This project is inspired by Gomes et al 2004 paper on infection, reinfection, and vaccination under sub-optimal immune protection. This project gives an introductory analysis on the first six models in Gomes et al paper, as well as the explanation of the mathematical methods, and a literary review of eight papers cited by Gomes et al paper. The models in this paper include the SIS (susceptible-infectious-susceptible) model, SIR (susceptible-infectious-recovered) model, and four extensions of the SIR model which allow for different types of immunity. I aimed to replicate the results and figures in Gomes et al paper as a framework for the project. From this framework I added a more in depth analysis of the figures and models including all the working out throughout the body of the project. In order to replicate the figures in the paper I scaled the equations for each model, and determined the corresponding steady-states and their stability. From the steady state analysis I was able to determine the long term behaviour of the models and different threshold values. At the conclusion of each model I produced numerical simulations of the solutions over time or the endemic equilibria as a function of the basic reproductive number and other significant parameters. The conclusion of the project is a literary review, reviewing eight papers cited by Gomes et al that I personally found interesting, not necessarily related to epidemiological models.
M.Gabriela M. Gomes, Lisa J. White, and Graham F. Medley. ``Infection, reinfection, and vaccination under suboptimal immune protection: epidemiological perspectives''. In: Journal of Theoretical Biology 228.4 (June 2004), pp. 539–549. doi: 10.1016/j.jtbi.2004.02.015. https://doi.org/10.1016/j.jtbi.2004.02.015.