View source for Epidemiology: The SIR model

Simulation of differential equations with turtle graphics using [http://jsxgraph.org 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} = -(\frac{dS}{dt}+\frac{dR}{dt}) </math>

====Example Hong Kong flu==== 
* initially 7.9 million people, 
* 10 infected, 
* 0 recovered. 
* estimated average period of infection: 3 days, so <math>\gamma = 1/3</math> 
* infection rate: one new person every other day, so <math>\beta = 1/2</math>

Thus S(0) = 1, I(0) = 1.27E-6, R(0) = 0, see [http://www.cs.princeton.edu/introcs/94diffeq/].

The lines in the JSXGraph-simulation below have the following meaning:
 * <span style="color:Blue">Blue: Rate of susceptible 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)
<html>
<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 a simulation of 100 days" onClick="clearturtle();run()">
<input type="button" value="stop" onClick="stop()">
<input type="button" value="continue" onClick="goOn()"></form>
<div id="box" class="jxgbox" style="width:600px; height:450px;"></div>
<script language="JavaScript">
var brd = JXG.JSXGraph.initBoard('box', {originX: 20, originY: 300, unitX: 6, 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], [30,-0.3],[0,1.27E-6,1]], {name:'s'});
brd.createElement('text', [40,-0.3, "initially infected population rate (on load: I(0)=1.27E-6)"]);
var beta = brd.createElement('slider', [[0,-0.4], [30,-0.4],[0,0.5,1]], {name:'&beta;'});
brd.createElement('text', [40,-0.4, "&beta;: infection rate"]);
var gamma = brd.createElement('slider', [[0,-0.5], [30,-0.5],[0,0.3,1]], {name:'&gamma;'});
brd.createElement('text', [40,-0.5, "&gamma;: recovery rate = 1/(days of infection)"]);

var t = 0; // global

brd.createElement('text', [40,-0.2, 
        function() {return "Day "+t+": infected="+brd.round(7900000*I.Y(),1)+" recovered="+brd.round(7900000*R.Y(),1);}]);
            
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.Value());
                
  delta = 1; // global
  t = 0;  // global
  loop();
}
             
function turtleMove(turtle,dx,dy) {
  turtle.moveTo([dx+turtle.X(),dy+turtle.Y()]);
}
             
function loop() {
  var dS = -beta.Value()*S.Y()*I.Y();
  var dR = gamma.Value()*I.Y();
  var dI = -(dS+dR);
  turtleMove(S,delta,dS);
  turtleMove(R,delta,dR);
  turtleMove(I,delta,dI);
                
  t += delta;
  if (t<100.0) {
    active = setTimeout(loop,10);
  }
}

function stop() {
  if (active) clearTimeout(active);
  active = null;
}
function goOn() {
   if (t>0) {
     if (active==null) {
       active = setTimeout(loop,10);
     }
   } else {
     run();
   }

}
</script>
</html>

===The underlying JavaScript code===
<source lang="html4strict">
<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 a simulation of 100 days" onClick="clearturtle();run()">
<input type="button" value="stop" onClick="stop()">
<input type="button" value="continue" onClick="goOn()"></form>
<div id="box" class="jxgbox" style="width:600px; height:450px;"></div>
</source>

<source lang="javascript">
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], [30,-0.3],[0,0.03,1]], {name:'s'});
brd.createElement('text', [40,-0.3, "initially infected population rate"]);
var beta = brd.createElement('slider', [[0,-0.4], [30,-0.4],[0,0.5,1]], {name:'&beta;'});
brd.createElement('text', [40,-0.4, "&beta;: infection rate"]);
var gamma = brd.createElement('slider', [[0,-0.5], [30,-0.5],[0,0.3,1]], {name:'&gamma;'});
brd.createElement('text', [40,-0.5, "&gamma;: recovery rate = 1/(days of infection)"]);

var t = 0; // global

brd.createElement('text', [40,-0.2, 
        function() {return "Day "+t+": infected="+brd.round(7900000*I.Y(),1)+" recovered="+brd.round(7900000*R.Y(),1);}]);
            
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.Value());
                
  delta = 1; // global
  t = 0;  // global
  loop();
}
             
function turtleMove(turtle,dx,dy) {
  turtle.moveTo([dx+turtle.X(),dy+turtle.Y()]);
}
             
function loop() {
  var dS = -beta.Value()*S.Y()*I.Y();
  var dR = gamma.Value()*I.Y();
  var dI = -(dS+dR);
  turtleMove(S,delta,dS);
  turtleMove(R,delta,dR);
  turtleMove(I,delta,dI);
                
  t += delta;
  if (t<100.0) {
    active = setTimeout(loop,10);
  }
}
function stop() {
  if (active) clearTimeout(active);
  active = null;
}
function goOn() {
   if (t>0) {
     if (active==null) {
       active = setTimeout(loop,10);
     }
   } else {
     run();
   }

}
</source>

===See also===
* [[Epidemiology: The SEIR model]]

===References===
* [http://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology http://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology]
* [http://mathworld.wolfram.com/SIRModel.html http://mathworld.wolfram.com/SIRModel.html]
* [http://www.cs.princeton.edu/introcs/94diffeq http://www.cs.princeton.edu/introcs/94diffeq]

[[Category:Examples]]
[[Category:Turtle Graphics]]
[[Category:Calculus]]