The Susceptible–Infectious–Removed (SIR) model is the foundational framework in mathematical epidemiology. Originally proposed by W. O. Kermack and A. G. McKendrick in 1927, this model provides a deterministic mechanism to analyze how infectious diseases evolve within a population over time.
1. Core Compartmental Structure
The model partitions the total population (N) into three mutually exclusive categories based on disease status:
- Susceptible (S): Individuals who are uninfected and lack immunity. They are “at risk” and exit this compartment by becoming infected.
- Infectious (I): Individuals currently carrying the disease and capable of transmitting it. In the basic model, there is no latent period; they are immediately contagious.
- Removed (R): Individuals removed from the transmission chain via recovery (with immunity), isolation, or death. They do not return to the susceptible state.
Conservation Law
The total population size is fixed, meaning the sum of individuals in all states remains constant at any time (t):
S(t) + I(t) + R(t) = N
2. Mathematical Specification and Dynamics
The dynamics of the SIR model are governed by a system of ordinary differential equations (ODEs).
The Governing Equations
Using β (beta) for transmission and γ (gamma) for recovery:
dS/dt = −β ⋅ (S ⋅ I) / N
dI/dt = β ⋅ (S ⋅ I) / N − γ ⋅ I
dR/dt = γ ⋅ I
Note: The negative sign in the first equation indicates the Susceptible population is constantly decreasing.
Parameter Definitions and Ranges
| Parameter | Symbol | Definition | Range |
| Transmission Rate | β | The product of contact rate and transmission probability. | β > 0 |
| Recovery Rate | γ | The rate at which individuals move to the Removed class. | γ > 0 |
| Infectious Period | 1/γ | Average duration an individual remains infectious. | Time > 0 |
| Total Population | N | The total number of individuals in the system. | Integer > 0 |
3. Threshold Phenomena and Râ‚€
The model identifies critical thresholds that determine if an outbreak will become an epidemic.
Basic Reproduction Number (Râ‚€)
This is the most critical metric in epidemiology. It represents the expected number of secondary infections generated by one case in a completely susceptible population.
R₀ = β / γ
- If Râ‚€ > 1: The disease will spread (Epidemic).
- If R₀ ≤ 1: The disease will die out (Disease-free equilibrium).
The Epidemic Threshold
An epidemic will terminate before everyone is infected. The spread stops when the density of susceptible people drops below a critical level, known as the Herd Immunity threshold:
S(t) / N < 1 / Râ‚€
4. Limitations
While powerful, the basic deterministic SIR model has limitations:
- Homogeneous Mixing: Assumes everyone has an equal chance of contacting everyone else, ignoring social structures.
- No Latent Period: Assumes immediate infectiousness (unlike SEIR models which add an “Exposed” phase).
- Deterministic: Ignores random (stochastic) effects, which are crucial when case numbers are low.
- No Vital Dynamics: Ignores births and natural deaths, making it suitable only for short-term outbreaks.
📚 Reference
Seminal Paper:
Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115(772), 700-721.