Kermack-McKendrick SIR Epidemic Model — 3D Lab Simulation
Compartmental transmission dynamics: Susceptible (S), Infectious (I), Recovered (R)
Theoretical Foundation & Historical Context
Published in 1927 by Kermack and McKendrick, the SIR model quantifies viral propagation and defines the basic reproduction number R0 and herd immunity thresholds.
Law: Kermack-McKendrick Epidemic Transmission Equations | Scientist: William Ogilvy Kermack & Anderson Gray McKendrick | Year: 1927
Key Learning Objectives & Competencies
- ✓ Evaluate the Kermack-McKendrick compartment equations: dS/dt = -βSI/N, dI/dt = βSI/N - γI, dR/dt = γI.
- ✓ Derive the basic reproduction number R₀ = β / γ and understand epidemic growth criteria (R₀ > 1).
- ✓ Calculate the herd immunity threshold HIT = 1 - 1/R₀ required to halt transmission.
Mathematical Formulations & Differential Equations
\begin{cases} \frac{dS}{dt} = -\beta \frac{S I}{N} \\ \frac{dI}{dt} = \beta \frac{S I}{N} - \gamma I \\ \frac{dR}{dt} = \gamma I \end{cases}
System of differential equations for population compartments
R_0 = \frac{\beta}{\gamma}
Basic reproduction number
Physical Parameters & Units of Measurement
| Symbol | Physical Quantity | SI Unit | Default |
|---|---|---|---|
| N | Total Population (N) | người | 1000 |
| beta | Transmission Rate (beta) | 1/ngày | 0.35 |
| gamma | Recovery Rate (gamma) | 1/ngày | 0.1 |
Real-World & Industrial Engineering Applications
Public health epidemic forecasting and vaccination program coverage planning (COVID-19, Influenza, Measles).
Computer virus and cyber-malware botnet propagation dynamics across internet networks.
Information cascade and viral marketing rumor spread modeling on social media platforms.
3-Step Controlled Experimental Protocol
- Step 1: Baseline Setup — Initialize the simulation in standard conditions and observe undisturbed system dynamics.
- Step 2: Parameter Variation — Gradually modulate physical parameters using the control panel sliders.
- Step 3: Verification & Phase Space Analysis — Compare phase space trajectories and conservation metrics against analytical ODE solutions.
Frequently Asked Questions (FAQ)
What does the basic reproduction number R₀ tell epidemiologists?
R₀ represents the average number of secondary infections caused by a single infected individual in a completely susceptible population. If R₀ > 1, the disease spreads exponentially; if R₀ < 1, it naturally dies out.
Why does the number of daily infections peak and decline before everyone gets infected?
Because as more people become infected and recover, the pool of susceptible individuals S(t) drops below the critical threshold N/R₀. At this point, each infected person transmits to fewer than one person on average, causing active cases to decline.
What does 'Flattening the Curve' mean mathematically in the SIR model?
Reducing the transmission rate β (via masks, social distancing, hygiene) lowers the peak height of I(t) and stretches the epidemic over a longer duration, preventing active cases from overwhelming hospital ICU capacities.
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Interactive Simulation: Interactive real-time 60 FPS WebGL simulation is active in your browser. Rotate, zoom, and inspect physical apparatus components with full camera freedom.