In this project you will have the opportunity to: 1) study some of the probabilistic models and stochastic processes used to analyse the behaviour of epidemics; 2) study the modelling of real-world networks using random graphs; 3) investigate the interplay between these two areas of study.
There are many interesting models for the spread of diseases (deterministic models such as the SIR and SIS models; their stochastic versions; and other probabilistic models such as the Reed–Frost model). Many of these "basic" models assume some homogeneity of the population, typically an assumption that the interactions between pairs of individuals are all equally likely to occur.
To model interactions more accurately one is lead to study the random graph models of networks (the famous Erdős–Rényi model; other models of small-world networks, such as the Barabási–Albert model).
There's plenty of interesting mathematics and directions of study for your project in either of these areas separately, as well as exciting questions about how to model epidemics on these networks. For those keen on doing some computation/coding, there is plenty of scope for running computer simulations of any of the epidemic/random graph models.
If this sounds interesting and you want to find out more, google some of the keywords above and/or have a look at the suggested resources below.
Discrete Mathematics and 2H Probability are essential.
3H Stochastic Processes is not necessary, but may be helpful with understanding of the probabilistic models (e.g. branching processes, random graphs).
The book Epidemics and Rumours in Complex Networks by Driaef and Massoulié, is fully available via the university login to Cambridge Core: https://doi.org/10.1017/CBO9780511806018
The book Networks: An Introduction by Newman, is also available online via Oxford Scholarship Online: https://doi.org/10.1093/acprof:oso/9780199206650.001.0001
Get in touch if you have any questions!