Difference between revisions of "Epidemiology: The SIR model"

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var s = brd.createElement('slider', [[0,-0.3], [10,-0.3],[0,0.03,1]], {name:'s'});
 
var s = brd.createElement('slider', [[0,-0.3], [10,-0.3],[0,0.03,1]], {name:'s'});
brd.createElement('text', [12,-0.5, "initially infected population rate"]);
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brd.createElement('text', [12,-0.3, "initially infected population rate"]);
 
var beta = brd.createElement('slider', [[0,-0.4], [10,-0.4],[0,0.5,1]], {name:'β'});
 
var beta = brd.createElement('slider', [[0,-0.4], [10,-0.4],[0,0.5,1]], {name:'β'});
brd.createElement('text', [12,-0.6, "β: infection rate"]);
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var gamma = brd.createElement('slider', [[0,-0.5], [10,-0.5],[0,0.3,1]], {name:'γ'});
 
var gamma = brd.createElement('slider', [[0,-0.5], [10,-0.5],[0,0.3,1]], {name:'γ'});
brd.createElement('text', [12,-0.7, "γ: recovery rate"]);
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brd.createElement('text', [12,-0.5, "γ: recovery rate"]);
  
brd.createElement('text', [12,-0.4,  
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brd.createElement('text', [12,-0.2,  
 
         function() {return "S(t)="+brd.round(S.pos[1],3) +", I(t)="+brd.round(I.pos[1],3) +", R(t)="+brd.round(R.pos[1],3);}]);
 
         function() {return "S(t)="+brd.round(S.pos[1],3) +", I(t)="+brd.round(I.pos[1],3) +", R(t)="+brd.round(R.pos[1],3);}]);
 
              
 
              

Revision as of 18:47, 21 January 2009

Simulation of differential equations with turtle graphics using JSXGraph.

SIR model without vital dynamics

A single epidemic outbreak is usually far more rapid than the vital dynamics of a population, thus, if the aim is to study the immediate consequences of a single epidemic, one may neglect the birth-death processes. In this case the SIR system described above can be expressed by the following set of differential equations:

[math] \frac{dS}{dt} = - \beta I S [/math]
[math] \frac{dR}{dt} = \gamma I [/math]
[math] \frac{dI}{dt} = -(dS+dR) [/math]

The lines in the JSXGraph-simulation below have the following meaning:

* Blue: Rate of susceptible population
* Red: Rate of infected population
* Green: Rate of recovered population (which means: immune, isolated or dead)