# SIR SEIR Compartmental Epidemic R0 and Intervention Modeling Coach

Run SIR / SEIR / SEIR+V / SEIR+H ODEs with RK4: R0, herd-immunity threshold, analytic peak and final-size equations, growth-rate conversion, WAIFW next-generation-matrix R0, incidence fitting and intervention simulation.

> Canonical page: https://elysiatools.com/en/tools/sir-seir-compartmental-epidemic-r0-and-intervention-modeling-coach

- **Category:** Health

- **Keywords:** sir model, seir model, r0 calculator, epidemic curve, herd immunity, final size equation, next generation matrix, intervention simulation

## Overview

A deterministic compartmental coach for learning and teaching infectious-disease dynamics.

Models: SIR, SEIR, SEIR+V (vaccination S→V) and SEIR+H (hospitalization arm I→H→R), solved with classic RK4 at a 0.05-day step. Derived quantities: R0 = β/γ, post-intervention effective R0, herd-immunity threshold 1−1/R0, analytic SIR peak I/N = 1−(1+lnR0)/R0, final size z = 1−e^(−R0·z), and the exact growth-rate relation R0 = 1+r/γ (SIR) or (1+r/σ)(1+r/γ) (SEIR, Wallinga–Lipsitch).

Optional inputs: a "day,incidence" series triggers a grid + coordinate-descent least-squares fit of (R0, reporting fraction, I0) with a log-linear growth-rate cross-check; a WAIFW matrix (e.g. "10 2; 2 1") computes the age-stratified next-generation-matrix R0 = ρ(β_ij/γ_j) by power iteration; intervention day + beta reduction, vaccination campaign, and a hospital arm with LOS show policy trade-offs on the curve.

## Inputs

- **Model** (select)
- **Population (N)** (number): 1000000
- **Basic Reproduction Number (R0)** (number): 2.5
- **Infectious Period (days)** (number): 5
- **Latent Period (days, SEIR)** (number): 3
- **Initial Infectious (I0)** (number): 10
- **Horizon (days)** (number): 180
- **Intervention Start (day)** (number): 0 = none
- **Beta Reduction (%)** (number): 0
- **Vaccination Start (day)** (number): 0 = none
- **Vaccination Rate (% of S per day)** (number): 0
- **Hospitalized Share of Infectious (%)** (number): 0
- **Hospital Length of Stay (days)** (number): 10
- **WAIFW Matrix (optional)** (textarea): e.g. 10 2; 2 1 (who-acquires-infection-from-whom, β per day)
- **Incidence Series to Fit (optional)** (textarea): day,incidence per line e.g. 1,12 2,19 3,31

## When to use

- Simulating infectious disease dynamics to project infection peaks, attack rates, and hospital bed occupancy across customizable horizons.
- Estimating the basic reproduction number (R0) and reporting rates from reported daily incidence time series using least-squares optimization.
- Evaluating the epidemiological impact of non-pharmaceutical interventions, vaccination rollouts, or structured age-contact matrices.

## How it works

- Select the compartmental model (SIR, SEIR, SEIR+V, or SEIR+H) and configure baseline parameters such as population size, R0, infectious period, and latent duration.
- Optionally enter intervention parameters (transmission beta reduction, vaccination rate, hospitalization fraction and LOS), a contact matrix (WAIFW), or empirical incidence data.
- The solver computes compartment trajectories via classic RK4 integration at a 0.05-day step size while evaluating analytic milestones and optimization routines.
- Review the interactive HTML report featuring dynamic compartment curves, effective Rt metrics, herd-immunity thresholds, and fitting diagnostics.

## Use cases

- Healthcare capacity planning to estimate peak hospitalizations and bed occupancy under varying lengths of stay.
- Epidemiological education to illustrate analytical final size relations, herd immunity thresholds, and Wallinga–Lipsitch growth conversions.
- Outbreak analysis to calibrate transmission rates (R0) and reporting fractions against observed case counts.

## Frequently asked questions

### Which numerical integration method does the simulator use?

It uses a classic 4th-order Runge-Kutta (RK4) integration algorithm solved at a fixed step size of 0.05 days.

### How are the SIR and SEIR models differentiated?

SIR transitions susceptible hosts directly to infectious status, while SEIR includes an explicit latent compartment (Exposed) parameterized by the latent period.

### How does incidence fitting estimate R0 and initial cases?

It performs grid and coordinate-descent least-squares fitting over R0, reporting fraction, and I0, accompanied by a log-linear growth-rate regression.

### What is calculated when supplying a WAIFW matrix?

The tool builds the next-generation matrix from age-stratified transmission rates and computes the spectral radius (R0) using power iteration.

### How do vaccination and hospitalization parameters alter the compartments?

SEIR-V transfers a daily percentage of susceptible individuals to a protected compartment (V), while SEIR-H diverts infectious cases into a hospital compartment (H) governed by length of stay.

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