Difference between revisions of "Epidemiology: The SEIR model"
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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}); | ||
| Line 46: | Line 46: | ||
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); | ||
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} | } | ||
</jsxgraph> | </jsxgraph> | ||
| + | |||
| + | ===See also=== | ||
| + | * [[Epidemiology: The SIR model]] | ||
| + | |||
| + | ===References=== | ||
| + | * [http://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology http://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology] | ||
| + | |||
| + | [[Category:Examples]] | ||
| + | [[Category:Turtle Graphics]] | ||
| + | [[Category:Calculus]] | ||
Latest revision as of 16:58, 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]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)
