What are compartmental models in epidemiology?

In this talk, we will give an overview, starting from very basic modeling principles, of compartmental models in epidemiology. We will start with simpler models developing relevant ideas in the SIR context and will gradually build more complicated models, including the SEIR and all the way to models of relevance to COVID-19, including features such as asymptomatic infections, vaccinations etc. We will try to focus on some of the mathematical techniques that are needed for the formulation and analysis of the models from ideas in Mathematical Modeling, and Dynamical Systems to simple (and often linear) Algebraic methods, and extending to ideas from PDEs and Numerical Analysis for spatially distributed models. The hope is to showcase how the relevant ideas can be appreciated even starting from a very modest mathematical background of calculus and linear algebra, while also demonstrating some of the ongoing challenges in this area.