# Difference between revisions of "Epidemiology: The SEIR model"

From JSXGraph Wiki

Jump to navigationJump to searchA WASSERMANN (talk | contribs) |
A WASSERMANN (talk | contribs) |
||

Line 1: | Line 1: | ||

+ | 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 10:02, 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]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 * Vellow: Rate of exposed population * Red: Rate of infectious population * Green: Rate of recovered population (which means: immune, isolated or dead)