As part of my Masters degree, I was involved in research into the mathematical modelling of how an infection spreads within a network of nodes.
Specifically, I was investigating how the probability of an epidemic changes when the regular network undergoes rewiring to increase randomness. Computer simulations were used extensively in order to highlight any interesting effects and back up any mathematical models produced. The real world applications of epidemiology obviously include finding ways to inhibit spreading of disease but it can also be used to model how social trends spread such as rumours or opinions.
Below I’ve given a brief description of some important terminology:
- Network: a network (or graph) refers to a collection of nodes (or vertices), some of which are connected by bonds (or edges). One example is a regular square lattice, which was the primary subject of investigation.
- Rewiring: this involves breaking some of the regular bonds and replacing them with connections to other random nodes in the network.
- SIR model: this model refers to the states that the nodes can be in: Susceptible to infection, Infected and capable of passing infection to Susceptible nodes, and Recovered and unable to be infected again. In the image above and videos below, the grey nodes are Susceptible, red are Infected and black are Recovered. Most of the investigations were carried out for SIR systems.
- I-r synergy: Infected-recipient synergy is where the probability of infection of a neighbouring node is affected by other neighbouring infected nodes. This synergy can be constructive, where having other neighbouring infected nodes increases the probability of infection, or destructive, where other neighbouring infected nodes decrease the chance of infection.
- Epidemic: an epidemic occurs when the entire network is Infected (or Recovered) at the end of the infection.
The main findings were as follows. For destructive I-r synergy, increasing the amount of rewiring within the network leads to systems which are less resilient to epidemics:
This simulation shows the spread of infection with destructive I-r synergy. The left panel shows a regular square lattice whilst the right panel shows a lattice which has undergone 10% rewiring. The rewired system is more susceptible to infection as rewiring increases the likelihood of the infection spreading to a non-infected area where the destructive synergy is weakest.
Interestingly, for highly constructive I-r synergy, the increasing randomness caused by rewiring actually hinders the spread of disease and results in a relatively higher resilience to epidemics:
This simulation shows the spread of infection with constructive I-r synergy. The left panel shows a regular square lattice whilst the right panel shows a lattice which has undergone 10% rewiring. The rewired system is less susceptible to infection as rewiring decreases the likelihood of the infection spreading to an infected area where the constructive synergy is strongest.
The simulations were carried out by a C++ application using kinetic monte carlo techniques. OpenCV was used to produce some graphing and the video outputs seen above. To produce reliable data points with low errors, several tens of thousands of simulations were carried out for each set of input parameters. Supercomputers within University of Cambridge’s Chemistry department were utilised for several months to complete these runs.
Results from this project were published in Physical Review Letters.