Epidemiology: The SIR model

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]\displaystyle{ S(t) }[/math] be the fraction that is susceptible to an infectious, but not deadly, disease at time t; let [math]\displaystyle{ I(t) }[/math] be the fraction that is infected at time [math]\displaystyle{ t }[/math]; and let [math]\displaystyle{ R(t) }[/math] be the fraction that has recovered. Let [math]\displaystyle{ \beta }[/math] be the rate at which an infected person infects a susceptible person. Let [math]\displaystyle{ \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]\displaystyle{ \frac{dS}{dt} = - \beta I S }[/math]

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

[math]\displaystyle{ \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]\displaystyle{ \gamma = 1/3 }[/math]
• infection rate: one new person every other day, so [math]\displaystyle{ \beta = 1/2 }[/math]

Thus S(0) = 1, I(0) = 1.27E-6, R(0) = 0, see [1].

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)

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 a simulation of 100 days" 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], [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:'β'});
brd.createElement('text', [40,-0.4, "β: infection rate"]);
var gamma = brd.createElement('slider', [[0,-0.5], [30,-0.5],[0,0.3,1]], {name:'γ'});
brd.createElement('text', [40,-0.5, "γ: 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.pos[1],1)+" recovered="+brd.round(7900000*R.pos[1],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.X());
                
  delta = 1; // global
  t = 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<100.0) {
    setTimeout(loop,10);
  }
}

References

• http://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology
• http://mathworld.wolfram.com/SIRModel.html
• http://www.cs.princeton.edu/introcs/94diffeq