SIR_SIMULATION
Simulation of Disease Propagation with the SIR Model

SIR_SIMULATION 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.

We consider the evolution of a disease in a hospital in which patients are arranged on an array of beds.

We assume that the beds form an array of M rows and N columns, so that there are a total of M * N patients.

We assume that the patients can be classified as Susceptible, Infected or Recovering, with the properties that:

• Susceptible: A patient who has never been infected with the disease. A susceptible patient can get the disease.
• Infected: 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".
• Recovered: 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.

We set up an M by N array A to represent the patients. A(I,J) contains information on the patient in row I, column J. A(I,J) will be

• 0, if the patient is susceptible.
• a value between 1 and K, if the patient is infected. The value is the number of days the patient has been infected.
• -1, if the patient is recovered.

The rules for transmission of the disease essentially update the patient array once a day. If patient A(I,J) was:

• 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.
• 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.
• -1, nothing happens.

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.

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.

Usage:

sir = sir_simulation ( m, n, a, k, tau, t_max )

• m is the number of rows of patients.
• n is the number of columns of patients.
• a is the M by N matrix of the initial patient states.
• k is the number of days a patient stays infected.
• tau is the probability that a susceptible patient will become infected because of one "nearby" infected patient (north, south, east or west) over one day.
• t_max is the total number of days to consider, counting the initial condition as day 1.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

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Reference:

1. Dianne OLeary,
   Models of Infection: Person to Person,
   Computing in Science and Engineering,
   Volume 6, Number 1, January/February 2004.
2. Dianne OLeary,
   Scientific Computing with Case Studies,
   SIAM, 2008,
   ISBN13: 978-0-898716-66-5,
   LC: QA401.O44.

Source Code:

• sir_area_display.m, displays an area plot of the SIR percentages over time.
• sir_line_display.m, displays a line plot of the SIR percentages over time.
• sir_simulation.m, 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.
• timestep_display.m, displays an image of the hospital room at each timestep, indicating the locations of suceptible, infected, and recovered patients.

Examples and Tests:

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:

• sir_day13.png, 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,

Using the command

      plot ( 1:t_max, sir(1,:), 'g', 1:t_max, sir(2,:), 'r', 1:t_max, sir(3,:), 'k' )
      

• sir_line1.png
• sir_line2.png
• sir_line3.png
• sir_line4.png
• sir_line5.png
• sir_line6.png

Using the commands

      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 ] );
      

• sir_area.png

You can go up one level to the MATLAB source codes.