# Difference between revisions of "Epidemiology: The SIR model"

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Given a fixed population, let <math>S(t)</math> be the fraction that is susceptible to an infectious, but not deadly, disease at time t; let <math>I(t)</math> be the fraction that is infected at time <math>t</math>; | Given a fixed population, let <math>S(t)</math> be the fraction that is susceptible to an infectious, but not deadly, disease at time t; let <math>I(t)</math> be the fraction that is infected at time <math>t</math>; | ||

and let <math>R(t)</math> be the fraction that has recovered. Let <math>\beta</math> be the rate at which an infected person infects a susceptible person. Let <math>\gamma</math> be the rate at which infected people recover from the disease. | and let <math>R(t)</math> be the fraction that has recovered. Let <math>\beta</math> be the rate at which an infected person infects a susceptible person. Let <math>\gamma</math> be the rate at which infected people recover from the disease. | ||

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A single epidemic outbreak is usually far more rapid than the vital dynamics of a population, thus, if the aim is to study the immediate consequences of a single epidemic, one may neglect birth-death processes. In this case the SIR system can be expressed by the following set of differential equations: | A single epidemic outbreak is usually far more rapid than the vital dynamics of a population, thus, if the aim is to study the immediate consequences of a single epidemic, one may neglect birth-death processes. In this case the SIR system can be expressed by the following set of differential equations: | ||

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* <span style="color:red">Red: Rate of infected population</span> | * <span style="color:red">Red: Rate of infected population</span> | ||

* <span style="color:green">Green: Rate of recovered population (which means: immune, isolated or dead) | * <span style="color:green">Green: Rate of recovered population (which means: immune, isolated or dead) | ||

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<html> | <html> | ||

<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" /> | <link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" /> |

## Revision as of 19:00, 21 January 2009

Simulation of differential equations with turtle graphics using JSXGraph.

## Contents

### SIR model without vital dynamics

The SIR model measures the number of susceptible, infected, and recovered individuals in a host population. Given a fixed population, let [math]S(t)[/math] be the fraction that is susceptible to an infectious, but not deadly, disease at time t; let [math]I(t)[/math] be the fraction that is infected at time [math]t[/math]; and let [math]R(t)[/math] be the fraction that has recovered. Let [math]\beta[/math] be the rate at which an infected person infects a susceptible person. Let [math]\gamma[/math] be the rate at which infected people recover from the disease.

A single epidemic outbreak is usually far more rapid than the vital dynamics of a population, thus, if the aim is to study the immediate consequences of a single epidemic, one may neglect birth-death processes. In this case the SIR system can be expressed by the following set of differential equations:

- [math] \frac{dS}{dt} = - \beta I S [/math]

- [math] \frac{dR}{dt} = \gamma I [/math]

- [math] \frac{dI}{dt} = -(dS+dR) [/math]

The lines in the JSXGraph-simulation below have the following meaning:

* Blue: Rate of susceptible population * Red: Rate of infected population * Green: Rate of recovered population (which means: immune, isolated or dead)

#### Example

**Hong Kong flu:** initially 7.9 million people, 10 infected, 0 recovered. Thus S(0) = 1, I(0) = 1.27E-6, R(0) = 0, see [1].

### The underlying JavaScript code

```
<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/prototype.js"></script>
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>
<form><input type="button" value="clear and run" onClick="clearturtle();run()"></form>
<div id="box" class="jxgbox" style="width:600px; height:450px;"></div>
```

```
var brd = JXG.JSXGraph.initBoard('box', {originX: 20, originY: 300, unitX: 20, unitY: 250});
var S = brd.createElement('turtle',[],{strokeColor:'blue',strokeWidth:3});
var I = brd.createElement('turtle',[],{strokeColor:'red',strokeWidth:3});
var R = brd.createElement('turtle',[],{strokeColor:'green',strokeWidth:3});
var xaxis = brd.createElement('axis', [[0,0], [1,0]], {});
var yaxis = brd.createElement('axis', [[0,0], [0,1]], {});
var s = brd.createElement('slider', [[0,-0.3], [10,-0.3],[0,0.03,1]], {name:'s'});
brd.createElement('text', [12,-0.3, "initially infected population rate"]);
var beta = brd.createElement('slider', [[0,-0.4], [10,-0.4],[0,0.5,1]], {name:'β'});
brd.createElement('text', [12,-0.4, "β: infection rate"]);
var gamma = brd.createElement('slider', [[0,-0.5], [10,-0.5],[0,0.3,1]], {name:'γ'});
brd.createElement('text', [12,-0.5, "γ: recovery rate"]);
brd.createElement('text', [12,-0.2,
function() {return "S(t)="+brd.round(S.pos[1],3) +", I(t)="+brd.round(I.pos[1],3) +", R(t)="+brd.round(R.pos[1],3);}]);
S.hideTurtle();
I.hideTurtle();
R.hideTurtle();
function clearturtle() {
S.cs();
I.cs();
R.cs();
S.hideTurtle();
I.hideTurtle();
R.hideTurtle();
}
function run() {
S.setPos(0,1.0-s.Value());
R.setPos(0,0);
I.setPos(0,s.X());
delta = 0.3; // global
t = 0.0; // global
loop();
}
function turtleMove(turtle,dx,dy) {
turtle.lookTo([1.0+turtle.pos[0],dy+turtle.pos[1]]);
turtle.fd(dx*Math.sqrt(1+dy*dy));
}
function loop() {
var dS = -beta.Value()*S.pos[1]*I.pos[1];
var dR = gamma.Value()*I.pos[1];
var dI = -(dS+dR);
turtleMove(S,delta,dS);
turtleMove(R,delta,dR);
turtleMove(I,delta,dI);
t += delta;
if (t<30.0) {
setTimeout(loop,10);
}
}
```