SIR Model

Definition

The SIR Model is a foundational mathematical model in epidemiology used to simulate the spread of an infectious disease through a population. It partitions the population into three mutually exclusive groups: Susceptible, Infected, and Recovered (or Removed).

Why It Matters

The SIR model is the ‘mathematical immune system’ of public health; it provides the predictive power needed to understand how a disease will spread through a population, allowing for evidence-based decisions on containment and vaccination.

Core Concepts

• Susceptible (S): Individuals who can contract the disease.

• Infected (I): Individuals who have the disease and can transmit it.

• Recovered (R): Individuals who have moved out of the infected group due to recovery (with immunity) or death.

• Model Parameters:

  • Infection Rate ($\lambda$): The rate at which the disease is transferred between S and I.

    • How to read: “Lambda.”
    • Meaning: Infection rate—controls how fast susceptible people become infected when they interact with infected individuals.

  • Recovery Rate ($\delta$): The rate at which I individuals move to R.

    • How to read: “Delta.”
    • Meaning: Recovery rate—how quickly infected people recover or are removed from the infected group.

• Difference Equations: In a continuous simulation, the populations are updated over time intervals ($\Delta t$):

  • $\Delta S = -\lambda S I \Delta t$
  • $\Delta I = (\lambda S I - \delta I) \Delta t$
  • $\Delta R = \delta I \Delta t$
  • How to read: “Delta-S equals negative lambda S I delta-t; delta-I equals lambda S I minus delta I, times delta-t; delta-R equals delta I delta-t.”
  • Meaning / when to use: Discrete-time update rules. New infections proportional to $S \times I$ (mass action); recoveries proportional to $I$. Total $S + I + R$ stays constant.

Connected Concepts

• Continuous Simulation
• System Dynamics
• Euler’s Method
• Interdependence