Difference between revisions of "Epidemiology: The SEIR model"

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Line 46: Line 46:
  
 
brd.createElement('text', [40,-0.2,  
 
brd.createElement('text', [40,-0.2,  
         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.Y(),1)+" recovered="+brd.round(7900000*R.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);

Revision as of 09:57, 23 June 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
* Black: Rate of exposed population
* Red: Rate of infectious population
* Green: Rate of recovered population (which means: immune, isolated or dead)

See also

References