SIR Epidemic Calculator

What is the SIR Model in Epidemiology?

The SIR model is a foundational mathematical framework in epidemiology developed by Kermack and McKendrick in 1927. It models the transmission dynamics of infectious diseases by categorizing a population into three distinct, mutually exclusive compartments:

Susceptible ($S$): Individuals who are healthy and uninfected but can contract the infection upon contact with an infected person.
Infected ($I$): Individuals who currently harbor the pathogen and are actively transmitting it to susceptible hosts.
Recovered / Immune ($R$): Individuals who have cleared the infection, developed immunity, or been isolated/vaccinated and are no longer transmitting or susceptible.

Mathematical Differential Equations

Assuming a closed population of constant size $N = S + I + R$, the transition rates between compartments over continuous time $t$ are described by three non-linear ordinary differential equations (ODEs):

$$\frac{dS}{dt} = -\frac{\beta S I}{N}$$ $$\frac{dI}{dt} = \frac{\beta S I}{N} - \gamma I$$ $$\frac{dR}{dt} = \gamma I$$

Where:

$\beta$ (Beta): The effective transmission rate per day (the contact rate multiplied by the transmission probability per contact).
$\gamma$ (Gamma): The recovery rate per day, representing the inverse of the mean infectious period ($D = \frac{1}{\gamma}$).
$N$: Total population size.

Key Epidemiological Metrics

Metric	Formula	Clinical Interpretation
Basic Reproduction Number ($R_0$)	R₀ = β / γ	Average secondary cases infected by one infected individual in a fully susceptible population. If $R_0 > 1$, an outbreak grows.
Herd Immunity Threshold (HIT)	HIT = 1 - (1 / R₀)	The minimum fraction of the population that must acquire immunity (via recovery or vaccination) to stop sustained transmission.
Mean Infectious Period	D = 1 / γ	Average duration in days that an infected person remains contagious.
Attack Rate	(N - S_final) / N	Cumulative percentage of the population infected by the end of the epidemic wave.

Explore our related clinical tools including the Vaccine Efficacy Calculator and Incidence Rate Calculator for comprehensive epidemiological risk modeling.

Frequently Asked Questions

What is the significance of R₀ in the SIR model?

The basic reproduction number ($R_0 = \beta / \gamma$) indicates whether a disease outbreak will expand exponentially ($R_0 > 1$) or fizzle out naturally ($R_0 \le 1$). For instance, measles has an $R_0 \approx 12-18$, seasonal influenza $R_0 \approx 1.3$, and early SARS-CoV-2 $R_0 \approx 2.5-3.0$.

How does herd immunity stop an epidemic?

When the proportion of immune individuals exceeds the herd immunity threshold ($1 - 1/R_0$), the effective reproduction number ($R_t = R_0 \times S/N$) drops below 1. Each infected person transmits to fewer than one susceptible host on average, causing transmission chains to collapse.

What are the core assumptions of the basic SIR model?

The standard SIR model assumes a closed population without births or natural deaths, homogeneous mixing (everyone has equal contact probability), immediate infectiousness upon exposure (no latent/incubation period), and permanent lifelong immunity following recovery.

How do public health interventions flatten the curve?

Interventions like social distancing, mask wearing, hand hygiene, and quarantine reduce the transmission parameter ($\beta$), which lowers $R_0$, delays the epidemic peak, and reduces peak active hospitalizations to preserve healthcare capacity.

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What is the SIR Model in Epidemiology?

Mathematical Differential Equations

Key Epidemiological Metrics

Frequently Asked Questions

Related Tools

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