Difference between revisions of "Epidemiology: The SEIR model"

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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).
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Assuming that the period of staying in the latent state is a random variable with exponential distribution with
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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:
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:<math> \frac{dS}{dt} = \mu N - \mu S - \beta \frac{I}{N} S </math>
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:<math> \frac{dE}{dt} = \beta \frac{I}{N} S - (\mu +a ) E </math>
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:<math> \frac{dI}{dt} = a E - (\gamma +\mu ) I </math>
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:<math> \frac{dR}{dt} = \gamma I  - \mu R. </math>
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Of course, we have that <math>S+E+I+R=N</math>.
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The lines in the JSXGraph-simulation below have the following meaning:
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* <span style="color:Blue">Blue: Rate of susceptible population</span>
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* <span style="color:yellow">Vellow: Rate of exposed population</span>
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* <span style="color:red">Red: Rate of infectious population</span>
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* <span style="color:green">Green: Rate of recovered population (which means: immune, isolated or dead)
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<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});
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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 10:02, 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]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:

* 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)