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:black">Black: 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', { | var brd = JXG.JSXGraph.initBoard('box', {axis: true, boundingbox: [-4, 1.25, 114, -1.25]}); | ||
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:'black',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 16: | Line 37: | ||
var gamma = brd.createElement('slider', [[0,-0.5], [30,-0.5],[0,0.3,1]], {name:'γ'}); | var gamma = brd.createElement('slider', [[0,-0.5], [30,-0.5],[0,0.3,1]], {name:'γ'}); | ||
var mu = brd.createElement('slider', [[0,-0.6], [30,-0.6],[0,0.0,1]], {name:'μ'}); | var mu = brd.createElement('slider', [[0,-0.6], [30,-0.6],[0,0.0,1]], {name:'μ'}); | ||
var a = brd.createElement('slider', [[0,-0.7], [30,-0.7],[0, | var a = brd.createElement('slider', [[0,-0.7], [30,-0.7],[0,1.0,1]], {name:'a'}); | ||
brd.createElement('text', [40,-0.3, "initially infected population rate (on load: I(0)=1.27E-6)"]); | brd.createElement('text', [40,-0.3, "initially infected population rate (on load: I(0)=1.27E-6)"]); | ||
| Line 25: | 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);}]); | ||
S.hideTurtle(); | S.hideTurtle(); | ||
E.hideTurtle(); | E.hideTurtle(); | ||
I.hideTurtle(); | I.hideTurtle(); | ||
R.hideTurtle(); | R.hideTurtle(); | ||
function clearturtle() { | function clearturtle() { | ||
S.cs(); | S.cs(); | ||
| Line 57: | Line 78: | ||
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-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(I,delta,dI); | turtleMove(I,delta,dI); | ||
turtleMove(R,delta,dR); | turtleMove(R,delta,dR); | ||
| Line 91: | Line 112: | ||
} | } | ||
</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 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)
