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<html>
<head>
<title>
SIR_SIMULATION - Simulation of Disease Propagation with the SIR Model
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SIR_SIMULATION <br> Simulation of Disease Propagation with the SIR Model
</h1>
<hr>
<p>
<b>SIR_SIMULATION</b>
is a MATLAB program which
simulates the spread of a disease through a hospital room of M by N beds,
using the SIR (Susceptible/Infected/Recovered) model.
</p>
<p>
We consider the evolution of a disease in a hospital
in which patients are arranged on an array of beds.
</p>
<p>
We assume that the beds form an array of M rows and N columns, so that
there are a total of M * N patients.
</p>
<p>
We assume that the patients can be classified as Susceptible, Infected or
Recovering, with the properties that:
<ul>
<li>
<i>Susceptible</i>: A patient who has never been infected with the
disease. A susceptible patient can get the disease.
</li>
<li>
<i>Infected</i>: A patient who has never gotten the disease.
A patient stays infected for K days. On the K+1 of the disease,
the patient "recovers".
</li>
<li>
<i>Recovered</i>: A patient who has had the disease, that is,
has caught the disease and been sick for a full K days. A recovered
patient never gets sick again.
</li>
</ul>
</p>
<p>
We set up an M by N array <b>A</b> to represent the patients.
A(I,J) contains information on the patient in row I, column J.
A(I,J) will be
<ul>
<li>
0, if the patient is susceptible.
</li>
<li>
a value between 1 and K, if the patient is infected. The value
is the number of days the patient has been infected.
</li>
<li>
-1, if the patient is recovered.
</li>
</ul>
</p>
<p>
The rules for transmission of the disease essentially update the
patient array once a day. If patient A(I,J) was:
<ul>
<li>
0, then check the four neighbors A(I-1,J), A(I+1,J), A(I,J-1)
and A(I,J+1). For each neighbor that is infected, pick a random
number, and if that random number is less than TAU, then patient
A(I,J) becomes infected, that is, we set A(I,J) to 1.
</li>
<li>
a value between 1 and K, then the value is increased by 1.
But if the value was already K, it is now reset to -1, because the
patient has recovered.
</li>
<li>
-1, nothing happens.
</li>
</ul>
</p>
<p>
Quantities of interest include an animation of the day to day status
of patients in the hospital (the "geometry") and the values of S, I, and R,
that is, the total number of patients in each category, as it evolves
over time.
</p>
<p>
Since this problem contains a probabilistic element in the transmission of
disease, the outcome of any single run has limited meaning. It is much
more valuable to run many simulations, and thus to get both average or "expected"
values, as well as a feeling for the variance of the data from these averages.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<i>sir</i> = <b>sir_simulation</b> ( <i>m</i>, <i>n</i>, <i>a</i>, <i>k</i>,
<i>tau</i>, <i>t_max</i> )
</blockquote>
where
<ul>
<li>
<i>m</i> is the number of rows of patients.
</li>
<li>
<i>n</i> is the number of columns of patients.
</li>
<li>
<i>a</i> is the M by N matrix of the initial patient states.
</li>
<li>
<i>k</i> is the number of days a patient stays infected.
</li>
<li>
<i>tau</i> is the probability that a susceptible patient will become infected
because of one "nearby" infected patient (north, south, east or west)
over one day.
</li>
<li>
<i>t_max</i> is the total number of days to consider, counting the initial
condition as day 1.
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SIR_SIMULATION</b> is available in
<a href = "../../m_src/sir_simulation/sir_simulation.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/brownian_motion_simulation/brownian_motion_simulation.html">
BROWNIAN_MOTION_SIMULATION</a>,
a MATLAB program which
simulates Brownian motion in an M-dimensional region.
</p>
<p>
<a href = "../../m_src/dice_simulation/dice_simulation.html">
DICE_SIMULATION</a>,
a MATLAB program which
simulates N tosses of M dice, making a histogram of the results.
</p>
<p>
<a href = "../../m_src/duel_simulation/duel_simulation.html">
DUEL_SIMULATION</a>,
a MATLAB program which
simulates N repetitions of a duel between two players, each of
whom has a known firing accuracy.
</p>
<p>
<a href = "../../c_src/forest_fire_simulation/forest_fire_simulation.html">
FOREST_FIRE_SIMULATION</a>,
a C program which
simulates the occurrence of fires and regrowth in a forest,
displaying the results using X Windows,
by Michael Creutz.
</p>
<p>
<a href = "../../m_src/gamblers_ruin_simulation/gamblers_ruin_simulation.html">
GAMBLERS_RUIN_SIMULATION</a>,
a MATLAB program which
simulates the game of gambler's ruin.
</p>
<p>
<a href = "../../m_src/high_card_simulation/high_card_simulation.html">
HIGH_CARD_SIMULATION</a>,
a MATLAB program which
simulates a situation in which you see the cards in a deck one by one,
and must select the one you think is the highest and stop.
</p>
<p>
<a href = "../../m_src/ising_2d_simulation/ising_2d_simulation.html">
ISING_2D_SIMULATION</a>,
a MATLAB program which
carries out a Monte Carlo simulation of an Ising model,
a 2D array of positive and negative charges,
each of which is likely to "flip" to be in agreement with neighbors.
</p>
<p>
<a href = "../../c_src/life_opengl/life_opengl.html">
LIFE_OPENGL</a>,
a C program which
uses OpenGL to display the evolution of John Conway's "Game of Life",
by Simon Green.
</p>
<p>
<a href = "../../m_src/lorenz_simulation/lorenz_simulation.html">
LORENZ_SIMULATION</a>,
a MATLAB program which
solves the Lorenz equations and displays the solution, for various
starting conditions.
</p>
<p>
<a href = "../../f_src/md1/md1.html">
MD1</a>,
a FORTRAN90 program which
carries out a simple molecular dynamics simulation;
</p>
<p>
<a href = "../../f_src/md2/md2.html">
MD2</a>,
a FORTRAN90 program which
carries out a simple molecular dynamics simulation;
</p>
<p>
<a href = "../../f_src/md3/md3.html">
MD3</a>,
a FORTRAN90 program which
carries out a simple molecular dynamics simulation;
</p>
<p>
<a href = "../../f_src/md3glue/md3glue.html">
MD3GLUE</a>,
a FORTRAN90 program which
carries out a simple molecular dynamics simulation;
</p>
<p>
<a href = "../../m_src/poisson_simulation/poisson_simulation.html">
POISSON_SIMULATION</a>,
a MATLAB library which
simulates a Poisson process in which events randomly occur with an
average waiting time of Lambda.
</p>
<p>
<a href = "../../m_src/random_walk_1d_simulation/random_walk_1d_simulation.html">
RANDOM_WALK_1D_SIMULATION</a>,
a MATLAB program which
simulates a random walk in a 1-dimensional region.
</p>
<p>
<a href = "../../m_src/random_walk_2d_avoid_simulation/random_walk_2d_avoid_simulation.html">
RANDOM_WALK_2D_AVOID_SIMULATION</a>,
a MATLAB program which
simulates a self-avoiding random walk in a 2-dimensional region.
</p>
<p>
<a href = "../../m_src/random_walk_2d_simulation/random_walk_2d_simulation.html">
RANDOM_WALK_2D_SIMULATION</a>,
a MATLAB program which
simulates a random walk in a 2-dimensional region.
</p>
<p>
<a href = "../../m_src/random_walk_3d_simulation/random_walk_3d_simulation.html">
RANDOM_WALK_3D_SIMULATION</a>,
a MATLAB program which
simulates a random walk in a 3-dimensional region.
</p>
<p>
<a href = "../../m_src/reactor_simulation/reactor_simulation.html">
REACTOR_SIMULATION</a>,
a MATLAB program which
a simple Monte Carlo simulation of the shielding effect of a slab
of a certain thickness in front of a neutron source. This program was
provided as an example with the book "Numerical Methods and Software."
</p>
<p>
<a href = "../../m_src/three_body_simulation/three_body_simulation.html">
THREE_BODY_SIMULATION</a>,
a MATLAB program which
simulates the behavior of three planets, constrained to lie in a plane,
and moving under the influence of gravity,
by Walter Gander and Jiri Hrebicek.
</p>
<p>
<a href = "../../m_src/traffic_simulation/traffic_simulation.html">
TRAFFIC_SIMULATION</a>,
a MATLAB program which
simulates the cars waiting to get through a traffic light.
</p>
<p>
<a href = "../../m_src/truel_simulation/truel_simulation.html">
TRUEL_SIMULATION</a>,
a MATLAB program which
simulates N repetitions of a duel between three players, each of
whom has a known firing accuracy.
</p>
<p>
<a href = "../../c_src/xising/xising.html">
XISING</a>,
a C program which
models the variations in ferromagnetism in a material, displaying
the results using X Windows.
</p>
<p>
<a href = "../../c_src/xwaves/xwaves.html">
XWAVES</a>,
a C program which
simulates the behavior of solutions of certain forms of the wave equation, displaying
the results using X Windows.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Dianne OLeary,<br>
Models of Infection: Person to Person,<br>
Computing in Science and Engineering,<br>
Volume 6, Number 1, January/February 2004.
</li>
<li>
Dianne OLeary,<br>
Scientific Computing with Case Studies,<br>
SIAM, 2008,<br>
ISBN13: 978-0-898716-66-5,<br>
LC: QA401.O44.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sir_area_display.m">sir_area_display.m</a>,
displays an area plot of the SIR percentages over time.
</li>
<li>
<a href = "sir_line_display.m">sir_line_display.m</a>,
displays a line plot of the SIR percentages over time.
</li>
<li>
<a href = "sir_simulation.m">sir_simulation.m</a>,
the main program, which takes user parameter values,
computes the configuration for each time step,
displays an image of the configuration for each time,
and returns the SIR percentages.
</li>
<li>
<a href = "timestep_display.m">timestep_display.m</a>,
displays an image of the hospital room at each timestep,
indicating the locations of suceptible, infected, and recovered
patients.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
As the SIR_SIMULATION program runs, it displays a plot of the
hospital room, with susceptibles in green, infecteds in shades of red,
and recovereds in gray. Here is an example:
<ul>
<li>
<a href = "sir_day13.png">sir_day13.png</a>,
a plot of the patient status array on day 13,
with M = N = 10, a single infected patient at A(5,5),
K = 4, TAU = 0.2, and T_MAX = 50,
</li>
</ul>
</p>
<p>
Using the command
<pre>
plot ( 1:t_max, sir(1,:), 'g', 1:t_max, sir(2,:), 'r', 1:t_max, sir(3,:), 'k' )
</pre>
we can make line plots of the S, I and R populations. For M = N = 10,
a single infected patient at A(5,5), K = 4, TAU = 0.2, and T_MAX = 50,
we did this several times:
<ul>
<li>
<a href = "sir_line1.png">sir_line1.png</a>
</li>
<li>
<a href = "sir_line2.png">sir_line2.png</a>
</li>
<li>
<a href = "sir_line3.png">sir_line3.png</a>
</li>
<li>
<a href = "sir_line4.png">sir_line4.png</a>
</li>
<li>
<a href = "sir_line5.png">sir_line5.png</a>
</li>
<li>
<a href = "sir_line6.png">sir_line6.png</a>
</li>
</ul>
</p>
<p>
Using the commands
<pre>
h = areas ( sir' );
set ( h(1), 'FaceColor', [ 0, 1, 0 ] );
set ( h(2), 'FaceColor', [ 1, 0, 0 ] );
set ( h(3), 'FaceColor', [ 0.8, 0.8, 0.8 ] );
</pre>
we can make an area plot of the S, I and R populations. For M = N = 10,
a single infected patient at A(5,5), K = 4, TAU = 0.2, and T_MAX = 50,
we did this:
<ul>
<li>
<a href = "sir_area.png">sir_area.png</a>
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 09 November 2009.
</i>
<!-- John Burkardt -->
</body>
</html>