Multiple disease model - Locally self-organized percolation

Tutorial

You can try to:
  • Click on the screen to start a new disease that you can follow in detail. A green node is infected with this disease. A red node has been infected, but is now recovered and immune. To remove the coloured disease again, press 'remove disease'.

  • Increase the update speed to make everything happen faster.

  • Increase the disease time to make nodes be sick for a longer period of time.

  • Increase alpha to make new diseases occur more frequently on the network.

  • Use absorbing boundary conditions such that the people on the left boundary are no longer friends with the people on the right boundary.

  • Use square coupling such it gets even harder for diseases to spread from people carrying many diseases.

  • Make fractal disease shapes by clicking on the screen when the number of diseases has converged to an equilibrium value with a disease time larger than 2.77.

Philosophy


Critical threshold
The probability that a person infects a given neighbour with a given disease before she is cured is called p. If that person carries k diseases, the probability can be found to be

where tau is the disease time. Thus, the more diseases a node carries, the less is the probability that each of them is transmitted. If p is smaller than 0.5, most diseases will not have time to spread from a person to person, before they are cured. Conversely, if p is larger than 0.5, diseases will have plenty of time to spread, and they will all grow to infect most of the network. If p is exactly 0.5, diseases will spread in fractal shapes, as shown below

Self organization
Initially, only few diseases are present on the network and all diseases will grow rapidly. Consequently, the average number of diseases per node will increase and the diseases will grow slower. If p becomes less than 0.5, most new diseases will only spread to a handful of nodes before dying out. Thus, the number of diseases will decrease and p will increase. This feedback mechanism will drive the system to a critical state, with p close to 0.5, where the fractal disease shapes occur.

Critical exponents
In the critical state, disease clusters of all sizes will occur. The size distribution will be scale-invariant with a critical exponent of -1.05. That is, the chance that a disease infects e.g. 1000 people is about half the chance that is infects 500. Likewise, the number of people infected will scale with the maximum distance between two infected persons with the critical exponent 1.896.

Percolation
The model belongs to the class of susceptible/infectious/recovered (SIR) epidemic models. On a network, such a model can be described by the mathematical theory of bond percolation, meaning that all nodes (persons) are linked (infect each other) with a probability p. Thus, the critical exponents of the model belong to the universality class of percolation. Many phenomena of interest within physics, mathematics, chemistry, biology, and materials science can be described using percolation theory.

Model by Jeppe Juul and Kim Sneppen