# Difference between revisions of "SIR model: swine flu"

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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 | + | :<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:'β'}); | var beta = brd.createElement('slider', [[0,-0.4], [60,-0.4],[0,0.2857,1]], {name:'β'}); | ||

var gamma = brd.createElement('slider', [[0,-0.5], [60,-0.5],[0,0.1428,1]], {name:'γ'}); | var gamma = brd.createElement('slider', [[0,-0.5], [60,-0.5],[0,0.1428,1]], {name:'γ'}); | ||

+ | 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 "β: infection rate, R<sub>0</sub>="+(beta.Value()/gamma.Value()).toFixed(2);}]); | brd.createElement('text', [90,-0.4, function(){ return "β: infection rate, R<sub>0</sub>="+(beta.Value()/gamma.Value()).toFixed(2);}]); | ||

Line 31: | Line 33: | ||

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 14:11, 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]\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.

- 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. 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

- Clinical Signs and Symptoms of Influenza
- Modeling influenza epidemics and pandemics: insights into the future of swine flu (H1N1)
- First analysis of swine flu spread supports pandemic plan

### 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:'β'});
var gamma = brd.createElement('slider', [[0,-0.5], [60,-0.5],[0,0.1428,1]], {name:'γ'});
brd.createElement('text', [90,-0.3, "initially infected population rate"]);
brd.createElement('text', [90,-0.4, function(){ return "β: infection rate, R<sub>0</sub>="+(beta.Value()/gamma.Value()).toFixed(2);}]);
brd.createElement('text', [90,-0.5, "γ: 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>
```