Epidemiology: The SIR model: Difference between revisions
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brd.createElement('text', [12,-0.7, "γ: recovery rate"]); | brd.createElement('text', [12,-0.7, "γ: recovery rate"]); | ||
brd.createElement('text', [12,-0.4, | |||
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(); | S.hideTurtle(); | ||
I.hideTurtle(); | I.hideTurtle(); | ||
| Line 45: | Line 47: | ||
function clearturtle() { | function clearturtle() { | ||
S.cs(); | S.cs(); | ||
I.cs(); | I.cs(); | ||
R.cs(); | R.cs(); | ||
S.hideTurtle(); | S.hideTurtle(); | ||
I.hideTurtle(); | I.hideTurtle(); | ||
R.hideTurtle(); | R.hideTurtle(); | ||
} | } | ||
function run() { | function run() { | ||
S.setPos(0,1.0-s.Value()); | S.setPos(0,1.0-s.Value()); | ||
R.setPos(0,0); | R.setPos(0,0); | ||
I.setPos(0,s.X()); | I.setPos(0,s.X()); | ||
delta = 0.1; // global | delta = 0.1; // global | ||
t = 0.0; // global | t = 0.0; // global | ||
loop(); | loop(); | ||
} | } | ||
function turtleMove(turtle,dx,dy) { | function turtleMove(turtle,dx,dy) { | ||
turtle.lookTo([1.0+turtle.pos[0],dy+turtle.pos[1]]); | turtle.lookTo([1.0+turtle.pos[0],dy+turtle.pos[1]]); | ||
turtle.fd(dx*Math.sqrt(1+dy*dy)); | turtle.fd(dx*Math.sqrt(1+dy*dy)); | ||
} | } | ||
function loop() { | function loop() { | ||
var dS = -beta.Value()*S.pos[1]*I.pos[1]; | var dS = -beta.Value()*S.pos[1]*I.pos[1]; | ||
var dR = gamma.Value()*I.pos[1]; | var dR = gamma.Value()*I.pos[1]; | ||
var dI = -(dS+dR); | var dI = -(dS+dR); | ||
turtleMove(S,delta,dS); | turtleMove(S,delta,dS); | ||
turtleMove(R,delta,dR); | turtleMove(R,delta,dR); | ||
turtleMove(I,delta,dI); | turtleMove(I,delta,dI); | ||
t += delta; | t += delta; | ||
if (t<20.0 && I.pos[1]) { | if (t<20.0 && I.pos[1]) { | ||
setTimeout(loop,10); | setTimeout(loop,10); | ||
} | } | ||
} | } | ||
</script> | </script> | ||
</html> | </html> | ||
Revision as of 17:40, 21 January 2009
Simulation of differential equations with turtle graphics using JSXGraph.
SIR model without vital dynamics
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 the birth-death processes. In this case the SIR system described above 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} = -(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)
