SEIR Model Calculator

Simulate infectious disease outbreaks using the Susceptible-Exposed-Infectious-Recovered compartmental model. Adjust parameters to explore epidemic dynamics and intervention strategies.

Model equations: dS/dt = -β·S·I/N, dE/dt = β·S·I/N - σ·E, dI/dt = σ·E - γ·I, dR/dt = γ·I

Basic reproduction number R₀ = β / γ (in a fully susceptible population)

contacts · transmission probability
avg incubation (exposed to infectious)
duration of infectiousness
COVID-19 (early) Influenza Measles SARS‑2003
Simulating...

Understanding the SEIR Model

The SEIR model is a deterministic compartmental framework used to simulate the transmission of infectious diseases. Individuals progress through four states: Susceptible (S), Exposed (E), Infectious (I), and Recovered (R).

Compartment details:

  • Susceptible (S): Individuals who can contract the disease. They become exposed after effective contact with an infectious person.
  • Exposed (E): Individuals who have been infected but are not yet infectious. This latent period accounts for diseases where transmission starts after a delay (e.g., COVID‑19, measles).
  • Infectious (I): Individuals capable of transmitting the pathogen to susceptibles.
  • Recovered (R): Individuals who have recovered and are assumed to have lifelong immunity (no re‑infection).

Key Parameters & Their Interpretation

Parameter Symbol Units Meaning
Transmission rate β per person per day Rate at which one infectious individual infects susceptibles, given contact. It combines contact rate and transmission probability.
Latent rate σ = 1 / latent period per day Rate at which exposed individuals become infectious. The mean latent period is 1/σ days.
Recovery rate γ = 1 / infectious period per day Rate at which infectious individuals recover. The mean infectious period is 1/γ days.
Basic reproduction number R₀ = β / γ dimensionless Average number of secondary infections caused by one infectious individual in a fully susceptible population. R₀ > 1 indicates potential outbreak; R₀ < 1 means the disease will die out.

Herd Immunity Threshold (HIT): The proportion of the population that must be immune (through vaccination or prior infection) to stop transmission. For a simple SEIR model, HIT = 1 – 1/R₀. For example, if R₀ = 3, about 67% immunity is needed.

Model Assumptions & Limitations

  • Homogeneous mixing: Every individual has equal probability of contacting any other. Real populations have age structure, social networks, and spatial heterogeneity.
  • Constant rates: β, σ, γ are fixed over time. In reality, behaviour changes (e.g., social distancing) alter β, and latent/infectious periods may vary.
  • No demographic effects: Births, deaths (other than disease‑related) and migration are ignored – suitable for short‑term outbreaks.
  • Lifelong immunity: Assumes recovered individuals never become susceptible again. For waning immunity, an SEIRS model would be needed.
  • No intervention measures: The model does not include vaccination, quarantine, or treatment. You can approximate these by reducing β or initial susceptibles.

Extensions of the SEIR Model

Epidemiologists often use more complex variants to capture real‑world dynamics:

  • SEIRS: Adds waning immunity – recovered individuals eventually become susceptible again.
  • SEIR with age structure: Splits compartments by age groups to model differential contact rates.
  • Stochastic SEIR: Includes randomness, useful for small populations or early outbreak dynamics.
  • Metapopulation models: Couple multiple SEIR models to represent spatial spread between regions.

Further Reading

For a deeper dive, consult:

  • Keeling, M. J., & Rohani, P. (2008). Modeling Infectious Diseases in Humans and Animals. Princeton University Press.
  • Anderson, R. M., & May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press.
  • World Health Organization – SEIR modelling resources (placeholder link).

Frequently Asked Questions

SEIR adds an "Exposed" compartment for diseases with a latent period where individuals are infected but not yet infectious. SIR is suitable for diseases with no latent phase (or very short).

R₀ = β / γ, assuming the entire population is susceptible at the start. It represents the average number of secondary cases produced by one infectious individual during their infectious period.

If R₀ ≤ 1 or initial infectious/exposed are very low, the disease may die out quickly. Increase β or initial cases to trigger an epidemic.

This basic SEIR doesn't include vaccination, but you can approximate by reducing initial susceptibles or lowering β. Advanced extensions (SEIRV) exist but are not implemented here.