Week 4 & 5

This post will be a two in one since I forgot to write a post last week. So we're about half way through the program and I've certainly learned a lot. The past couple of weeks we spent looking into ways to simulate disease spread on our graphs. I made a nice basic simulator to model diseases. We've also made plans of how to further explore this subject as is pertains to our large graphs.

Disease Simulator

The big achievement I've made this week is creating our mini simulator so that we may begin studying how a disease will spread in these graphs. The simulator implements a basic SIR model. The SIR model stands for Susceptible, Infectious, and Recovered. Each of these represents a state that an agent can be in. When an agent is susceptible, they are healthy uninfected individuals. When a susceptible agent comes in contact with an infectious agent, there is a chance the agent will transition to the infectious state themselves. In the infectious state, an agent may infect other susceptible agents until their infectious period elapses. The infectious period is a pre-defined amount of time that an agent will remain infectious. At the end of this period, the agent will transition to the recovered state. In this state, they will no longer infect other agents, and they themselves can never be infected again. This state diagram can be represented as the following:

[ Susceptible ] -> [ Infectious ] -> [ Recovered ]

There are variations of this model where agents can be reinfected. Also we can add cases where an agent is a carrier, as in they carry the disease and infect others, but do not show symptoms. This is also represented by different infectious periods. We could also introduce vaccines, where some agents are much more immune to the infection and are therefore less likely to become infected. This simulation I have built only deals with the simple SIR model. However it can be expanded in the future without much issue.

The simulation runs in the following manner:

• All agents start as susceptible
• One agent is randomly selected as patient zero and becomes infectious
• Loop for the duration of the simulation:
  • For each infected agent:
    • Try to infect each neighbor
    • If infectious period has elapsed, set state to recovered.

When the simulation is run, the user is asked to input some parameters for the experiment. The first is the infectious probability. This is a number between 0 and 1 that determines how likely an individual is to be infected. The next parameter is how long to run the simulation. This is an arbitrary time unit that you can imagine as days, weeks, months or whatever. The last parameter is the infectious period. Again this is how many time steps an agent will remain in the infectious state.

The program will take these parameters and run the simulation using the steps above. When it is done, it will dump the results into a file which we can use to display some nice charts in the web browser.

Now as nice as it would be to display the infection in real time visually, this is extremely difficult due to the complexity of these large graphs. Perhaps in the future I can display some very rudimentary visualization technique.

Results

So now that we have this simulation, we can run it on some of our graphs and begin to make assumptions about the data. Let's take a look at one I ran:

For this experiment, I chose a graph that was fairly evenly distributed. If you recall my last post, that means that most nodes had a good distribution of how many neighbors each had. I gave the infection a chance of 15% to infect another agent, set the duration for 500 time steps, and the infectious period for 8 timesteps. The chart on the left shows how many new infections appeared at each time step. This averaged around 450-500 infections per time step. The graph on the right shows how the population changed over time. If you look at the stats in the box, you can see that about 99.94% of the population was infected with the disease before everyone recovered. Remember this simulation is run without the assumption of any preventative measures. So think of it as no one washes their hands, and people are way too comfortable with each other.

So with this in mind, let's take a look at an example where agents are less connected. We can think of this as a case where people are more distant and don't connect with as many people on a day to day basis. However, still nobody washes their hands. (when will they learn?)

As we can see, this graph paints a bit of a different picture. Due to the nature of this graph I had to alter the parameters a bit to get some good data. Using the old parameters didn't change the outcome much. As we can see from this picture, we were only able to infect 83.13% of the population. The left chart shows us that by time step 12, we weren't infecting any new agents. However if you look at the right chart, you can clearly see people were still infectious at that time. From that data, we can make the assumption that the infectious individuals were not connected to any susceptible individuals.

Next Steps

Now that we have this small simulation, we want to begin looking at something called clusters in the graphs. This is where a group of nodes is more tightly connected with each other than other nodes. However these clusters (or communities) are also connected to each other usually by one or more nodes. As a real world example, think of it this way: I have a group of friends I hang out with at school. But I also have a group of friends I hang out with outside of school. I'm friends with all of these people in both groups, but my school friends are not friends with my non-school friends. What we would like to see is what happens when say one of my school friends gets sick and spreads it to our community and observe say how I could get sick then spread that infection to my non-school community. This can give us an idea of how these communities aid in the spread of disease throughout the greater area.