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]
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.
- estimated average period of infection: 3 days, so [math]\displaystyle{ \gamma = 1/3 }[/math]
- infection rate: one new person every day, so [math]\displaystyle{ \beta = 1/2 }[/math]
Thus S(0) = 1, I(0) = 1.27E-6, R(0) = 0, see 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], [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"]);
brd.createElement('text', [40,-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 = 1; // 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<100.0) {
setTimeout(loop,10);
}
}
