SIR Model — Epidemiological Simulator

Infectious Disease Dynamics — SIR Model (Susceptible, Infected, Recovered) · R₀ = β / γ

Input Parameters

Historical Presets

Total Population (N)

Total number of individuals in the population

10001.000e+7

Initial Infected (I₀)

Number of infected individuals at the start

1100000

Transmission Rate (β)

Infective contacts per day

Recovery Rate (γ)

Daily recovery rate

Simulation Time

Simulation duration in days

days

10730

Results

R₀ (Basic Reproduction Number)

2.5

Herd Immunity Threshold (HIT)

Infection Peak

23350

Peak Day

days

Incidence Peak

2936

Peak Incidence Day

days

Total Infected (cumulative)89.3K

Epidemic Duration214 days

Epidemic expansion phase: R₀ > 1 — each infected person transmits to more than one on average

dS/dt = −β·S·I/N | dI/dt = β·S·I/N − γ·I | dR/dt = γ·I

Fundamentals & Explanation

What is the SIR model?

The SIR model is a mathematical tool that describes how an infectious disease spreads through a population. It divides people into three groups that change over time: Susceptible (S) — those who can catch the disease, Infected (I) — those currently sick and contagious, and Recovered (R) — those who have recovered and are immune.

People move through these groups in one direction only: S → I → R. The model assumes a closed population of constant size N, so S + I + R = N at all times.

It was formalized by Kermack & McKendrick (1927) and remains the foundation of modern epidemiological modeling, including more complex variants like SEIR (adds an Exposed period) and SIRS (allows reinfection).

How does the spread work?

The rate of change of each group is governed by two parameters: β (how easily the disease spreads per day) and γ (how quickly people recover — γ = 1 / average days sick). The equations below determine exactly how fast each group grows or shrinks each day:

\frac{dS}{dt} = -\frac{\beta \cdot S \cdot I}{N}

\frac{dI}{dt} = \frac{\beta \cdot S \cdot I}{N} - \gamma \cdot I

\frac{dR}{dt} = \gamma \cdot I

The term β·S·I/N captures the "collision" between susceptible and infected individuals — the more of either group there is, the more new infections occur. As S falls (fewer people left to infect), the epidemic naturally slows down.

Limitations of this model

The SIR model is a simplification of reality. It is most useful for understanding general epidemic dynamics, not for precise forecasting. Key assumptions to keep in mind:

•Homogeneous mixing: everyone is equally likely to contact everyone else — no geography, age groups, or social networks.
•No incubation period: infected individuals are immediately contagious. The SEIR model adds an Exposed (E) compartment to fix this.
•Permanent immunity: once recovered, individuals cannot be reinfected. Not true for diseases like influenza or COVID-19 variants.
•Constant β and γ: the model does not account for interventions like lockdowns, vaccines, or behavioral changes over time.

Basic reproduction number (R₀)

R_0 = \frac{\beta}{\gamma}

R₀ answers: "how many people does one sick person infect on average?"It is the single most important number in the model:

R₀ > 1— epidemic grows. Each person infects more than one.
R₀ < 1— epidemic fades. The disease dies out naturally.
R₀ = 1— stable state. Each person infects exactly one other.

Reference: Delamater et al. (2019), Emerging Infectious Diseases

Herd immunity threshold (HIT)

HIT = \left(1 - \frac{1}{R_0}\right) \times 100\%

The minimum percentage of the population that must be immune (through vaccination or prior infection) for the epidemic to start declining. The higher the R₀, the harder it is to reach herd immunity.

Reference: WHO — Herd immunity and COVID-19

Reference values

Disease	β	γ	R₀	HIT	Source
COVID-19 (Original)	0.25	0.10	2.5	60%	↗
Seasonal Influenza	0.26	0.20	1.3	23%	↗
Measles	1.50	0.10	15	93%	↗

β and γ values are illustrative approximations for this simulator. Real-world values vary by region, variant, and study methodology.