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Epidemiology: SIR model
This is an example of simulation of differential equations with __turtle graphics__.
### 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 $S(t)$ be the fraction that is susceptible to an infectious, but not deadly, disease at time $t$,
let $I(t)$ be the fraction that is infected at time $t$
and let $R(t)$ be the fraction that has recovered. Let $\beta$ be the rate at which an infected person infects a susceptible person. Let $\gamma$ 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:
$$
\begin{aligned}
\frac{dS}{dt} &= - \beta I S \\
\frac{dR}{dt} &= \gamma I \\
\frac{dI}{dt} &= -(\frac{dS}{dt}+\frac{dR}{dt})
\end{aligned}
$$
### Example Hong Kong flu
- initially 7.9 million people,
- 10 infected,
- 0 recovered.
- estimated average period of infection: 3 days, so $\gamma = 1/3$
- infection rate: one new person every other day, so $\beta = 1/2$
Thus $S(0) = 1$, $I(0) = 1.27E-6$, $R(0) = 0$, see [https://www.cs.princeton.edu/introcs/94diffeq/](https://www.cs.princeton.edu/introcs/94diffeq/).
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)
<input type="button" value="clear and run a simulation of 100 days" onClick="clearTurtle(); run()"><br/>
<input type="button" value="stop" onClick="stop()"><br/>
<input type="button" value="continue" onClick="goOn()">// Define the id of your board in BOARDID
const board = JXG.JSXGraph.initBoard(BOARDID, {
axis: true,
boundingbox: [-5, 1.2, 100, -1.2],
showNavigation: false
});
// Turtles
var S = board.create('turtle', [], {
strokeColor: 'blue',
strokeWidth: 3
});
var I = board.create('turtle', [], {
strokeColor: 'red',
strokeWidth: 3
});
var R = board.create('turtle', [], {
strokeColor: 'green',
strokeWidth: 3
});
// Sliders
var s = board.create('slider', [
[0, -0.6],
[30, -0.6],
[0, 1.27E-6, 1]
], {
name: 's'
});
board.create('text', [10, -0.3, "initially infected population rate (on load: I(0)=1.27E-6)"]);
var beta = board.create('slider', [
[0, -0.7],
[30, -0.7],
[0, 0.5, 1]
], {
name: 'β'
});
board.create('text', [45, -0.7, "β: infection rate"]);
var gamma = board.create('slider', [
[0, -0.8],
[30, -0.8],
[0, 0.3, 1]
], {
name: 'γ'
});
board.create('text', [45, -0.8, "γ: recovery rate = 1/(days of infection)"]);
var t = 0; // global
board.create('text', [10, -0.2,
function() {
return "Day " + t + ": infected=" + (7900000 * I.Y()).toFixed(1) + " recovered=" + (7900000 * R.Y()).toFixed(1);
}
]);
// Reset the turtles
var clearTurtle = function () {
S.cs();
I.cs();
R.cs();
S.hideTurtle();
I.hideTurtle();
R.hideTurtle();
};
// Start the animation
var run = function() {
S.setPos(0, 1.0 - s.Value());
R.setPos(0, 0);
I.setPos(0, s.Value());
delta = 1; // global
t = 0; // global
loop();
};
var turtleMove = function (turtle, dx, dy) {
turtle.moveTo([dx + turtle.X(), dy + turtle.Y()]);
};
// Animation
var loop = function() {
var dS = -beta.Value() * S.Y() * I.Y();
var dR = gamma.Value() * I.Y();
var dI = -(dS + dR);
turtleMove(S, delta, dS);
turtleMove(R, delta, dR);
turtleMove(I, delta, dI);
t += delta;
if (t < 100.0) {
active = setTimeout(loop, 10);
}
};
// Stop animation
var stop = function() {
if (active) clearTimeout(active);
active = null;
};
// Continue
var goOn = function() {
if (t > 0) {
if (active === null) {
active = setTimeout(loop, 10);
}
} else {
run();
}
};
clearTurtle();
This example is licensed under a
Creative Commons Attribution 4.0 International License.
Please note you have to mention
The Center of Mobile Learning with Digital Technology
in the credits.