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SIR Epidemic Model

Explore the dynamics of disease spread through interactive differential equations

Time Evolution

Susceptible
Infected
Recovered

Phase Portrait (S-I)

Understanding the SIR Model

Before You Start

To understand this model, you'll need to know:

  • Basic algebra (working with variables and simple equations)
  • Fractions and percentages
  • How to read graphs

Simple Explanation

Imagine a school with 1000 students. When someone gets the flu:

  • Susceptible (S): These are healthy students who could get sick
  • Infected (I): These are students who have the flu and can spread it
  • Recovered (R): These are students who had the flu and got better

How It Works

The model uses three special numbers to predict how a disease spreads:

  • β (beta): How easily people catch the disease. Like how quickly the flu spreads when a sick student sits next to a healthy one.
  • γ (gamma): How quickly people recover. For example, if the flu typically lasts 5 days, γ would be 1/5 per day.
  • μ (mu): How often new people join or leave the group. Like when new students join the school or others graduate.

The Math Behind It

The equations tell us how these three groups change over time:

dSdt=μ−βSI−μSdIdt=βSI−γI−μIdRdt=γI−μR\begin{align*} \frac{dS}{dt} &= \mu - \beta SI - \mu S \\ \frac{dI}{dt} &= \beta SI - \gamma I - \mu I \\ \frac{dR}{dt} &= \gamma I - \mu R \end{align*}

Don't worry if these equations look scary! The graphs above show what they mean:

  • The blue line shows healthy people who might get sick
  • The red line shows sick people who can spread the disease
  • The green line shows people who got better

Try It Yourself!

Use the sliders above to experiment:

  • Increase β to make the disease spread faster
  • Increase γ to make people recover faster
  • Watch how the curves change!

Real World Examples

This model helps scientists understand and predict how diseases spread, like:

  • The seasonal flu in schools
  • COVID-19 in cities
  • Chicken pox in communities

Understanding this helps doctors and scientists decide how to protect people, like when to use masks or when to develop new medicines!