SIR model: swine flu

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The SIR model (see also Epidemiology: The SIR model) tries to predict influenza epidemics. Here, we try to model the spreading of the H1N1 virus, aka swine flu.

  • According to the CDC Centers of Disease Control and Prevention: "Adults shed influenza virus from the day before symptoms begin through 5-10 days after illness onset. However, the amount of virus shed, and presumably infectivity, decreases rapidly by 3-5 days after onset in an experimental human infection model." So, here we set [math]\gamma=1/7=0.1428[/math] as the recovery rate. This means, on average an infected person sheds the virus for 7 days.
  • In Modeling influenza epidemics and pandemics: insights into the future of swine flu (H1N1) the authors estimate the reproduction rate [math]R_0[/math] of the virus to be about [math]2[/math]. For the SIR model this means: the reproduction rate [math]R_0[/math] for influenza is equal to the infection rate of the strain ([math]\beta[/math]) multiplied by the duration of the infectious period ([math]1/\gamma[/math]), i.e.
[math]\beta = R_0\cdot \gamma[/math]. Therefore, we set [math]\beta = 2\cdot 1/7 = 0.2857.[/math] For the 1918–1919 pandemic [math]R_0[/math] is estimated to be between 2 and 3, whereas for the seasonal flu the range for [math]R_0[/math] is 0.9 to 2.1.
  • In [1] the mortality is estimated to be approximately 0.4 per cent.
  • We run the simulation for a population of 1 million people, where 1 person is infected initially, i.e. [math]s=1E{-6}[/math]. Thus, S(0) = 1, I(0) = 1.E-6, R(0) = 0.

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

External links

The underlying JavaScript code

<html>
<form><input type="button" value="clear and run a simulation of 200 days" onClick="clearturtle();run()">
<input type="button" value="stop" onClick="stop()">
<input type="button" value="continue" onClick="goOn()"></form>
</html>
var brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-6.66, 1.2, 226.66, -0.8], axis:true});
 
var S = brd.create('turtle',[],{strokeColor:'blue',strokeWidth:3});
var I = brd.create('turtle',[],{strokeColor:'red',strokeWidth:3});
var R = brd.create('turtle',[],{strokeColor:'green',strokeWidth:3});
 
var s = brd.create('slider', [[0,-0.3], [60,-0.3],[0,1E-6,1]], {name:'s'});
var beta = brd.create('slider', [[0,-0.4], [60,-0.4],[0,0.2857,1]], {name:'&beta;'});
var gamma = brd.create('slider', [[0,-0.5], [60,-0.5],[0,0.1428,0.5]], {name:'&gamma;'});
var mort = brd.create('slider', [[0,-0.6], [60,-0.6],[0,0.4,10.0]], {name:'% mortality'});
brd.create('text', [90,-0.3, "initially infected population rate"]);
brd.create('text', [90,-0.4, function(){ return "&beta;: infection rate, R<sub>0</sub>="+(beta.Value()/gamma.Value()).toFixed(2);}]);
brd.create('text', [90,-0.5, function(){ return "&gamma;: recovery rate = 1/(days of infection), days of infection= "+(1/gamma.Value()).toFixed(1);}]);
 
var t = 0; // global
 
brd.create('text', [100,-0.2, 
        function() {return "Day "+t+
                           ": infected="+(1000000*I.Y()).toFixed(1)+
                           " recovered="+(1000000*R.Y()).toFixed(1)+
                           " dead="+(1000000*R.Y()*mort.Value()*0.01).toFixed(0);}]);
 
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<200.0) {
    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();
   }
 
}