# Epidemiology: The SIR model

Simulation of differential equations with turtle graphics using JSXGraph.

### SIR model without vital dynamics

The SIR model measures the number of susceptible, infected, and recovered individuals in a host population. Given a fixed population, let $\displaystyle{ S(t) }$ be the fraction that is susceptible to an infectious, but not deadly, disease at time t; let $\displaystyle{ I(t) }$ be the fraction that is infected at time $\displaystyle{ t }$; and let $\displaystyle{ R(t) }$ be the fraction that has recovered. Let $\displaystyle{ \beta }$ be the rate at which an infected person infects a susceptible person. Let $\displaystyle{ \gamma }$ be the rate at which infected people recover from the disease.

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 birth-death processes. In this case the SIR system can be expressed by the following set of differential equations:

$\displaystyle{ \frac{dS}{dt} = - \beta I S }$
$\displaystyle{ \frac{dR}{dt} = \gamma I }$
$\displaystyle{ \frac{dI}{dt} = -(\frac{dS}{dt}+\frac{dR}{dt}) }$

#### Example Hong Kong flu

• initially 7.9 million people,
• 10 infected,
• 0 recovered.
• estimated average period of infection: 3 days, so $\displaystyle{ \gamma = 1/3 }$
• infection rate: one new person every other day, so $\displaystyle{ \beta = 1/2 }$

Thus S(0) = 1, I(0) = 1.27E-6, R(0) = 0, see [1].

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)


### The underlying JavaScript code

<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/prototype.js"></script>
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>
<form><input type="button" value="clear and run a simulation of 100 days" onClick="clearturtle();run()">
<input type="button" value="stop" onClick="stop()">
<input type="button" value="continue" onClick="goOn()"></form>
<div id="box" class="jxgbox" style="width:600px; height:450px;"></div>

var brd = JXG.JSXGraph.initBoard('box', {originX: 20, originY: 300, unitX: 20, unitY: 250});

var S = brd.createElement('turtle',[],{strokeColor:'blue',strokeWidth:3});
var I = brd.createElement('turtle',[],{strokeColor:'red',strokeWidth:3});
var R = brd.createElement('turtle',[],{strokeColor:'green',strokeWidth:3});

var xaxis = brd.createElement('axis', [[0,0], [1,0]], {});
var yaxis = brd.createElement('axis', [[0,0], [0,1]], {});

var s = brd.createElement('slider', [[0,-0.3], [30,-0.3],[0,0.03,1]], {name:'s'});
brd.createElement('text', [40,-0.3, "initially infected population rate"]);
var beta = brd.createElement('slider', [[0,-0.4], [30,-0.4],[0,0.5,1]], {name:'&beta;'});
brd.createElement('text', [40,-0.4, "&beta;: infection rate"]);
var gamma = brd.createElement('slider', [[0,-0.5], [30,-0.5],[0,0.3,1]], {name:'&gamma;'});
brd.createElement('text', [40,-0.5, "&gamma;: recovery rate = 1/(days of infection)"]);

var t = 0; // global

brd.createElement('text', [40,-0.2,
function() {return "Day "+t+": infected="+brd.round(7900000*I.pos[1],1)+" recovered="+brd.round(7900000*R.pos[1],1);}]);

S.hideTurtle();
I.hideTurtle();
R.hideTurtle();

function clearturtle() {
S.cs();
I.cs();
R.cs();

S.hideTurtle();
I.hideTurtle();
R.hideTurtle();
}

function run() {
S.setPos(0,1.0-s.Value());
R.setPos(0,0);
I.setPos(0,s.X());

delta = 1; // global
t = 0;  // global
loop();
}

function turtleMove(turtle,dx,dy) {
turtle.moveTo([dx+turtle.pos[0],dy+turtle.pos[1]]);
}

function loop() {
var dS = -beta.Value()*S.pos[1]*I.pos[1];
var dR = gamma.Value()*I.pos[1];
var dI = -(dS+dR);
turtleMove(S,delta,dS);
turtleMove(R,delta,dR);
turtleMove(I,delta,dI);

t += delta;
if (t<100.0) {
setTimeout(loop,10);
}
}
function stop() {
if (active) clearTimeout(active);
active = null;
}
function goOn() {
if (t>0) {
if (active==null) {
active = setTimeout(loop,10);
}
} else {
run();
}

}