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.  
 +
 
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)
 
 
 
<html>
 
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<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 20:00, 21 January 2009

Simulation of differential equations with turtle graphics using JSXGraph.

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:'&beta;'});
brd.createElement('text', [12,-0.4, "&beta;: infection rate"]);
var gamma = brd.createElement('slider', [[0,-0.5], [10,-0.5],[0,0.3,1]], {name:'&gamma;'});
brd.createElement('text', [12,-0.5, "&gamma;: 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);
  }
}

References