max=100)

# Bad - missing E0
df = run_seir_simulation(beta, sigma, gamma, S0, I0=5, t_max=100)

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📚 Key Takeaways

Concept	Summary
SEIR	S → E → I → R (adds Exposed compartment)
σ	Incubation rate = 1/incubation_days
Incubation	Time from infection to infectiousness
Peak	Later and lower than SIR
Best for	COVID-19, Ebola, Measles

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🔗 Next Steps

• Network Simulation Tutorial - Social network spread
• Parameter Optimization - Fit to real data
• Scenario Comparison - Intervention strategies

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