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()"> | ||
| Line 4: | Line 25: | ||
<input type="button" value="continue" onClick="goOn()"></form> | <input type="button" value="continue" onClick="goOn()"></form> | ||
</html> | </html> | ||
<jsxgraph width=" | <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:' | var S = brd.createElement('turtle',[],{strokeColor:'blue',strokeWidth:3}); | ||
var E = brd.createElement('turtle',[],{strokeColor:' | 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(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)
