SIR model: swine flu: Difference between revisions

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* According to the [http://www.cdc.gov/ 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.
* According to the [http://www.cdc.gov/ 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 [http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2715422 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.  
* In [http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2715422 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 the :<math>\beta = 2\cdot 1/7 = 0.2857</math>
:<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.
* We run the simulation for a population of 1 million people, where 1 person is infected initially, i.e. <math>s=1E{-6}</math>.
* 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
Thus S(0) = 1, I(0) = 1.E-6, R(0) = 0.
In [http://www.newscientist.com/article/dn17109-first-analysis-of-swine-flu-spread-supports-pandemic-plan.html] the mortality is estimated to be approximately 0.4 per cent.
The lines in the JSXGraph-simulation below have the following meaning:
The lines in the JSXGraph-simulation below have the following meaning:
  * <span style="color:Blue">Blue: Rate of susceptible population</span>
  * <span style="color:Blue">Blue: Rate of susceptible population</span>
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var beta = brd.createElement('slider', [[0,-0.4], [60,-0.4],[0,0.2857,1]], {name:'&beta;'});
var beta = brd.createElement('slider', [[0,-0.4], [60,-0.4],[0,0.2857,1]], {name:'&beta;'});
var gamma = brd.createElement('slider', [[0,-0.5], [60,-0.5],[0,0.1428,1]], {name:'&gamma;'});
var gamma = brd.createElement('slider', [[0,-0.5], [60,-0.5],[0,0.1428,1]], {name:'&gamma;'});
var mort = brd.createElement('slider', [[0,-0.6], [60,-0.6],[0,0.004,0.1]], {name:'mortality'});
brd.createElement('text', [90,-0.3, "initially infected population rate"]);
brd.createElement('text', [90,-0.3, "initially infected population rate"]);
brd.createElement('text', [90,-0.4, function(){ return "&beta;: infection rate, R<sub>0</sub>="+(beta.Value()/gamma.Value()).toFixed(2);}]);
brd.createElement('text', [90,-0.4, function(){ return "&beta;: infection rate, R<sub>0</sub>="+(beta.Value()/gamma.Value()).toFixed(2);}]);
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brd.createElement('text', [120,-0.2,  
brd.createElement('text', [120,-0.2,  
         function() {return "Day "+t+": infected="+brd.round(1000000*I.Y(),1)+" recovered="+brd.round(1000000*R.Y(),1);}]);
         function() {return "Day "+t+
                          ": infected="+brd.round(1000000*I.Y(),1)+
                          " recovered="+brd.round(1000000*R.Y(),1)+
                          " dead="+brd.round(1000000*R.Y()*mort.Value(),1)+;}]);
   
   
S.hideTurtle();
S.hideTurtle();

Revision as of 12:11, 10 August 2009

The SIR model (see also Epidemiology: The SIR model) tries to predict influenza epidemics. Here, we try to model the spreading of the 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]\displaystyle{ \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]\displaystyle{ R_0 }[/math] of the virus to be about [math]\displaystyle{ 2 }[/math]. For the SIR model this means: the reproduction rate [math]\displaystyle{ R_0 }[/math] for influenza is equal to the infection rate of the strain ([math]\displaystyle{ \beta }[/math]) multiplied by the duration of the infectious period ([math]\displaystyle{ 1/\gamma }[/math]), i.e.
[math]\displaystyle{ \beta = R_0\cdot \gamma }[/math]. Therefore, we set [math]\displaystyle{ \beta = 2\cdot 1/7 = 0.2857. }[/math] For the 1918–1919 pandemic [math]\displaystyle{ R_0 }[/math] is estimated to be between 2 and 3, whereas for the seasonal flu the range for [math]\displaystyle{ R_0 }[/math] is 0.9 to 2.1.
  • We run the simulation for a population of 1 million people, where 1 person is infected initially, i.e. [math]\displaystyle{ s=1E{-6} }[/math].

Thus S(0) = 1, I(0) = 1.E-6, R(0) = 0. In [1] the mortality is estimated to be approximately 0.4 per cent. 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>
<jsxgraph width="700" height="500">
var brd = JXG.JSXGraph.initBoard('jxgbox', {originX: 20, originY: 300, unitX: 3, unitY: 250, axis:true});
 
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 s = brd.createElement('slider', [[0,-0.3], [60,-0.3],[0,1E-6,1]], {name:'s'});
var beta = brd.createElement('slider', [[0,-0.4], [60,-0.4],[0,0.2857,1]], {name:'&beta;'});
var gamma = brd.createElement('slider', [[0,-0.5], [60,-0.5],[0,0.1428,1]], {name:'&gamma;'});
brd.createElement('text', [90,-0.3, "initially infected population rate"]);
brd.createElement('text', [90,-0.4, function(){ return "&beta;: infection rate, R<sub>0</sub>="+(beta.Value()/gamma.Value()).toFixed(2);}]);
brd.createElement('text', [90,-0.5, "&gamma;: recovery rate = 1/(days of infection)"]);
 
var t = 0; // global
 
brd.createElement('text', [120,-0.2, 
        function() {return "Day "+t+": infected="+brd.round(1000000*I.Y(),1)+" recovered="+brd.round(1000000*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<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();
   }
 
}
</jsxgraph>
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