Critical HexSIRSize

by Dirk Brockmann

This explorable illustrates the behavior of a contagion process near its critical point. Contagion processes, for example transmissable infectious diseases, typically exhibit a critical point, a threshold below which the disease will die out, and above which the disease is sustained in a population. Interesting dynamical things happen when the system is near its critical point.

First press the Play button. While the system is doing its thing, keep on reading....

This is how it works

The example above is a spatial SIRS model, similar to the system discussed in the Epidemonic Explorable. The backbone of the system is a hexagonal lattice, with 7651 individual sites. Each site only interacts with its 6 neighboring sites (apart from the boundary sites, that have fewer neighbors).

Each site can be in one of three states:

  • Susceptible: In this state a site can be infected by an infectious neighbor.
  • Infectious: In this state a site can transmit the infection to a susceptible neighbor.
  • Recovered: In this state a site is immune to infection. Infected sites become recovered spontaneously at a defined rate.

Recovered sites do not remain immune indefinitely (life long immunity) but become susceptible again after some time. A site therefore cycles through the states S, I, R, S which is why the model is called SIRS model.

In the simulation, all sites are susceptible initially, except for a small seed group in the center that this infected (red blob).

What are the parameters?

The parameters of the system are the transmission rate (the propensity that an infected transmits the disease), the recovery rate (the likelihood that an infected individual recovers and acquires immunity), and the rate at which this immunity is lost (waning immunity rate).

It the transmission rate is too small or the recovery rate is too large or the waning immunity rate is too small, the system is below the critical point, and the disease cannot prevail.

The initial parameters (slider positions) are chosen such that the system is slightly above the critical threshold. This usually means that the systems exhibits strong fluctuations, spatially reminiscent of forest fire dynamics, a related phenomenon.

Try this

First, turn down the recovery rate and observe that the system goes into a fairly homogeneous state with a fairly high concentration of infecteds. When you (very slowly) increase the recovery rate you will see that as you approach the critical point spatial random wave fronts of infecteds will emerge. When when you pass the critical point the disease will die out.

Likewise you can try the same by changing the transmission rate or the waning immunity rate.