Age of infection disease modeling: from Kermack and McKendrick to multi-compartment models

In the last few years, there has raised a renewed interest in the mathematical modeling of epidemiological diseases. The seminal work of Kermack and McKendrick (KMK) in 1929 served as the very first theoretical framework to model disease spreading comprehensively. However, its mathematical complexity makes it intractable from a practical perspective, leading to the development of simpler models such as the well-known SIR model. Unfortunately, these simplifications omit the age-of-infection dependence on the original KMK model, making the resulting models less realistic and versatile. This thesis investigates the construction of new mathematical models that, while keeping it relatively simple, allow to introduce the dependence of transmission rates on the age of infection. Specifically, we apply this formalism to a vector-borne model, where hosts contract the disease from infected vectors. By incorporating the concept of age of infection, our models provide a more accurate representation of the dynamics of the disease. Furthermore, with the assistance of the reproduction number, we prove that incorporating the age of infection in the model strongly impacts disease progression.