Epidemiology: The SEIR model - Revision history

A WASSERMANN at 14:58, 20 February 2013

2013-02-20T14:58:10Z

← Older revision		Revision as of 14:58, 20 February 2013
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	brd.createElement('text', [40,-0.2,		brd.createElement('text', [40,-0.2,
	function() {return "Day "+t+": infected="+~~brd.round~~(7900000I.Y(),1)+" recovered="+~~brd.round~~(7900000R.Y(),1);}]);		function() {return "Day "+t+": infected="+(7900000I.Y()).toFixed(1)+" recovered="+(7900000R.Y()).toFixed(1);}]);

Michael at 13:51, 7 June 2011

2011-06-07T13:51:38Z

← Older revision		Revision as of 13:51, 7 June 2011
Line 26:		Line 26:
	</html>		</html>
	<jsxgraph width="700" height="600" box="box">		<jsxgraph width="700" height="600" box="box">
	var brd = JXG.JSXGraph.initBoard('box', {~~originX: 20,~~ axis: true, ~~originY~~: ~~300~~, ~~unitX: 6~~, ~~unitY: 250~~});		var brd = JXG.JSXGraph.initBoard('box', {axis: true, boundingbox: [-4, 1.25, 114, -1.25]});

	var S = brd.createElement('turtle',[],{strokeColor:'blue',strokeWidth:3});		var S = brd.createElement('turtle',[],{strokeColor:'blue',strokeWidth:3});

A WASSERMANN at 07:57, 23 June 2009

2009-06-23T07:57:00Z

← Older revision		Revision as of 07:57, 23 June 2009
Line 46:		Line 46:

	brd.createElement('text', [40,-0.2,		brd.createElement('text', [40,-0.2,
	function() {return "Day "+t+": infected="+brd.round(7900000I.~~pos[1]~~,1)+" recovered="+brd.round(7900000R.~~pos[1]~~,1);}]);		function() {return "Day "+t+": infected="+brd.round(7900000I.Y(),1)+" recovered="+brd.round(7900000R.Y(),1);}]);


Line 78:		Line 78:

	function turtleMove(turtle,dx,dy) {		function turtleMove(turtle,dx,dy) {
	turtle.moveTo([dx+turtle.~~pos[0]~~,dy+turtle.~~pos[1]~~]);		turtle.moveTo([dx+turtle.X(),dy+turtle.Y()]);
	}		}

	function loop() {		function loop() {
	var dS = mu.Value()(1.0-S.~~pos[1]~~)-beta.Value()I.~~pos[1]~~*S.~~pos[1]~~;		var dS = mu.Value()(1.0-S.Y())-beta.Value()I.Y()*S.Y();
	var dE = beta.Value()I.~~pos[1]~~S.~~pos[1]~~-(mu.Value()+a.Value())*E.~~pos[1]~~;		var dE = beta.Value()I.Y()S.Y()-(mu.Value()+a.Value())*E.Y();
	var dI = a.Value()E.~~pos[1]~~-(gamma.Value()+mu.Value())I.~~pos[1]~~;		var dI = a.Value()E.Y()-(gamma.Value()+mu.Value())I.Y();
	var dR = gamma.Value()I.~~pos[1]~~-mu.Value()R.~~pos[1]~~;		var dR = gamma.Value()I.Y()-mu.Value()R.Y();
	turtleMove(S,delta,dS);		turtleMove(S,delta,dS);
	turtleMove(E,delta,dE);		turtleMove(E,delta,dE);

A WASSERMANN at 08:08, 27 April 2009

2009-04-27T08:08:31Z

← Older revision		Revision as of 08:08, 27 April 2009
Line 112:		Line 112:
	}		}
	</jsxgraph>		</jsxgraph>

			===See also===
			* [[Epidemiology: The SIR model]]

			===References===
			* [http://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology http://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology]

			[[Category:Examples]]
			[[Category:Turtle Graphics]]
			[[Category:Calculus]]

A WASSERMANN at 08:04, 27 April 2009

2009-04-27T08:04:00Z

← Older revision		Revision as of 08:04, 27 April 2009
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	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>
	* <span style="color:~~yellow~~">~~Vellow~~: Rate of exposed population</span>		* <span style="color:black">Black: Rate of exposed population</span>
	* <span style="color:red">Red: Rate of infectious population</span>		* <span style="color:red">Red: Rate of infectious population</span>
	* <span style="color:green">Green: Rate of recovered population (which means: immune, isolated or dead)		* <span style="color:green">Green: Rate of recovered population (which means: immune, isolated or dead)
Line 29:		Line 29:

	var S = brd.createElement('turtle',[],{strokeColor:'blue',strokeWidth:3});		var S = brd.createElement('turtle',[],{strokeColor:'blue',strokeWidth:3});
	var E = brd.createElement('turtle',[],{strokeColor:'~~yellow~~',strokeWidth:3});		var E = brd.createElement('turtle',[],{strokeColor:'black',strokeWidth:3});
	var I = brd.createElement('turtle',[],{strokeColor:'red',strokeWidth:3});		var I = brd.createElement('turtle',[],{strokeColor:'red',strokeWidth:3});
	var R = brd.createElement('turtle',[],{strokeColor:'green',strokeWidth:3});		var R = brd.createElement('turtle',[],{strokeColor:'green',strokeWidth:3});

A WASSERMANN at 08:03, 27 April 2009

2009-04-27T08:03:22Z

← Older revision		Revision as of 08:03, 27 April 2009
Line 37:		Line 37:
	var gamma = brd.createElement('slider', [[0,-0.5], [30,-0.5],[0,0.3,1]], {name:'γ'});		var gamma = brd.createElement('slider', [[0,-0.5], [30,-0.5],[0,0.3,1]], {name:'γ'});
	var mu = brd.createElement('slider', [[0,-0.6], [30,-0.6],[0,0.0,1]], {name:'μ'});		var mu = brd.createElement('slider', [[0,-0.6], [30,-0.6],[0,0.0,1]], {name:'μ'});
	var a = brd.createElement('slider', [[0,-0.7], [30,-0.7],[0,0.0,1]], {name:'a'});		var a = brd.createElement('slider', [[0,-0.7], [30,-0.7],[0,1.0,1]], {name:'a'});

	brd.createElement('text', [40,-0.3, "initially infected population rate (on load: I(0)=1.27E-6)"]);		brd.createElement('text', [40,-0.3, "initially infected population rate (on load: I(0)=1.27E-6)"]);

A WASSERMANN at 08:02, 27 April 2009

2009-04-27T08:02:37Z

← Older revision		Revision as of 08:02, 27 April 2009
Line 1:		Line 1:
			For many important infections there is a significant period of time during which the individual has been infected but is not yet infectious himself. During this latent period the individual is in compartment E (for exposed).

			Assuming that the period of staying in the latent state is a random variable with exponential distribution with
			parameter a (i.e. the average latent period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate equal to death rate, we have the model:

			:<math> \frac{dS}{dt} = \mu N - \mu S - \beta \frac{I}{N} S </math>

			:<math> \frac{dE}{dt} = \beta \frac{I}{N} S - (\mu +a ) E </math>

			:<math> \frac{dI}{dt} = a E - (\gamma +\mu ) I </math>

			:<math> \frac{dR}{dt} = \gamma I - \mu R. </math>

			Of course, we have that <math>S+E+I+R=N</math>.

			The lines in the JSXGraph-simulation below have the following meaning:
			* <span style="color:Blue">Blue: Rate of susceptible population</span>
			* <span style="color:yellow">Vellow: Rate of exposed population</span>
			* <span style="color:red">Red: Rate of infectious population</span>
			* <span style="color:green">Green: Rate of recovered population (which means: immune, isolated or dead)

	<html>		<html>
	<form><input type="button" value="clear and run a simulation of 100 days" onClick="clearturtle();run()">		<form><input type="button" value="clear and run a simulation of 100 days" onClick="clearturtle();run()">
Line 4:		Line 25:
	<input type="button" value="continue" onClick="goOn()"></form>		<input type="button" value="continue" onClick="goOn()"></form>
	</html>		</html>
	<jsxgraph width="~~600~~" height="600" box="box">		<jsxgraph width="700" height="600" box="box">
	var brd = JXG.JSXGraph.initBoard('box', {originX: 20, axis: true, originY: 300, unitX: 6, unitY: 250});		var brd = JXG.JSXGraph.initBoard('box', {originX: 20, axis: true, originY: 300, unitX: 6, unitY: 250});

	var S = brd.createElement('turtle',[],{strokeColor:'~~yellow~~',strokeWidth:3});		var S = brd.createElement('turtle',[],{strokeColor:'blue',strokeWidth:3});
	var E = brd.createElement('turtle',[],{strokeColor:'~~blue~~',strokeWidth:3});		var E = brd.createElement('turtle',[],{strokeColor:'yellow',strokeWidth:3});
	var I = brd.createElement('turtle',[],{strokeColor:'red',strokeWidth:3});		var I = brd.createElement('turtle',[],{strokeColor:'red',strokeWidth:3});
	var R = brd.createElement('turtle',[],{strokeColor:'green',strokeWidth:3});		var R = brd.createElement('turtle',[],{strokeColor:'green',strokeWidth:3});
Line 27:		Line 48:
	function() {return "Day "+t+": infected="+brd.round(7900000I.pos[1],1)+" recovered="+brd.round(7900000R.pos[1],1);}]);		function() {return "Day "+t+": infected="+brd.round(7900000I.pos[1],1)+" recovered="+brd.round(7900000R.pos[1],1);}]);

	/*
	S.hideTurtle();		S.hideTurtle();
	E.hideTurtle();		E.hideTurtle();
	I.hideTurtle();		I.hideTurtle();
	R.hideTurtle();		R.hideTurtle();
	*/
	function clearturtle() {		function clearturtle() {
	S.cs();		S.cs();

A WASSERMANN at 07:57, 27 April 2009

2009-04-27T07:57:47Z

← Older revision		Revision as of 07:57, 27 April 2009
Line 61:		Line 61:

	function loop() {		function loop() {
	var dS = mu.Value()(1-S.pos[1])-beta.Value()I.pos[1]*S.pos[1];		var dS = mu.Value()(1.0-S.pos[1])-beta.Value()I.pos[1]*S.pos[1];
	var dE = beta.Value()I.pos[1]S.pos[1]-(mu.Value()+a.Value())*E.pos[1];		var dE = beta.Value()I.pos[1]S.pos[1]-(mu.Value()+a.Value())*E.pos[1];
	var dI = a.Value()E.pos[1]-(gamma.Value()+mu.Value())I.pos[1];		var dI = a.Value()E.pos[1]-(gamma.Value()+mu.Value())I.pos[1];
	var dR = gamma.Value()I.pos[1]-mu.Value()R.pos[1];		var dR = gamma.Value()I.pos[1]-mu.Value()R.pos[1];
	turtleMove(S,delta,dS);		turtleMove(S,delta,dS);
	// turtleMove(E,delta,dE);		turtleMove(E,delta,dE);
	turtleMove(I,delta,dI);		turtleMove(I,delta,dI);
	turtleMove(R,delta,dR);		turtleMove(R,delta,dR);

A WASSERMANN at 07:55, 27 April 2009

2009-04-27T07:55:42Z

← Older revision		Revision as of 07:55, 27 April 2009
Line 61:		Line 61:

	function loop() {		function loop() {
	var dS = /mu.Value()(1-S.pos[1])/-beta.Value()I.pos[1]*S.pos[1];		var dS = mu.Value()(1-S.pos[1])-beta.Value()I.pos[1]*S.pos[1];
	var dE = beta.Value()I.pos[1]S.pos[1]-(mu.Value()+a.Value())*E.pos[1];		var dE = beta.Value()I.pos[1]S.pos[1]-(mu.Value()+a.Value())*E.pos[1];
	var dI = /a.Value()E.pos[1]/-(gamma.Value()+mu.Value())I.pos[1];		var dI = a.Value()E.pos[1]-(gamma.Value()+mu.Value())I.pos[1];
	var dR = gamma.Value()I.pos[1]/-mu.Value()R.pos[1]/;		var dR = gamma.Value()I.pos[1]-mu.Value()R.pos[1];
	turtleMove(S,delta,dS);		turtleMove(S,delta,dS);
	// turtleMove(E,delta,dE);		// turtleMove(E,delta,dE);

A WASSERMANN at 07:54, 27 April 2009

2009-04-27T07:54:21Z

← Older revision		Revision as of 07:54, 27 April 2009
Line 62:		Line 62:
	function loop() {		function loop() {
	var dS = /mu.Value()(1-S.pos[1])/-beta.Value()I.pos[1]*S.pos[1];		var dS = /mu.Value()(1-S.pos[1])/-beta.Value()I.pos[1]*S.pos[1];
	var dE = betaI.pos[1]S.pos[1]-(mu.Value()+a.Value())*E.pos[1];		var dE = beta.Value()I.pos[1]S.pos[1]-(mu.Value()+a.Value())*E.pos[1];
	var dI = /a.Value()E.pos[1]/-(gamma.Value()+mu.Value())I.pos[1];		var dI = /a.Value()E.pos[1]/-(gamma.Value()+mu.Value())I.pos[1];
	var dR = gamma.Value()I.pos[1]/-mu.Value()R.pos[1]/;		var dR = gamma.Value()I.pos[1]/-mu.Value()R.pos[1]/;

← Older revision		Revision as of 07:54, 27 April 2009
Line 62:		Line 62:
	function loop() {		function loop() {
	var dS = /mu.Value()(1-S.pos[1])/-beta.Value()I.pos[1]*S.pos[1];		var dS = /mu.Value()(1-S.pos[1])/-beta.Value()I.pos[1]*S.pos[1];
	var dE = betaI.pos[1]S.pos[1]-(mu.Value()+a.Value())*E.pos[1];		var dE = beta.Value()I.pos[1]S.pos[1]-(mu.Value()+a.Value())*E.pos[1];
	var dI = /a.Value()E.pos[1]/-(gamma.Value()+mu.Value())I.pos[1];		var dI = /a.Value()E.pos[1]/-(gamma.Value()+mu.Value())I.pos[1];
	var dR = gamma.Value()I.pos[1]/-mu.Value()R.pos[1]/;		var dR = gamma.Value()I.pos[1]/-mu.Value()R.pos[1]/;