Epidemiology: The SEIR model: Difference between revisions

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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()">
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<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});
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         function() {return "Day "+t+": infected="+brd.round(7900000*I.pos[1],1)+" recovered="+brd.round(7900000*R.pos[1],1);}]);
         function() {return "Day "+t+": infected="+brd.round(7900000*I.pos[1],1)+" recovered="+brd.round(7900000*R.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();

Revision as of 08:02, 27 April 2009

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]\displaystyle{ a^{-1} }[/math]), and also assuming the presence of vital dynamics with birth rate equal to death rate, we have the model:

[math]\displaystyle{ \frac{dS}{dt} = \mu N - \mu S - \beta \frac{I}{N} S }[/math]
[math]\displaystyle{ \frac{dE}{dt} = \beta \frac{I}{N} S - (\mu +a ) E }[/math]
[math]\displaystyle{ \frac{dI}{dt} = a E - (\gamma +\mu ) I }[/math]
[math]\displaystyle{ \frac{dR}{dt} = \gamma I - \mu R. }[/math]

Of course, we have that [math]\displaystyle{ S+E+I+R=N }[/math].

The lines in the JSXGraph-simulation below have the following meaning:

* Blue: Rate of susceptible population
* Vellow: Rate of exposed population
* Red: Rate of infectious population
* Green: Rate of recovered population (which means: immune, isolated or dead)