In this course, we revisit the classical compartmental models of mathematical epidemiology and present some of their recent advances. The topics can be organized into three main modules. In the first module, we trace the history of epidemiological models starting from the pioneering work of Bernoulli (1766) up to the well-known Susceptible-Infected-Removed (SIR) model by Kermack and McKendrick (1927) and its subsequent variants. We study in detail the SIR-like models by using tools from stability theory and bifurcation theory of dynamical systems. The second module is devoted to the use of epidemiological models for planning optimal control strategies against disease spreading. To this aim, we introduce the general concept of optimal control problem in the sense of Pontryagin theory and the correlated analytical results. Numerical methods for the optimal control of epidemiological models are addressed in exercise sessions. In the third module, we introduce the recent field of behavioural epidemiology of infectious diseases. In particular, we illustrate some different approaches to incorporate the role of human behavioural changes in epidemiological models. Also, contributions from the specialist literature are discussed, including recent models of the COVID-19 pandemic.

**References and Textbooks**:

- Bacaër N.
*A Short History of Mathematical Population Dynamics*. Springer, London, 2011. - Grass D., Caulkins J.P., Feichtinger G., Tragler G., Behrens D.A.
*Optimal Control of Nonlinear Processes*. Springer, Berlin, 2008. - Guckenheimer J., Holmes P.
*Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields*. Springer, New York, 1983. - Lenhart S., Workman J.T.
*Optimal Control Applied to Biological Models*. Chapman and Hall/CRC, Boca Raton, 2007. - Martcheva M.
*An introduction to Mathematical Epidemiology*. Springer, New York, 2015. - van den Driessche P., Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.
*Mathematical Biosciences*, 180:29-48, 2002. - Wang Z., Bauch C.T., Bhattacharyya S., d'Onofrio A., Manfredi P., Perc M., Perra N., Salathé M., Zhao D. Statistical physics of vaccination.
*Physics Reports*, 664:1-113, 2016.