Epidemiology: The SEIR model: Difference between revisions
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</html> | </html> | ||
<jsxgraph width="700" height="600" box="box"> | <jsxgraph width="700" height="600" box="box"> | ||
var brd = JXG.JSXGraph.initBoard('box', { | 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}); | ||
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brd.createElement('text', [40,-0.2, | brd.createElement('text', [40,-0.2, | ||
function() {return "Day "+t+": infected="+ | function() {return "Day "+t+": infected="+(7900000*I.Y()).toFixed(1)+" recovered="+(7900000*R.Y()).toFixed(1);}]); | ||
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function turtleMove(turtle,dx,dy) { | function turtleMove(turtle,dx,dy) { | ||
turtle.moveTo([dx+turtle. | turtle.moveTo([dx+turtle.X(),dy+turtle.Y()]); | ||
} | } | ||
function loop() { | function loop() { | ||
var dS = mu.Value()*(1.0-S. | var dS = mu.Value()*(1.0-S.Y())-beta.Value()*I.Y()*S.Y(); | ||
var dE = beta.Value()*I. | var dE = beta.Value()*I.Y()*S.Y()-(mu.Value()+a.Value())*E.Y(); | ||
var dI = a.Value()*E. | var dI = a.Value()*E.Y()-(gamma.Value()+mu.Value())*I.Y(); | ||
var dR = gamma.Value()*I. | 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); | ||
Latest revision as of 14:58, 20 February 2013
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 * Black: Rate of exposed population * Red: Rate of infectious population * Green: Rate of recovered population (which means: immune, isolated or dead)
