Question

I am trying to implement the SIR epidemic model. Basically, the way I understand the model is that at each time step, it tells us how many nodes get infected and how many nodes are recovered. I am now trying to convert this into an discrete-event simulation and am confused at a point.

The model is generally solved using Euler’s method. Now, when I am converting it into a discrete event simulation, I am doing something like this (the numbers are used for clarity):

Initialize 100 members
At every time step t,
  //Determine how many get infected
  for i = 1 to 100
     let i pass a message to its neighbors
     When the neighbor receives the message from an infected member, it generates a random number and if it is less than beta*(infected/total), where beta is the infection rate, then the member gets infected
     Update the count for infected, recovered, susceptible

  //Determine how many are recovered
  for i = 1 to 100
      Generate a random number from a uniform distribution and check if it is less than gamma*infected. If it is, then this member is recovered.
      Update the count for infected, recovered, susceptible

I was essentially wondering if the above approach is right. Any suggestions?

Answer 1

Looks pretty good as a start, except that for the first loop you need to remember that only susceptible individuals can become infected, and for the second one, that only infected individuals can become recovered. I also believe the probabilities of transition on each event (susceptible receiving message from infected neighbor, infected possibly recovering) are not a function of the current number of infected individuals — they’re constants (I think you’re misleading the “mass effect” probabilities as applying to each individual episode at a time step — they don’t).

A bit subtler is how you do the first loop (not obvious to me from the SIR model): I think you want to determine all “messages” first, then which ones cause transitions susceptible -> infected — i.e., two loops rather than one — because an individual that’s just gotten infected at this time step cannot infect yet others in this same time step but only in the future; also, the transition infected -> recovered is not possible for individual who just became infected at this very time step, so you’d have to arrange your loops a bit differently!

Consider modeling each individual with two “state” attributes:

-- nummsgs, number of "messages" received this time step
-- compartment (susceptible, infected or recovered)

as well as a fixed set of neighbors. Then:

for each individual:
    if individual.compartment != infected:
        continue
    for each neighbor of the individual:
        neighbor.nummsgs += 1
    if (random number says so):
        individual.compartment = recovered

for each individual:
    if individual.compartment != susceptible:
        continue
    maybe (depending on random number & nummsgs):
        individual.compartment = infected

for each individual:
    individual.nummsgs = 0

this seems to better capture the overall flow (net of collecting and logging overall counts, which you can conceptually do as part of the last loop).